{"id":5443,"date":"2023-02-06T07:09:07","date_gmt":"2023-02-06T07:09:07","guid":{"rendered":"https:\/\/www.storyofmathematics.com\/?page_id=5443"},"modified":"2023-03-17T19:23:47","modified_gmt":"2023-03-17T19:23:47","slug":"exponential-function","status":"publish","type":"page","link":"https:\/\/www.storyofmathematics.com\/exponential-function\/","title":{"rendered":"Exponential function &#8211; Properties, Graphs, &#038; Applications"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"5443\" class=\"elementor elementor-5443\" data-elementor-post-type=\"page\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0e24de3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0e24de3\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-51045de\" data-id=\"51045de\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-01e4b6b elementor-widget elementor-widget-text-editor\" data-id=\"01e4b6b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h1>Exponential Function &#8211; Properties, Graphs, &amp; Applications<\/h1><p><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter size-full wp-image-307306\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/03\/Exponential-Function-Title.png\" alt=\"Exponential Function\" width=\"1800\" height=\"900\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/03\/Exponential-Function-Title.png 1800w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/03\/Exponential-Function-Title-300x150.png 300w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/03\/Exponential-Function-Title-1024x512.png 1024w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/03\/Exponential-Function-Title-768x384.png 768w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/03\/Exponential-Function-Title-1536x768.png 1536w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/03\/Exponential-Function-Title-600x300.png 600w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2023\/03\/Exponential-Function-Title-1298x649.png 1298w\" sizes=\"(max-width: 1800px) 100vw, 1800px\" \/>You may have encountered expressions where $x$ or any other variable is an exponent in your Algebra class. If this is true, then you may have your first encounters with exponential functions. These functions help construct real-world models such as bacterial growth, compound interest rates, and exponential decay.<\/p><p>This is why learning about exponential functions, knowing how to manipulate them, and graphing exponential functions is essential. Here\u2019s a quick definition of exponential functions to get us started.<\/p><p><strong><em>Exponential functions are functions that contain a constant base and algebraic expressions (or variables) on their exponents.<\/em><\/strong><\/p><p>Make sure to take notes while going through this article as we\u2019ll discuss the following elements of exponential functions:<\/p><ul><li>Identifying exponential functions and learning their definition.<\/li><li>Learning the components of exponential functions\u2019 graphs.<\/li><li>Studying real-world examples that can be modeled through exponential functions.<\/li><\/ul><p>Let\u2019s begin with a more thorough understanding of what makes up an exponential function.<\/p><h2><strong>What is an exponential function?<\/strong><\/h2><p>As we have mentioned in the earlier section, exponential functions contain variables in their exponents. This section will show why we have the general form of the exponential function as shown below.<\/p><p>\\begin{aligned}y &amp;= {\\color{green}b}^{\\color{blue}x }\\\\ \\color{green} b&amp;: \\color{green}\\text{base }\\\\\\color{blue} x&amp;: \\color{blue}\\text{exponent }\\end{aligned}<\/p><p>It\u2019s important to keep in mind that $b$ must be a positive number except for $1$ (meaning, $b &gt;0$ but $b \\neq 1$).<\/p><p>Let\u2019s go ahead and break down this general form and find examples of exponential functions using this definition.<\/p><h3><strong>Exponential functions definition and examples<\/strong><\/h3><p>From the general form of exponential functions, we can see that they are the functions that contain a base, $b$, such that it is any real and positive number other than one. This means $b$ can be $3$, $8$, and even $e$.<\/p><p>As for the exponent part, as long as the function contains variables and algebraic expressions in the exponent such as $x$, $x \u2013 1$, and $4x \u2013 2$, the exponential function will be valid.<\/p><p>Let\u2019s use these bases and exponents to construct some examples of exponential functions.<\/p><ul><li>$y = 3^x$<\/li><li>$y = 8^{x &#8211; 1}$<\/li><li>$f(x) = e^{4x \u2013 2}$<\/li><li>$g(x) = 3^x \u2013 2$<\/li><\/ul><p>These four examples of exponential functions show expressions that satisfy the general form\u2019s condition, $y = b^x$, where $b&gt;0$ but $b \\neq 1$.<\/p><h2><strong>How to graph exponential functions?\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong><\/h2><p>We can graph any form of exponential functions by creating a table of values and tracing the curve through the points. Here are some helpful steps to remember when graphing exponential functions:<\/p><ul><li>Assign some values of $x$ and $y$ that satisfy the given exponential function.<\/li><li>Use the table of values to plot the different ordered pairs, $(x, y)$.<\/li><li>Connect the points representing the ordered pairs with a curve to graph the function.<\/li><\/ul><p>Why don\u2019t we go ahead and apply these steps for $y = 2^x$?<\/p><p>Let\u2019s assign some values for $x$ then determine the corresponding values of $y$.<\/p><table><tbody><tr><td width=\"266\"><p>$\\boldsymbol{x}$<\/p><\/td><td width=\"266\"><p>$\\boldsymbol{y}$<\/p><\/td><td width=\"266\"><p>$\\boldsymbol{(x, y)}$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$-2$<\/p><\/td><td width=\"266\"><p>$2^{\\color{blue}-2} = \\dfrac{1}{4}$<\/p><\/td><td width=\"266\"><p>$\\left(-2, \\dfrac{1}{4}\\right)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$-1$<\/p><\/td><td width=\"266\"><p>$2^{\\color{blue}-1} = \\dfrac{1}{2}$<\/p><\/td><td width=\"266\"><p>$\\left(-1, \\dfrac{1}{2}\\right)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$0$<\/p><\/td><td width=\"266\"><p>$2^{\\color{blue}0} = 1$<\/p><\/td><td width=\"266\"><p>$(0, 1)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$1$<\/p><\/td><td width=\"266\"><p>$2^{\\color{blue}1} = 2$<\/p><\/td><td width=\"266\"><p>$(1, 2)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$2$<\/p><\/td><td width=\"266\"><p>$2^{\\color{blue}2} = 4$<\/p><\/td><td width=\"266\"><p>$(2, 4)$<\/p><\/td><\/tr><\/tbody><\/table><p>Now that we have some points to guide us in graphing $y = 2^x$, let\u2019s first plot these points on an $xy$-coordinate system.