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As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events. Graphing Exponential FunctionsBefore we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form[latex]\,f\left(x\right)={b}^{x}\,[/latex]whose base is greater than one. We’ll use the function[latex]\,f\left(x\right)={2}^{x}.\,[/latex]Observe how the output values in (Figure) change as the input increases by[latex]\,1.[/latex]
Each output value is the product of the previous output and the base,[latex]\,2.\,[/latex]We call the base[latex]\,2\,[/latex]the constant ratio. In fact, for any exponential function with the form[latex]\,f\left(x\right)=a{b}^{x},[/latex][latex]\,b\,[/latex]is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of[latex]\,a.[/latex] Notice from the table that
(Figure) shows the exponential growth function [latex]\,f\left(x\right)={2}^{x}.[/latex] Figure 1. Notice that the graph gets close to the x-axis, but never touches it. The domain of[latex]\,f\left(x\right)={2}^{x}\,[/latex]is all real numbers, the range is[latex]\,\left(0,\infty \right),[/latex] and the horizontal asymptote is[latex]\,y=0.[/latex] To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form[latex]\,f\left(x\right)={b}^{x}\,[/latex]whose base is between zero and one. We’ll use the function[latex]\,g\left(x\right)={\left(\frac{1}{2}\right)}^{x}.\,[/latex]Observe how the output values in (Figure) change as the input increases by[latex]\,1.[/latex]
Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio[latex]\,\frac{1}{2}.[/latex] Notice from the table that
(Figure) shows the exponential decay function,[latex]\,g\left(x\right)={\left(\frac{1}{2}\right)}^{x}.[/latex] Figure 2. The domain of[latex]\,g\left(x\right)={\left(\frac{1}{2}\right)}^{x}\,[/latex]is all real numbers, the range is[latex]\,\left(0,\infty \right),[/latex]and the horizontal asymptote is[latex]\,y=0.[/latex] Characteristics of the Graph of the Parent Function f(x) = bxAn exponential function with the form[latex]\,f\left(x\right)={b}^{x},[/latex][latex]\,b>0,[/latex][latex]\,b\ne 1,[/latex]has these characteristics:
(Figure) compares the graphs of exponential growth and decay functions. Figure 3. How ToGiven an exponential function of the form[latex]\,f\left(x\right)={b}^{x},[/latex]graph the function.
Sketching the Graph of an Exponential Function of the Form f(x) = bxSketch a graph of[latex]\,f\left(x\right)={0.25}^{x}.\,[/latex]State the domain, range, and asymptote. Try ItSketch the graph of[latex]\,f\left(x\right)={4}^{x}.\,[/latex]State the domain, range, and asymptote. Graphing Transformations of Exponential FunctionsTransformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function[latex]\,f\left(x\right)={b}^{x}\,[/latex]without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied. Graphing a Vertical ShiftThe first transformation occurs when we add a constant[latex]\,d\,[/latex]to the parent function[latex]\,f\left(x\right)={b}^{x},[/latex] giving us a vertical shift[latex]\,d\,[/latex]units in the same direction as the sign. For example, if we begin by graphing a parent function,[latex]\,f\left(x\right)={2}^{x},[/latex] we can then graph two vertical shifts alongside it, using[latex]\,d=3:\,[/latex]the upward shift,[latex]\,g\left(x\right)={2}^{x}+3\,[/latex]and the downward shift,[latex]\,h\left(x\right)={2}^{x}-3.\,[/latex]Both vertical shifts are shown in (Figure). Figure 5. Observe the results of shifting[latex]\,f\left(x\right)={2}^{x}\,[/latex]vertically:
Graphing a Horizontal ShiftThe next transformation occurs when we add a constant[latex]\,c\,[/latex]to the input of the parent function[latex]\,f\left(x\right)={b}^{x},[/latex] giving us a horizontal shift[latex]\,c\,[/latex]units in the opposite direction of the sign. For example, if we begin by graphing the parent function[latex]\,f\left(x\right)={2}^{x},[/latex] we can then graph two horizontal shifts alongside it, using[latex]\,c=3:\,[/latex]the shift left,[latex]\,g\left(x\right)={2}^{x+3},[/latex] and the shift right,[latex]\,h\left(x\right)={2}^{x-3}.\,[/latex]Both horizontal shifts are shown in (Figure). Figure 6. Observe the results of shifting[latex]\,f\left(x\right)={2}^{x}\,[/latex]horizontally:
Shifts of the Parent Function f(x) = bxFor any constants[latex]\,c\,[/latex]and[latex]\,d,[/latex]the function[latex]\,f\left(x\right)={b}^{x+c}+d\,[/latex]shifts the parent function[latex]\,f\left(x\right)={b}^{x}[/latex]
How ToGiven an exponential function with the form[latex]\,f\left(x\right)={b}^{x+c}+d,[/latex]graph the translation.