<\/p><p><img decoding=\"async\" class=\"aligncenter size-full wp-image-8839\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-points-of-exponential-function-y-2^x.png\" alt=\"\" width=\"511\" height=\"476\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-points-of-exponential-function-y-2^x.png 511w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-points-of-exponential-function-y-2^x-300x279.png 300w\" sizes=\"(max-width: 511px) 100vw, 511px\" \/><\/p><p>We can then graph the function, $y = 2^x$, by connecting the curve using the points as a guide.<\/p><p><img decoding=\"async\" class=\"aligncenter size-full wp-image-8840\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-exponential-function-y-2^x.png\" alt=\"\" width=\"500\" height=\"475\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-exponential-function-y-2^x.png 500w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-exponential-function-y-2^x-300x285.png 300w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><\/p><p>Hence, we have a graph of $y = 2^x$ as shown by the image above. Notice how the graph never crosses the $x$-axis? That\u2019s because $2^x$ can never be equal to $0$. This means that $y = 2^x$ has a horizontal asymptote at $y = 0$.<\/p><p>We can apply the same process to graph $y = \\dfrac{1}{2}^x$. Once we have the graph of $y = \\dfrac{1}{2}^x$, let\u2019s compare the graphs of $y = 2^x$ and $y = \\dfrac{1}{2}^x$.<\/p><p>First, construct another set of table of values that satisfy the function, $y = \\dfrac{1}{2}^x$.<\/p><table><tbody><tr><td width=\"266\"><p>$\\boldsymbol{x}$<\/p><\/td><td width=\"266\"><p>$\\boldsymbol{y}$<\/p><\/td><td width=\"266\"><p>$\\boldsymbol{(x, y)}$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$-2$<\/p><\/td><td width=\"266\"><p>$\\dfrac{1}{2}^{\\color{blue}-2} = 4$<\/p><\/td><td width=\"266\"><p>$(-2, 4)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$-1$<\/p><\/td><td width=\"266\"><p>$\\dfrac{1}{2}^{\\color{blue}-1} =2 $<\/p><\/td><td width=\"266\"><p>$(-1, 2)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$0$<\/p><\/td><td width=\"266\"><p>$\\dfrac{1}{2}^{\\color{blue}0} = 1$<\/p><\/td><td width=\"266\"><p>$(0, 1)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$1$<\/p><\/td><td width=\"266\"><p>$\\dfrac{1}{2}^{\\color{blue}1} = \\dfrac{1}{2}$<\/p><\/td><td width=\"266\"><p>$\\left(1, \\dfrac{1}{2}\\right)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$2$<\/p><\/td><td width=\"266\"><p>$\\dfrac{1}{2}^{\\color{blue}2} = \\dfrac{1}{4}$<\/p><\/td><td width=\"266\"><p>$\\left(2, \\dfrac{1}{4}\\right)$<\/p><\/td><\/tr><\/tbody><\/table><p>Plot the five points, then connect them with a curve representing $y = \\dfrac{1}{2}^x$.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-8843\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-exponential-function-when-b-0.5.png\" alt=\"\" width=\"480\" height=\"420\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-exponential-function-when-b-0.5.png 480w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-exponential-function-when-b-0.5-300x263.png 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/p><p>Hence, we have the graph of $y = \\dfrac{1}{2}^x$, as shown above. For this graph, we can also see that the function has a horizontal asymptote at $y = 0$.<\/p><p>Why don\u2019t we go ahead and compare the graphs of the two functions?<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-8844\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/exponential-growth-and-decay.png\" alt=\"\" width=\"526\" height=\"456\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/exponential-growth-and-decay.png 526w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/exponential-growth-and-decay-300x260.png 300w\" sizes=\"(max-width: 526px) 100vw, 526px\" \/><\/p><p>Let\u2019s compare the two graphs &#8211; the two graphs are also symmetric along the $y$-axis. This is true for all functions of the form $y = k^x$ and $\\dfrac{1}{k}^x$, where $k$ is a positive constant.<\/p><p>In addition, $y = 2^x$ is drastically increasing as $x$ increases while $y = \\dfrac{1}{2}^x$\u2019s value drastically decreases as $x$ increases. The behaviors of the two graphs are good examples of exponential growth and decay.<\/p><p><strong>Exponential growth<\/strong><\/p><p>Functions reflect them with a greater base than $1$, such as $y = 2^x$. When the graph is significantly increasing as $x$ is increasing, the exponential function is exhibiting exponential growth.<\/p><p>The direction of the graph should be similar to that of $y = 2^x$.<\/p><p><strong>Exponential Decay<\/strong><\/p><p>As you may have guessed, these are the functions that have values that are significantly decreasing as $x$ increases. When the base is less than $1$ or in short, there are fractions such as $y = \\dfrac{1}{2}^x$.<\/p><p>The graph of $y = \\dfrac{1}{2}^x$ exhibits exponential decay and graphs of functions that exhibit exponential decay.<\/p><p>Here are some other important properties to keep in mind when graphing exponential functions:<\/p><ul><li>If the function is of the form, $y = b^x$, the graph of the exponential function will always pass through $(0, 1)$ and $(1, b)$. It will also have a horizontal asymptote at $y=0$.<\/li><li>If the function is of the form, $y = ab^x$, the exponential function graph will always pass through $(0, a)$ and $(1, ab)$. The graph will also have a horizontal asymptote at $y = 0$.<\/li><li>If the function is of a more complex form, $y = ab^{x \u2013 c} + d$, the graph of the exponential function will pass through $(c, a +d)$. In addition, the graph will have a horizontal asymptote at $y = d$.<\/li><\/ul><h3><strong>How to find an exponential function from a graph?\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong><\/h3><p>What if we need to do the reverse? When given the graph of an exponential function, we can also determine the graph&#8217;s expression by inspecting the points the curve passes through. Here are some important reminders when trying to find the exponential function:<\/p><ul><li>Inspect the $y$-coordinate of the exponential function. If it\u2019s $y = 1$, the function is of the form $y = b^x$.<\/li><li>If the $y$-coordinate is a constant or equal to $y = a$, then it may be of the form $y = ab^x$. Find the value of $b$ by observing the value of $y$ when $x =1$.<\/li><li>When the function\u2019s $y$-coordinate doesn\u2019t pass through these, it\u2019s best to slowly inspect the other values and see if they satisfy a more complex form.<\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-8845\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/example-on-finding-the-exponential-function-given-a-graph.png\" alt=\"\" width=\"493\" height=\"447\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/example-on-finding-the-exponential-function-given-a-graph.png 493w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/example-on-finding-the-exponential-function-given-a-graph-300x272.png 300w\" sizes=\"(max-width: 493px) 100vw, 493px\" \/><\/p><p>Since the curve of $f(x)$ passes through $\\left(0, -\\dfrac{1}{2}\\right)$ and $\\left(1, -\\dfrac{3}{2}\\right)$, we know that $f(x)$ is of the form, $y = ab^x$.