Graphing a Shift of an Exponential FunctionGraph[latex]\,f\left(x\right)={2}^{x+1}-3.\,[/latex]State the domain, range, and asymptote. Try ItGraph[latex]\,f\left(x\right)={2}^{x-1}+3.\,[/latex]State domain, range, and asymptote. How ToGiven an equation of the form[latex]\,f\left(x\right)={b}^{x+c}+d\,[/latex]for[latex]\,x,[/latex] use a graphing calculator to approximate the solution.
Approximating the Solution of an Exponential EquationSolve[latex]\,42=1.2{\left(5\right)}^{x}+2.8\,[/latex]graphically. Round to the nearest thousandth. Try ItSolve[latex]\,4=7.85{\left(1.15\right)}^{x}-2.27\,[/latex]graphically. Round to the nearest thousandth. Graphing a Stretch or CompressionWhile horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function[latex]\,f\left(x\right)={b}^{x}\,[/latex]by a constant[latex]\,|a|>0.\,[/latex]For example, if we begin by graphing the parent function[latex]\,f\left(x\right)={2}^{x},[/latex]we can then graph the stretch, using[latex]\,a=3,[/latex]to get[latex]\,g\left(x\right)=3{\left(2\right)}^{x}\,[/latex]as shown on the left in (Figure), and the compression, using[latex]\,a=\frac{1}{3},[/latex]to get[latex]\,h\left(x\right)=\frac{1}{3}{\left(2\right)}^{x}\,[/latex]as shown on the right in (Figure). Figure 8. (a)[latex]\,g\left(x\right)=3{\left(2\right)}^{x}\,[/latex]stretches the graph of[latex]\,f\left(x\right)={2}^{x}\,[/latex]vertically by a factor of[latex]\,3.\,[/latex](b)[latex]\,h\left(x\right)=\frac{1}{3}{\left(2\right)}^{x}\,[/latex]compresses the graph of[latex]\,f\left(x\right)={2}^{x}\,[/latex]vertically by a factor of[latex]\,\frac{1}{3}.[/latex] Stretches and Compressions of the Parent Function f(x) = bxFor any factor[latex]\,a>0,[/latex]the function[latex]\,f\left(x\right)=a{\left(b\right)}^{x}[/latex]
Graphing the Stretch of an Exponential FunctionSketch a graph of[latex]\,f\left(x\right)=4{\left(\frac{1}{2}\right)}^{x}.\,[/latex]State the domain, range, and asymptote. Try ItSketch the graph of[latex]\,f\left(x\right)=\frac{1}{2}{\left(4\right)}^{x}.\,[/latex]State the domain, range, and asymptote. Graphing ReflectionsIn addition to shifting, compressing, and stretching a graph, we can also reflect it about the x-axis or the y-axis. When we multiply the parent function[latex]\,f\left(x\right)={b}^{x}\,[/latex]by[latex]\,-1,[/latex]we get a reflection about the x-axis. When we multiply the input by[latex]\,-1,[/latex]we get a reflection about the y-axis. For example, if we begin by graphing the parent function[latex]\,f\left(x\right)={2}^{x},[/latex] we can then graph the two reflections alongside it. The reflection about the x-axis,[latex]\,g\left(x\right)={-2}^{x},[/latex]is shown on the left side of (Figure), and the reflection about the y-axis[latex]\,h\left(x\right)={2}^{-x},[/latex] is shown on the right side of (Figure). Figure 10. (a)[latex]\,g\left(x\right)=-{2}^{x}\,[/latex]reflects the graph of[latex]\,f\left(x\right)={2}^{x}\,[/latex]about the x-axis. (b)[latex]\,g\left(x\right)={2}^{-x}\,[/latex]reflects the graph of[latex]\,f\left(x\right)={2}^{x}\,[/latex]about the y-axis. Reflections of the Parent Function f(x) = bxThe function[latex]\,f\left(x\right)=-{b}^{x}[/latex]
The function[latex]\,f\left(x\right)={b}^{-x}[/latex]
Writing and Graphing the Reflection of an Exponential FunctionFind and graph the equation for a function,[latex]\,g\left(x\right),[/latex]that reflects[latex]\,f\left(x\right)={\left(\frac{1}{4}\right)}^{x}\,[/latex]about the x-axis. State its domain, range, and asymptote. Try ItFind and graph the equation for a function,[latex]\,g\left(x\right),[/latex] that reflects[latex]\,f\left(x\right)={1.25}^{x}\,[/latex]about the y-axis. State its domain, range, and asymptote. Summarizing Translations of the Exponential FunctionNow that we have worked with each type of translation for the exponential function, we can summarize them in (Figure) to arrive at the general equation for translating exponential functions. 1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b>1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).”>
Translations of Exponential FunctionsA translation of an exponential function has the form [latex] f\left(x\right)=a{b}^{x+c}+d[/latex] Where the parent function,[latex]\,y={b}^{x},[/latex][latex]\,b>1,[/latex]is
Note the order of the shifts, transformations, and reflections follow the order of operations. Writing a Function from a DescriptionWrite the equation for the function described below. Give the horizontal asymptote, the domain, and the range.