<\/p><p>We can find the value of $a$ by using the $y$-coordinate of the graph, so $a = -\\dfrac{1}{2}$. We can use the second ordered pair to calculate the value of $b$.<\/p><p>\\begin{aligned}y &amp;= ab^x\\\\-\\dfrac{3}{2}&amp;= -\\dfrac{1}{2}(b)^1\\\\-\\dfrac{3}{2}&amp;=-\\dfrac{b}{2}\\\\b&amp;=3 \\end{aligned}<\/p><p>Now that we have $a = -\\dfrac{1}{2}$ and $b = 3$, the exponential function\u2019s expression is $f(x) = -\\dfrac{1}{2}\\cdot 3^x$.<\/p><p>These are the concepts and topics that we need to learn for now about exponential functions. Let\u2019s check our knowledge by trying out the problems below!<\/p><p><strong><em>Example 1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em><\/strong><\/p><p>Use the fact that $h(x) = \\dfrac{1}{3}^x \u2013 2$ and answer the following questions.<\/p><p>a. What is the value of $h(-3)$? What would the point on the graph be?<br \/>b. If $h(x) = -\\dfrac{17}{9}$, what is the value of $x$? What would the point on the graph be?<br \/>c. Use these two points to make a quick sketch of the graph of $h(x)$.<br \/>d. Will $h(x)$ be increasing? At what value does it approach as $x$ approaches infinity?<\/p><p><u>Solution<\/u><\/p><p>To find $h(-3)$, we simply substitute $x = -3$ into the expression for $h(x)$.<\/p><p>\\begin{aligned}h(x)&amp;= \\dfrac{1}{3}^x &#8211; 2\\\\h(-3)&amp;= \\dfrac{1}{3}^{-3} &#8211; 2\\\\&amp;= 3^{-1 \\cdot -3} &#8211; 2\\\\&amp;=3^3 &#8211; 2\\\\&amp;= 27 &#8211; 2\\\\&amp;=25\\end{aligned}<\/p><p>This means that $h(-3) = 25$ and the point that represents this on the graph will be $(-3, 25)$.<\/p><p>Now, we\u2019re given the value of $h(x)$. To find the value of $x$, we equate the expression of $h(x)$ to $-\\dfrac{17}{9}$.<\/p><p>\\begin{aligned}h(x)&amp;= -\\dfrac{17}{9}\\\\\\dfrac{1}{3}^x \u2013 2 &amp;= -\\dfrac{17}{9}\\end{aligned}<\/p><p>To isolate $x$, we add $2$ to both sides of the equation. Express the right-hand side as a power of $\\dfrac{1}{3}$<\/p><p>\\begin{aligned}h(x)&amp;= -\\dfrac{17}{9}\\\\\\dfrac{1}{3}^{\\displaystyle{x}}-2 &amp;= -\\dfrac{17}{9}\\\\\\dfrac{1}{3}^{\\displaystyle{x}} &amp;= -\\dfrac{17}{9}+2\\\\\\dfrac{1}{3}^{\\displaystyle{x}} &amp;= \\dfrac{1}{9}\\\\\\dfrac{1}{3}^{\\displaystyle{x}} &amp;= \\dfrac{1}{3}^{\\displaystyle{2}}\\\\x &amp;= 2\\end{aligned}<\/p><p>Hence, $x$ is equal to $2$ when $h(x) = -\\dfrac{17}{9}$. On the graph, this will be represented by $\\left(2, -\\dfrac{17}{9}\\right)$.<\/p><p>Let\u2019s use the two points, $(-3, 25)$ and $\\left(2, -\\dfrac{17}{9}\\right)$, and plot these two on graph. Once we have the points on the graph, we can connect the two points with a curve to graph $h(x)$. Here are some helpful reminders when graphing $h(x)$:<\/p><ul><li>The graph will have a horizontal asymptote at $y = -2$.<\/li><li>It will also pass through the $y$-intercept at $(0, -1)$.<\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-8847\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-exponential-functions.png\" alt=\"\" width=\"376\" height=\"407\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-exponential-functions.png 376w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-exponential-functions-277x300.png 277w\" sizes=\"(max-width: 376px) 100vw, 376px\" \/><\/p><p>Hence, we have the graph of $h(x)$ as shown above.<\/p><p>From this, we can confirm that as $x$ increases, the value of $h(x)$ drastically decreases. This actually makes sense since the base is a fraction, and we\u2019re expecting $h(x)$ to be exponentially decaying.<\/p><p>Since $h(x)$ has a horizontal asymptote at $y = -2$, the value of $h(x)$ approaches $-2$ as $x$ approaches infinity.<\/p><p><strong><em>Example 2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em><\/strong><\/p><p>Graph the exponential function, $y = 3^x$, by first creating its table of values. Use the resulting graph of $y= 3^x$ to graph the following exponential functions.<\/p><ul><li>$y = 3^{x \u2013 2}$<\/li><li>$y = 3^x + 2$<\/li><li>$y = \\dfrac{1}{3}^x$<\/li><\/ul><p><u>Solution<\/u><\/p><p>We can begin by constructing the table of values for the initial function, $y = 3^x$. We can do this by assigning appropriate values for $x$.<\/p><table><tbody><tr><td width=\"266\"><p>$\\boldsymbol{x}$<\/p><\/td><td width=\"266\"><p>$\\boldsymbol{y}$<\/p><\/td><td width=\"266\"><p>$\\boldsymbol{(x, y)}$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$-2$<\/p><\/td><td width=\"266\"><p>$3^{\\color{blue}-2} = \\dfrac{1}{9}$<\/p><\/td><td width=\"266\"><p>$\\left(-2, \\dfrac{1}{9}\\right)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$-1$<\/p><\/td><td width=\"266\"><p>$3^{\\color{blue}-1} = \\dfrac{1}{3}$<\/p><\/td><td width=\"266\"><p>$\\left(-1, \\dfrac{1}{3}\\right)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$0$<\/p><\/td><td width=\"266\"><p>$3^{\\color{blue}0} = 1$<\/p><\/td><td width=\"266\"><p>$(0, 1)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$1$<\/p><\/td><td width=\"266\"><p>$3^{\\color{blue}1} = 3$<\/p><\/td><td width=\"266\"><p>$(1, 3)$<\/p><\/td><\/tr><tr><td width=\"266\"><p>$2$<\/p><\/td><td width=\"266\"><p>$3^{\\color{blue}2} = 9$<\/p><\/td><td width=\"266\"><p>$(2, 9)$<\/p><\/td><\/tr><\/tbody><\/table><p>Plot the five ordered pairs on the $xy$-coordinate system. Connect these points with a curve to graph $y = 3^x$.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-8848\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-points-of-exponential-function-y-3^x.png\" alt=\"\" width=\"640\" height=\"434\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-points-of-exponential-function-y-3^x.png 640w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/graphing-the-points-of-exponential-function-y-3^x-300x203.png 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" \/><\/p><p>To graph $y = 3^{x \u2013 2}$ and $y = 3^x + 2$, we simply apply <a href=\"https:\/\/www.storyofmathematics.com\/transformations-of-functions\">transformations<\/a> we\u2019ve learned in the past- in particular, horizontal and vertical translations.<\/p><p>This means that the graph of $y = 3^{x \u2013 2}$ can be plotted by simply translating $y = 3^x$ <strong>two units to the right<\/strong>.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-8849\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/horizontally-translating-exponential-functions.png\" alt=\"\" width=\"722\" height=\"404\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/horizontally-translating-exponential-functions.png 722w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/horizontally-translating-exponential-functions-300x168.png 300w\" sizes=\"(max-width: 722px) 100vw, 722px\" \/><\/p><p>Similarly, we can graph $y = 3^x + 2$\u00a0 by shifting $y = 3^x$ <strong>two units upward<\/strong>.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-8849\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/horizontally-translating-exponential-functions.png\" alt=\"\" width=\"722\" height=\"404\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/horizontally-translating-exponential-functions.png 722w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/horizontally-translating-exponential-functions-300x168.png 300w\" sizes=\"(max-width: 722px) 100vw, 722px\" \/><\/p><p>Recall that given two exponential functions, $y = k^x$ and $y = \\dfrac{1}{k}^x$, their graphs will be symmetric along the $y$-axis. Hence, to graph $y = \\dfrac{1}{3}^x$, <strong>we simply reflect <\/strong>$y = 3^x$ <strong>along the<\/strong> $y$<strong>-axis<\/strong>.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-8850\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/reflecting-exponential-functions-along-the-y-axis.png\" alt=\"\" width=\"658\" height=\"417\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/reflecting-exponential-functions-along-the-y-axis.png 658w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/reflecting-exponential-functions-along-the-y-axis-300x190.png 300w\" sizes=\"(max-width: 658px) 100vw, 658px\" \/><\/p><p>This problem shows that we can also graph exponential functions by transforming the graphs of simpler exponential functions.<\/p><p><strong><em>Example 3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em><\/strong><\/p><p>Find an exponential function of the form, $y = ab^x$, that passes through the points, $(-2, 1)$ and $(3, 32)$.<\/p><p><u>Solution<\/u><\/p><p>Since we\u2019re given two ordered pairs, we can substitute the corresponding values of $x$ and $y$ into the exponential function&#8217;s general form.<\/p><table><tbody><tr><td width=\"399\"><p>$\\boldsymbol{(x, y)}$<\/p><\/td><td width=\"399\"><p>$\\boldsymbol{y = ab^x}$<\/p><\/td><\/tr><tr><td width=\"399\"><p>$(-2, 1)$<\/p><\/td><td width=\"399\"><p>\\begin{aligned}y &amp;= ab^x\\\\1&amp;= ab^{-2}\\\\a&amp;=b^2 \\end{aligned}<\/p><\/td><\/tr><tr><td width=\"399\"><p>$(3, 32)$<\/p><\/td><td width=\"399\"><p>\\begin{aligned}y &amp;= ab^x\\\\32&amp;= ab^{3}\\\\a&amp;= \\dfrac{32}{b^3} \\end{aligned}<\/p><\/td><\/tr><\/tbody><\/table><p>Let\u2019s replace $a = b^2$ into the second equation, $a = \\dfrac{32}{b^2}$, to find the values of $b$.<\/p><p>\\begin{aligned}b^2 &amp;= \\dfrac{32}{b^3}\\\\b^5 &amp;= 32\\\\b^5 &amp;= 2^5\\\\b&amp;= 2 \\end{aligned}<\/p><p>Now that we have $b = 2$, we can find $a$ using either of the equations. We\u2019ll use the first one since it\u2019s simpler.<\/p><p>\\begin{aligned}a &amp;= b^2 \\\\&amp;= 2^2\\\\&amp;= 4 \\end{aligned}<\/p><p>Substitute these values into the general form to find the exponential function that we need.<\/p><p>\\begin{aligned}y &amp;= ab^x\\\\y&amp;= 4 \\cdot 2^x\\end{aligned}<\/p><p>This means that the exponential function we\u2019re looking at is equal to $y = 4\\cdot 2^x$.<\/p><p><strong><em>Example 4\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em><\/strong><\/p><p>Determine the expressions for the exponential functions, $f(x)$, $g(x)$, and $h(x)$ using their graphs shown below.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-8853\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/determining-the-exponential-function-given-its-graph.png\" alt=\"\" width=\"560\" height=\"487\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/determining-the-exponential-function-given-its-graph.png 560w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/01\/determining-the-exponential-function-given-its-graph-300x261.png 300w\" sizes=\"(max-width: 560px) 100vw, 560px\" \/><\/p><p>Take note that when extended, $g(x)$ passes through $(1, -6)$ and $h(x)$ passes through $(0, 8)$.<\/p><p><u>Solution<\/u><\/p><p>Let\u2019s begin by finding the expression for $f(x)$. Since the graph passes through $(0, 1)$, we\u2019re expecting it to be of the form, $f(x) = b^x$.<\/p><p>To find the value of $b$, let\u2019s inspect the function\u2019s value when $x = 1$. Since $x = 1$ when $f(x) = 3$, so $\\boldsymbol{f(x) = 3^x}$.<\/p><p>As for the graph of $g(x)$, we can see that its $y$-coordinate passes through $(0, -3)$, so its general form must be of $y = ab^x$ and $a = -3$.<\/p><p>Since $g(x)$ also passes through $(1, -6)$, we can use this to find the value of $b$.<\/p><p>\\begin{aligned}y &amp;= (-3)b^x\\\\-6&amp;=(-3)b^1\\\\-6&amp;= -3b\\\\b&amp;= 2\\end{aligned}<\/p><p>This means that $\\boldsymbol{g(x)}$ <strong>is equal to <\/strong>$\\boldsymbol{-3\\cdot 2^x}$.<\/p><p>Moving along with $h(x)$, we can see that it must be of a complex form, so we start by first taking note of the horizontal asymptote and making using of the exponential function\u2019s general form, $y = ab^{x \u2013 c} + d$.<\/p><p>Since the horizontal asymptote is equal to $y = -1$, so $d$ must be equal to $-1$. Let\u2019s also observe the values that the graph passes through: $(1, 2)$, $(2, 0)$, and $(0, 8)$.<\/p><p>Let\u2019s substitute these values of $x$ and $y$ and see if we can the values of $a$, $b$, and $c$.<\/p><p>We can begin by seeing the result when we substitute $x = 0$ and $y = 8$.<\/p><p>\\begin{aligned} y &amp;= ab^{x \u2013 c} -1\\\\8&amp;= ab^{0 &#8211; c} &#8211; 1\\\\9&amp;=ab^{-c}\\\\9&amp;=\\dfrac{a}{b^c}\\end{aligned}<\/p><p>Now, let\u2019s move on to observing how the expressions behave when $x = 2$ and $y = 0$.<\/p><p>\\begin{aligned} y &amp;= ab^{x \u2013 c} -1\\\\0&amp;= ab^{2 &#8211; c} &#8211; 1\\\\1&amp;=ab^{2 -c}\\end{aligned}<\/p><p>Manipulating $ab^{2 \u2013 c}$ and using $9 = \\dfrac{a}{b^c}$, we have the following:<\/p><p>\\begin{aligned} 1&amp;=ab^{2 -c}\\\\ &amp;= ab^2 b^{-c}\\\\&amp;=b^2 \\dfrac{a}{b^c}\\\\\\\\1&amp;=b^2(9)\\\\b^2&amp;= \\dfrac{1}{9}\\end{aligned}<\/p><p>Since $b$ can never be negative $b = \\dfrac{1}{3}$. Since the graph does not stretch vertically, we\u2019re expecting $a$ to be $1$.<\/p><p>We can now find the value of $c$ using the equations, $1=ab^{2 -c}$.<\/p><p>\\begin{aligned}1&amp;=ab^{2 -c}\\\\1&amp;= 1\\left(\\dfrac{1}{3}\\right)^{2 &#8211; c}\\\\1&amp;= \\left(\\dfrac{1}{3}\\right)^{2 &#8211; c}\\end{aligned}<\/p><p>For this to be true, $2 \u2013 c$ must be equal to $0$, so $c = 2$.<\/p><p>Let\u2019s now substitute the necessary values into the general expression, $y = ab^{x \u2013 c} + d$.<\/p><p>\\begin{aligned}a &amp;= 1\\\\b&amp;= \\dfrac{1}{3}\\\\c&amp;= 2\\\\d &amp;=-1\\\\\\\\y &amp;= ab^{x &#8211; c} + d\\\\&amp;=1\\left(\\dfrac{1}{3} \\right )^{x-2} &#8211; 1\\\\&amp;= \\dfrac{1}{3}^{x &#8211; 2} &#8211; 1\\end{aligned}<\/p><p>This means that $\\boldsymbol{h(x) = \\dfrac{1}{3}^{x &#8211; 2} \u2013 1}$.