Try ItWrite the equation for function described below. Give the horizontal asymptote, the domain, and the range.
Key Equations
Key Concepts
Section ExercisesVerbalWhat role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph? What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically? AlgebraicThe graph of[latex]\,f\left(x\right)={3}^{x}\,[/latex]is reflected about the y-axis and stretched vertically by a factor of[latex]\,4.\,[/latex]What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept, domain, and range. The graph of[latex]\,f\left(x\right)={\left(\frac{1}{2}\right)}^{-x}\,[/latex]is reflected about the y-axis and compressed vertically by a factor of[latex]\,\frac{1}{5}.\,[/latex]What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept, domain, and range. The graph of[latex]\,f\left(x\right)={10}^{x}\,[/latex]is reflected about the x-axis and shifted upward[latex]\,7\,[/latex]units. What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept, domain, and range. The graph of[latex]\,f\left(x\right)={\left(1.68\right)}^{x}\,[/latex]is shifted right[latex]\,3\,[/latex]units, stretched vertically by a factor of[latex]\,2,[/latex]reflected about the x-axis, and then shifted downward[latex]\,3\,[/latex]units. What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept (to the nearest thousandth), domain, and range. The graph of[latex]\,f\left(x\right)=2{\left(\frac{1}{4}\right)}^{x-20}[/latex] is shifted left[latex]\,2\,[/latex]units, stretched vertically by a factor of[latex]\,4,[/latex]reflected about the x-axis, and then shifted downward[latex]\,4\,[/latex]units. What is the equation of the new function,[latex]\,g\left(x\right)?\,[/latex]State its y-intercept, domain, and range. GraphicalFor the following exercises, graph the function and its reflection about the y-axis on the same axes, and give the y-intercept. [latex]f\left(x\right)=3{\left(\frac{1}{2}\right)}^{x}[/latex] [latex]g\left(x\right)=-2{\left(0.25\right)}^{x}[/latex] [latex]h\left(x\right)=6{\left(1.75\right)}^{-x}[/latex] For the following exercises, graph each set of functions on the same axes. [latex]f\left(x\right)=3{\left(\frac{1}{4}\right)}^{x},[/latex][latex]g\left(x\right)=3{\left(2\right)}^{x},[/latex]and[latex]\,h\left(x\right)=3{\left(4\right)}^{x}[/latex] [latex]f\left(x\right)=\frac{1}{4}{\left(3\right)}^{x},[/latex][latex]g\left(x\right)=2{\left(3\right)}^{x},[/latex]and[latex]\,h\left(x\right)=4{\left(3\right)}^{x}[/latex] For the following exercises, match each function with one of the graphs in (Figure). Figure 12. [latex]f\left(x\right)=2{\left(0.69\right)}^{x}[/latex] [latex]f\left(x\right)=2{\left(1.28\right)}^{x}[/latex] [latex]f\left(x\right)=2{\left(0.81\right)}^{x}[/latex] [latex]f\left(x\right)=4{\left(1.28\right)}^{x}[/latex] [latex]f\left(x\right)=2{\left(1.59\right)}^{x}[/latex] [latex]f\left(x\right)=4{\left(0.69\right)}^{x}[/latex] For the following exercises, use the graphs shown in (Figure). All have the form[latex]\,f\left(x\right)=a{b}^{x}.[/latex] Figure 13. Which graph has the largest value for[latex]\,b?[/latex] Which graph has the smallest value for[latex]\,b?[/latex] Which graph has the largest value for[latex]\,a?[/latex] Which graph has the smallest value for[latex]\,a?[/latex] For the following exercises, graph the function and its reflection about the x-axis on the same axes. [latex]f\left(x\right)=\frac{1}{2}{\left(4\right)}^{x}[/latex] [latex]f\left(x\right)=3{\left(0.75\right)}^{x}-1[/latex] [latex]f\left(x\right)=-4{\left(2\right)}^{x}+2[/latex] For the following exercises, graph the transformation of[latex]\,f\left(x\right)={2}^{x}.