<\/p><h4 style=\"text-align: center;\" data-fontsize=\"18\" data-lineheight=\"23\">\u00a0<\/h4>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3b0b2aa elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3b0b2aa\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3010325\" data-id=\"3010325\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dd320b2 elementor-widget elementor-widget-text-editor\" data-id=\"dd320b2\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h3>Practice Questions<\/h3><script type=\"text\/javascript\" >\ndocument.addEventListener(\"DOMContentLoaded\", function(event) { \nif(!window.jQuery) alert(\"The important jQuery library is not properly loaded in your site. Your WordPress theme is probably missing the essential wp_head() call. You can switch to another theme and you will see that the plugin works fine and this notice disappears. If you are still not sure what to do you can contact us for help.\");\n});\n<\/script>  \n  \n<div  id=\"watupro_quiz\" class=\"quiz-area single-page-quiz\">\n<p id=\"submittingExam356\" style=\"display:none;text-align:center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"\"><\/p>\n\n\n\n<form action=\"\" method=\"post\" class=\"quiz-form\" id=\"quiz-356\"  enctype=\"multipart\/form-data\" >\n<div class='watu-question ' id='question-1' style=';'><div id='questionWrap-1'  class='   watupro-question-id-2818'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>1. <\/span>Given that $h(x) = -2\\cdot 3^x \u2013 1$, what is the value of $h(-4)$?<\/div><input type='hidden' name='question_id[]' id='qID_1' value='2818' class='watupro-question-id'\/><input type='hidden' id='answerType2818' class='answerTypeCnt1' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2818[]' id='answer-id-10547' class='answer   answerof-2818 ' value='10547'   \/><label for='answer-id-10547' id='answer-label-10547' class=' answer'><span>$h(-4) = -163$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2818[]' id='answer-id-10548' class='answer   answerof-2818 ' value='10548'   \/><label for='answer-id-10548' id='answer-label-10548' class=' answer'><span>$h(-4) = -\\dfrac{83}{81}$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2818[]' id='answer-id-10549' class='answer   answerof-2818 ' value='10549'   \/><label for='answer-id-10549' id='answer-label-10549' class=' answer'><span>$h(-4) = \\dfrac{83}{81}$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2818[]' id='answer-id-10550' class='answer   answerof-2818 ' value='10550'   \/><label for='answer-id-10550' id='answer-label-10550' class=' answer'><span>$h(-4) = 163$<\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-1'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2818\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2818\" class=\"liveResultBtn1 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2818, 1);\"><\/p>\n\t   <div class='watu-question ' id='question-2' style=';'><div id='questionWrap-2'  class='   watupro-question-id-2819'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>2. <\/span>Given that $h(x) = -2\\cdot 3^x \u2013 1$, what is the value of $h(3)$?<\/div><input type='hidden' name='question_id[]' id='qID_2' value='2819' class='watupro-question-id'\/><input type='hidden' id='answerType2819' class='answerTypeCnt2' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2819[]' id='answer-id-10551' class='answer   answerof-2819 ' value='10551'   \/><label for='answer-id-10551' id='answer-label-10551' class=' answer'><span>$h(-3) = -55$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2819[]' id='answer-id-10552' class='answer   answerof-2819 ' value='10552'   \/><label for='answer-id-10552' id='answer-label-10552' class=' answer'><span>$h(-3) = -12$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2819[]' id='answer-id-10553' class='answer   answerof-2819 ' value='10553'   \/><label for='answer-id-10553' id='answer-label-10553' class=' answer'><span>$h(-3) = 12$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2819[]' id='answer-id-10554' class='answer   answerof-2819 ' value='10554'   \/><label for='answer-id-10554' id='answer-label-10554' class=' answer'><span>$h(-3) = 55$<\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-2'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2819\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2819\" class=\"liveResultBtn2 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2819, 2);\"><\/p>\n\t   <div class='watu-question ' id='question-3' style=';'><div id='questionWrap-3'  class='   watupro-question-id-2820'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>3. <\/span>Given that $h(x) = -2\\cdot 3^x \u2013 1$ and $h(x) = -19$, what is the value of $x$?<\/div><input type='hidden' name='question_id[]' id='qID_3' value='2820' class='watupro-question-id'\/><input type='hidden' id='answerType2820' class='answerTypeCnt3' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2820[]' id='answer-id-10555' class='answer   answerof-2820 ' value='10555'   \/><label for='answer-id-10555' id='answer-label-10555' class=' answer'><span>$x = &#8211; 9$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2820[]' id='answer-id-10556' class='answer   answerof-2820 ' value='10556'   \/><label for='answer-id-10556' id='answer-label-10556' class=' answer'><span>$x =-2$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2820[]' id='answer-id-10557' class='answer   answerof-2820 ' value='10557'   \/><label for='answer-id-10557' id='answer-label-10557' class=' answer'><span>$x =2$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2820[]' id='answer-id-10558' class='answer   answerof-2820 ' value='10558'   \/><label for='answer-id-10558' id='answer-label-10558' class=' answer'><span>$x =9$<\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-3'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2820\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2820\" class=\"liveResultBtn3 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2820, 3);\"><\/p>\n\t   <div class='watu-question ' id='question-4' style=';'><div id='questionWrap-4'  class='   watupro-question-id-2821'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>4. <\/span>Given that $h(x) = -2\\cdot 3^x \u2013 1$ and $h(x) = -\\dfrac{5}{3}$, what is the value of $x$?<\/div><input type='hidden' name='question_id[]' id='qID_4' value='2821' class='watupro-question-id'\/><input type='hidden' id='answerType2821' class='answerTypeCnt4' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2821[]' id='answer-id-10559' class='answer   answerof-2821 ' value='10559'   \/><label for='answer-id-10559' id='answer-label-10559' class=' answer'><span>$x =-3$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2821[]' id='answer-id-10560' class='answer   answerof-2821 ' value='10560'   \/><label for='answer-id-10560' id='answer-label-10560' class=' answer'><span>$x =-1$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2821[]' id='answer-id-10561' class='answer   answerof-2821 ' value='10561'   \/><label for='answer-id-10561' id='answer-label-10561' class=' answer'><span>$x =1$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2821[]' id='answer-id-10562' class='answer   answerof-2821 ' value='10562'   \/><label for='answer-id-10562' id='answer-label-10562' class=' answer'><span>$x =3$<\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-4'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2821\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2821\" class=\"liveResultBtn4 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2821, 4);\"><\/p>\n\t   <div class='watu-question ' id='question-5' style=';'><div id='questionWrap-5'  class='   watupro-question-id-2822'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>5. <\/span>Using the points from the previous problem, which of the following shows the graph of $h(x) = -2\\cdot 3^x \u2013 1$ ?<\/div><input type='hidden' name='question_id[]' id='qID_5' value='2822' class='watupro-question-id'\/><input type='hidden' id='answerType2822' class='answerTypeCnt5' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2822[]' id='answer-id-10563' class='answer   answerof-2822 ' value='10563'   \/><label for='answer-id-10563' id='answer-label-10563' class=' answer'><span><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-26375\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-2-1.png\" alt=\"\" width=\"450\" height=\"407\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-2-1.png 748w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-2-1-300x272.png 300w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2822[]' id='answer-id-10564' class='answer   answerof-2822 ' value='10564'   \/><label for='answer-id-10564' id='answer-label-10564' class=' answer'><span><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-26376\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-3-1.png\" alt=\"\" width=\"450\" height=\"314\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-3-1.png 822w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-3-1-300x209.png 300w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-3-1-768x535.png 768w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2822[]' id='answer-id-10565' class='answer   answerof-2822 ' value='10565'   \/><label for='answer-id-10565' id='answer-label-10565' class=' answer'><span><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-26373\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-1-1.png\" alt=\"\" width=\"450\" height=\"425\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-1-1.png 859w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-1-1-300x284.png 300w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-1-1-768x726.png 768w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2822[]' id='answer-id-10566' class='answer   answerof-2822 ' value='10566'   \/><label for='answer-id-10566' id='answer-label-10566' class=' answer'><span><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-26377\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-4-1.png\" alt=\"\" width=\"450\" height=\"425\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-4-1.png 816w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-4-1-300x283.png 300w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2022\/01\/graphing-exponential-functions-4-1-768x726.png 768w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-5'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2822\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2822\" class=\"liveResultBtn5 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2822, 5);\"><\/p>\n\t   <div class='watu-question ' id='question-6' style=';'><div id='questionWrap-6'  class='   watupro-question-id-2823'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>6. <\/span>Using the resulting graph from the previous problem, is the function, $h(x) = -2\\cdot 3^x \u2013 1$, increasing or decreasing?<\/div><input type='hidden' name='question_id[]' id='qID_6' value='2823' class='watupro-question-id'\/><input type='hidden' id='answerType2823' class='answerTypeCnt6' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2823[]' id='answer-id-10567' class='answer   answerof-2823 ' value='10567'   \/><label for='answer-id-10567' id='answer-label-10567' class=' answer'><span>$h(x)$ is decreasing<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2823[]' id='answer-id-10568' class='answer   answerof-2823 ' value='10568'   \/><label for='answer-id-10568' id='answer-label-10568' class=' answer'><span>$h(x)$ is increasing<\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-6'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2823\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2823\" class=\"liveResultBtn6 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2823, 6);\"><\/p>\n\t   <div class='watu-question ' id='question-7' style=';'><div id='questionWrap-7'  class='   watupro-question-id-2824'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>7. <\/span>What is an exponential function of the form, $y = ab^x$, that passes through the points, $\\left(-3, -\\dfrac{1}{9}\\right)$ and $(2, -27)$?<\/div><input type='hidden' name='question_id[]' id='qID_7' value='2824' class='watupro-question-id'\/><input type='hidden' id='answerType2824' class='answerTypeCnt7' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2824[]' id='answer-id-10569' class='answer   answerof-2824 ' value='10569'   \/><label for='answer-id-10569' id='answer-label-10569' class=' answer'><span>$y = -3\\cdot 3^x$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2824[]' id='answer-id-10570' class='answer   answerof-2824 ' value='10570'   \/><label for='answer-id-10570' id='answer-label-10570' class=' answer'><span>$y = -3\\cdot 3^{x &#8211; 1}$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2824[]' id='answer-id-10571' class='answer   answerof-2824 ' value='10571'   \/><label for='answer-id-10571' id='answer-label-10571' class=' answer'><span>$y = 3\\cdot 3^{x &#8211; 1}$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2824[]' id='answer-id-10572' class='answer   answerof-2824 ' value='10572'   \/><label for='answer-id-10572' id='answer-label-10572' class=' answer'><span>$y = 3\\cdot 3^x$<\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-7'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2824\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2824\" class=\"liveResultBtn7 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2824, 7);\"><\/p>\n\t   <div class='watu-question ' id='question-8' style=';'><div id='questionWrap-8'  class='   watupro-question-id-2825'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>8. <\/span><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-10306 alignleft\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/finding-the-exponential-functions-given-their-graphs-on-one-coordinate-system.png\" alt=\"\" width=\"575\" height=\"447\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/finding-the-exponential-functions-given-their-graphs-on-one-coordinate-system.png 575w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/finding-the-exponential-functions-given-their-graphs-on-one-coordinate-system-300x233.png 300w\" sizes=\"(max-width: 575px) 100vw, 575px\" \/>&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\nThe graphs of $f(x)$, $g(x)$, and $h(x)$ are shown above. Which of the following shows the expression for $f(x)$?