\,[/latex]Give the horizontal asymptote, the domain, and the range. [latex]f\left(x\right)={2}^{-x}[/latex] [latex]h\left(x\right)={2}^{x}+3[/latex] [latex]f\left(x\right)={2}^{x-2}[/latex] For the following exercises, describe the end behavior of the graphs of the functions. [latex]f\left(x\right)=-5{\left(4\right)}^{x}-1[/latex] [latex]f\left(x\right)=3{\left(\frac{1}{2}\right)}^{x}-2[/latex] [latex]f\left(x\right)=3{\left(4\right)}^{-x}+2[/latex] For the following exercises, start with the graph of[latex]\,f\left(x\right)={4}^{x}.\,[/latex]Then write a function that results from the given transformation. Shift [latex]f\left(x\right)[/latex] 4 units upward Shift[latex]\,f\left(x\right)\,[/latex]3 units downward Shift[latex]\,f\left(x\right)\,[/latex]2 units left Shift[latex]\,f\left(x\right)\,[/latex]5 units right Reflect[latex]\,f\left(x\right)\,[/latex]about the x-axis Reflect[latex]\,f\left(x\right)\,[/latex]about the y-axis For the following exercises, each graph is a transformation of[latex]\,y={2}^{x}.\,[/latex]Write an equation describing the transformation. For the following exercises, find an exponential equation for the graph. NumericFor the following exercises, evaluate the exponential functions for the indicated value of[latex]\,x.[/latex] [latex]g\left(x\right)=\frac{1}{3}{\left(7\right)}^{x-2}\,[/latex]for[latex]\,g\left(6\right).[/latex] [latex]f\left(x\right)=4{\left(2\right)}^{x-1}-2\,[/latex]for[latex]\,f\left(5\right).[/latex] [latex]h\left(x\right)=-\frac{1}{2}{\left(\frac{1}{2}\right)}^{x}+6\,[/latex]for[latex]\,h\left(-7\right).[/latex] TechnologyFor the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. [latex]-50=-{\left(\frac{1}{2}\right)}^{-x}[/latex] [latex]116=\frac{1}{4}{\left(\frac{1}{8}\right)}^{x}[/latex] [latex]12=2{\left(3\right)}^{x}+1[/latex] [latex]5=3{\left(\frac{1}{2}\right)}^{x-1}-2[/latex] [latex]-30=-4{\left(2\right)}^{x+2}+2[/latex] ExtensionsExplore and discuss the graphs of[latex]\,F\left(x\right)={\left(b\right)}^{x}\,[/latex]and[latex]\,G\left(x\right)={\left(\frac{1}{b}\right)}^{x}.\,[/latex]Then make a conjecture about the relationship between the graphs of the functions[latex]\,{b}^{x}\,[/latex]and[latex]\,{\left(\frac{1}{b}\right)}^{x}\,[/latex]for any real number[latex]\,b>0.[/latex] Prove the conjecture made in the previous exercise. Explore and discuss the graphs of[latex]\,f\left(x\right)={4}^{x},[/latex][latex]\,g\left(x\right)={4}^{x-2},[/latex]and[latex]\,h\left(x\right)=\left(\frac{1}{16}\right){4}^{x}.\,[/latex]Then make a conjecture about the relationship between the graphs of the functions[latex]\,{b}^{x}\,[/latex]and[latex]\,\left(\frac{1}{{b}^{n}}\right){b}^{x}\,[/latex]for any real number n and real number[latex]\,b>0.[/latex] Prove the conjecture made in the previous exercise. How do the xThe distance between the x-intercepts in function A is half the distance between the x-intercepts of function B. The distance between the x-intercepts in function A is twice the distance between the x-intercepts of function B.
What are 2 other names for the XX-intercepts are also called zeros, roots, solutions, or solution sets.
How do you know if a function has two xIf the value of the discriminant is positive, there are two real solutions for x, meaning the graph of the solution has two distinct x-intercepts. If the value of the discriminant is zero, there is one real solution for x, meaning the graph of the solution has one x-intercept.
How are XThe x-intercept is where a line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.
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