<\/div><input type='hidden' name='question_id[]' id='qID_8' value='2825' class='watupro-question-id'\/><input type='hidden' id='answerType2825' class='answerTypeCnt8' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2825[]' id='answer-id-10573' class='answer   answerof-2825 ' value='10573'   \/><label for='answer-id-10573' id='answer-label-10573' class=' answer'><span>$f(x) =3^{-x}$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2825[]' id='answer-id-10574' class='answer   answerof-2825 ' value='10574'   \/><label for='answer-id-10574' id='answer-label-10574' class=' answer'><span>$f(x) =3^x &#8211; 1$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2825[]' id='answer-id-10575' class='answer   answerof-2825 ' value='10575'   \/><label for='answer-id-10575' id='answer-label-10575' class=' answer'><span>$f(x) =3^x$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2825[]' id='answer-id-10576' class='answer   answerof-2825 ' value='10576'   \/><label for='answer-id-10576' id='answer-label-10576' class=' answer'><span>$f(x) =3^x + 1$<\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-8'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2825\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2825\" class=\"liveResultBtn8 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2825, 8);\"><\/p>\n\t   <div class='watu-question ' id='question-9' style=';'><div id='questionWrap-9'  class='   watupro-question-id-2826'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>9. <\/span><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-10306 alignleft\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/finding-the-exponential-functions-given-their-graphs-on-one-coordinate-system.png\" alt=\"\" width=\"575\" height=\"447\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/finding-the-exponential-functions-given-their-graphs-on-one-coordinate-system.png 575w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/finding-the-exponential-functions-given-their-graphs-on-one-coordinate-system-300x233.png 300w\" sizes=\"(max-width: 575px) 100vw, 575px\" \/>&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\nThe graphs of $f(x)$, $g(x)$, and $h(x)$ are shown above. Which of the following shows the expression for $g(x)$?<\/div><input type='hidden' name='question_id[]' id='qID_9' value='2826' class='watupro-question-id'\/><input type='hidden' id='answerType2826' class='answerTypeCnt9' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2826[]' id='answer-id-10577' class='answer   answerof-2826 ' value='10577'   \/><label for='answer-id-10577' id='answer-label-10577' class=' answer'><span>$g(x) =3^x &#8211; 2$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2826[]' id='answer-id-10578' class='answer   answerof-2826 ' value='10578'   \/><label for='answer-id-10578' id='answer-label-10578' class=' answer'><span>$y=3^x &#8211; \\dfrac{1}{2}$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2826[]' id='answer-id-10579' class='answer   answerof-2826 ' value='10579'   \/><label for='answer-id-10579' id='answer-label-10579' class=' answer'><span>$y=3^x$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2826[]' id='answer-id-10580' class='answer   answerof-2826 ' value='10580'   \/><label for='answer-id-10580' id='answer-label-10580' class=' answer'><span>$g(x) =3^x  + 2$<\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-9'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2826\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2826\" class=\"liveResultBtn9 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2826, 9);\"><\/p>\n\t   <div class='watu-question ' id='question-10' style=';'><div id='questionWrap-10'  class='   watupro-question-id-2827'>\n\t\t\t<div class='question-content'><div><span class='watupro_num'>10. <\/span><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-10306 alignleft\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/finding-the-exponential-functions-given-their-graphs-on-one-coordinate-system.png\" alt=\"\" width=\"575\" height=\"447\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/finding-the-exponential-functions-given-their-graphs-on-one-coordinate-system.png 575w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/finding-the-exponential-functions-given-their-graphs-on-one-coordinate-system-300x233.png 300w\" sizes=\"(max-width: 575px) 100vw, 575px\" \/>&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\n&nbsp;<br \/>\r\n<br \/>\r\nThe graphs of $f(x)$, $g(x)$, and $h(x)$ are shown above. Which of the following shows the expression for $h(x)$?<\/div><input type='hidden' name='question_id[]' id='qID_10' value='2827' class='watupro-question-id'\/><input type='hidden' id='answerType2827' class='answerTypeCnt10' value='radio'><!-- end question-content--><\/div><div class='question-choices watupro-choices-columns '><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2827[]' id='answer-id-10581' class='answer   answerof-2827 ' value='10581'   \/><label for='answer-id-10581' id='answer-label-10581' class=' answer'><span>$h(x) = \\dfrac{1}{3}^x + 2$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2827[]' id='answer-id-10582' class='answer   answerof-2827 ' value='10582'   \/><label for='answer-id-10582' id='answer-label-10582' class=' answer'><span>$h(x) = 3^x + 2$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2827[]' id='answer-id-10583' class='answer   answerof-2827 ' value='10583'   \/><label for='answer-id-10583' id='answer-label-10583' class=' answer'><span>$h(x) = 3^x &#8211; 2$<\/span><\/label><\/div><div class='watupro-question-choice  ' dir='auto' ><input type='radio' name='answer-2827[]' id='answer-id-10584' class='answer   answerof-2827 ' value='10584'   \/><label for='answer-id-10584' id='answer-label-10584' class=' answer'><span>$h(x) = \\dfrac{1}{3}^x &#8211; 2$<\/span><\/label><\/div><!-- end question-choices--><\/div><!-- end questionWrap--><\/div>\t\t\t\t\t<div style=\"display:none;\" id='liveResult-10'>\t\t   \n\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t\t\t\t\t<\/div>\t\n\t\t\t<\/div> \n\t   \t<p id=\"liveResBtn-2827\" style=\"clear:both;\"><input type=\"button\" id=\"liveResultBtn2827\" class=\"liveResultBtn10 watupro-live-result-btn\" value=\"See Answer\" onclick=\"WatuPRO.liveResult(2827, 10);\"><\/p>\n\t   <div style='display:none' id='question-11'>\n\t<div class='question-content'>\n\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/img\/loading.gif\" width=\"16\" height=\"16\" alt=\"Loading...\" title=\"Loading...\" \/>&nbsp;Loading&#8230;\t<\/div>\n<\/div>\n\n<br \/>\n\t\n\t\t\t<div class=\"watupro_buttons flex \" id=\"watuPROButtons356\" >\n\t\t  <div id=\"prev-question\" style=\"display:none;\"><input type=\"button\" value=\"&lt; Previous\" onclick=\"WatuPRO.nextQuestion(event, 'previous');\"\/><\/div>\t\t  \t\t  \t\t   \n\t\t   \t  \t\t<div><input type=\"button\" name=\"action\" class=\"watupro-submit-button\" onclick=\"WatuPRO.submitResult(event)\" id=\"action-button\" value=\"View Results\"  \/>\n\t\t<\/div>\n\t\t<\/div>\n\t\t\n\t<input type=\"hidden\" name=\"quiz_id\" value=\"356\" id=\"watuPROExamID\"\/>\n\t<input type=\"hidden\" name=\"start_time\" id=\"startTime\" value=\"2025-03-11 02:32:42\" \/>\n\t<input type=\"hidden\" name=\"start_timestamp\" id=\"startTimeStamp\" value=\"1741660362\" \/>\n\t<input type=\"hidden\" name=\"question_ids\" value=\"\" \/>\n\t<input type=\"hidden\" name=\"watupro_questions\" value=\"2818:10547,10548,10549,10550 | 2819:10551,10552,10553,10554 | 2820:10555,10556,10557,10558 | 2821:10559,10560,10561,10562 | 2822:10563,10564,10565,10566 | 2823:10567,10568 | 2824:10569,10570,10571,10572 | 2825:10573,10574,10575,10576 | 2826:10577,10578,10579,10580 | 2827:10581,10582,10583,10584\" \/>\n\t<input type=\"hidden\" name=\"no_ajax\" value=\"0\">\t\t\t<\/form>\n\t<p>&nbsp;<\/p>\n<\/div>\n\n<script type=\"text\/javascript\">\n\/\/jQuery(document).ready(function(){\ndocument.addEventListener(\"DOMContentLoaded\", function(event) { \t\nvar question_ids = \"2818,2819,2820,2821,2822,2823,2824,2825,2826,2827\";\nWatuPROSettings[356] = {};\nWatuPRO.qArr = question_ids.split(',');\nWatuPRO.exam_id = 356;\t    \nWatuPRO.post_id = 5443;\nWatuPRO.store_progress = 0;\nWatuPRO.curCatPage = 1;\nWatuPRO.requiredIDs=\"0\".split(\",\");\nWatuPRO.hAppID = \"0.33418300 1741660362\";\nvar url = \"https:\/\/www.storyofmathematics.com\/wp-content\/plugins\/watupro\/show_exam.php\";\nWatuPRO.examMode = 1;\nWatuPRO.siteURL=\"https:\/\/www.storyofmathematics.com\/wp-admin\/admin-ajax.php\";\nWatuPRO.emailIsNotRequired = 0;\nWatuPRO.perDecade = 10;\nWatuPROIntel.init(356);\nWatuPRO.inCategoryPages=1;});    \t \n<\/script><h3>Open Problem<\/h3><p>Graph the exponential function, $y = 4^x$, by first creating its table of values. Use the resulting graph of $y= 4^x$ to graph the following exponential functions.<\/p><ul><li>$y = 4^{x + 2}$<\/li><li>$y = 4^x &#8211; 1$<\/li><li>$y = \\dfrac{1}{4}^x$<\/li><li>$y = \\dfrac{1}{4}^x \u2013 2$<\/li><\/ul><h3>Open Problem Solution<\/h3><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-10308\" src=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/graphing-multiple-transformations-of-y-4^x.png\" alt=\"\" width=\"480\" height=\"384\" srcset=\"https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/graphing-multiple-transformations-of-y-4^x.png 480w, https:\/\/www.storyofmathematics.com\/wp-content\/uploads\/2021\/02\/graphing-multiple-transformations-of-y-4^x-300x240.png 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/p><p><em>Images\/mathematical drawings are created with GeoGebra.<\/em><\/p><h4 style=\"text-align: center;\" data-fontsize=\"18\" data-lineheight=\"23\"><strong><a href=\"https:\/\/www.storyofmathematics.com\/complex-rational-expressions\" target=\"_blank\" rel=\"noopener\">Previous Lesson<\/a>\u00a0|\u00a0<a href=\"https:\/\/www.storyofmathematics.com\/precalculus\" target=\"_blank\" rel=\"noopener\">Main Page<\/a> | <a href=\"https:\/\/www.storyofmathematics.com\/change-of-base\" target=\"_blank\" rel=\"noopener\">Next Lesson<\/a><\/strong><\/h4>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Exponential Function &#8211; Properties, Graphs, &amp; Applications You may have encountered expressions where $x$ or any other variable is an exponent in your Algebra class. If this is true, then you may have your first encounters with exponential functions. These functions help construct real-world models such as bacterial growth, compound interest rates, and exponential decay. [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"page-sidebar-2.php","meta":{"_acf_changed":false,"footnotes":""},"categories":[],"tags":[],"class_list":["post-5443","page","type-page","status-publish","hentry","entry"],"acf":[],"rankMath":{"parentDomain":"storyofmathematics.com","noFollowDomains":[],"noFollowExcludeDomains":[],"noFollowExternalLinks":false,"featuredImageNotice":"The featured image should be at least 200 by 200 pixels to be picked up by Facebook and other social media sites.","pluginReviewed":false,"postSettings":{"linkSuggestions":true,"useFocusKeyword":false},"frontEndScore":false,"postName":"exponential-function","permalinkFormat":"https:\/\/www.storyofmathematics.com\/%pagename%\/","showLockModifiedDate":true,"assessor":{"focusKeywordLink":"https:\/\/www.storyofmathematics.com\/wp-admin\/edit.php?focus_keyword=%focus_keyword%&post_type=%post_type%","hasTOCPlugin":true,"primaryTaxonomy":false,"serpData":{"title":"","description":"Exponential functions are functions with a constant base and variables on their exponents. 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here!","focusKeywords":"","pillarContent":false,"canonicalUrl":"","breadcrumbTitle":"","advancedRobots":{"max-snippet":"-1","max-video-preview":"-1","max-image-preview":"large"},"facebookTitle":"","facebookDescription":"","facebookImage":"","facebookImageID":"","facebookHasOverlay":false,"facebookImageOverlay":"","facebookAuthor":"","twitterCardType":"","twitterUseFacebook":true,"twitterTitle":"","twitterDescription":"","twitterImage":"","twitterImageID":"","twitterHasOverlay":false,"twitterImageOverlay":"","twitterPlayerUrl":"","twitterPlayerSize":"","twitterPlayerStream":"","twitterPlayerStreamCtype":"","twitterAppDescription":"","twitterAppIphoneName":"","twitterAppIphoneID":"","twitterAppIphoneUrl":"","twitterAppIpadName":"","twitterAppIpadID":"","twitterAppIpadUrl":"","twitterAppGoogleplayName":"","twitterAppGoogleplayID":"","twitterAppGoogleplayUrl":"","twitterAppCountry":"","robots":{"index":true},"twitterAuthor":"StoryofMathema1","primaryTerm":0,"authorName":"William 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