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39 Graphs of Exponential Functions

Learning Objectives

  • Graph exponential functions.
  • Graph exponential functions using transformations.

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 Functions

Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the formf(x)=bxwhose base is greater than one. We’ll use the functionf(x)=2x.Observe how the output values in (Figure) change as the input increases by1.

x 3 2 1 0 1 2 3
f(x)=2x 18 14 12 1 2 4 8

Each output value is the product of the previous output and the base,2.We call the base2the constant ratio. In fact, for any exponential function with the formf(x)=abx,bis 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 ofa.

Notice from the table that

  • the output values are positive for all values of x;
  • asxincreases, the output values increase without bound; and
  • asxdecreases, the output values grow smaller, approaching zero.

(Figure) shows the exponential growth function f(x)=2x.

Graph of the exponential function, 2^(x), with labeled points at (-3, 1/8), (-2, ¼), (-1, ½), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote.
Figure 1. Notice that the graph gets close to the x-axis, but never touches it.

The domain off(x)=2xis all real numbers, the range is(0,), and the horizontal asymptote isy=0.

To get a sense of the behavior of exponential decay, we can create a table of values for a function of the formf(x)=bxwhose base is between zero and one. We’ll use the functiong(x)=(12)x.Observe how the output values in (Figure) change as the input increases by1.

x 3 2 1 0 1 2 3
g(x)=(12)x 8 4 2 1 12 14 18

Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio12.

Notice from the table that

  • the output values are positive for all values ofx;
  • asxincreases, the output values grow smaller, approaching zero; and
  • asxdecreases, the output values grow without bound.

(Figure) shows the exponential decay function,g(x)=(12)x.

Graph of decreasing exponential function, (1/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1/2), (2, 1/4), and (3, 1/8). The graph notes that the x-axis is an asymptote.
Figure 2.

The domain ofg(x)=(12)xis all real numbers, the range is(0,),and the horizontal asymptote isy=0.

Characteristics of the Graph of the Parent Function f(x) = bx

An exponential function with the formf(x)=bx,b>0,b1,has these characteristics:

  • one-to-one function
  • horizontal asymptote:y=0
  • domain:(,)
  • range:(0,)
  • x-intercept: none
  • y-intercept:(0,1)
  • increasing ifb>1
  • decreasing ifb<1

(Figure) compares the graphs of exponential growth and decay functions.

Figure 3.

How To

Given an exponential function of the formf(x)=bx,graph the function.

  1. Create a table of points.
  2. Plot at least3point from the table, including the y-intercept(0,1).
  3. Draw a smooth curve through the points.
  4. State the domain,(,),the range,(0,),and the horizontal asymptote, y=0.

Sketching the Graph of an Exponential Function of the Form f(x) = bx

Sketch a graph off(x)=0.25x.State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165135696740″]Show Solution[/reveal-answer]
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Before graphing, identify the behavior and create a table of points for the graph.

  • Sinceb=0.25is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptotey=0.
  • Create a table of points as in (Figure).
    x 3 2 1 0 1 2 3
    f(x)=0.25x 64 16 4 1 0.25 0.0625 0.015625
  • Plot the y-intercept,(0,1),along with two other points. We can use(1,4)and(1,0.25).

Draw a smooth curve connecting the points as in (Figure).

Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25).
Figure 4.

The domain is(,);the range is(0,);the horizontal asymptote isy=0.

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Sketch the graph off(x)=4x.State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137731723″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137731723″]

The domain is(,);the range is(0,);the horizontal asymptote isy=0.

Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).[/hidden-answer]

Graphing Transformations of Exponential Functions

Transformations 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 functionf(x)=bxwithout 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 Shift

The first transformation occurs when we add a constantdto the parent functionf(x)=bx, giving us a vertical shiftdunits in the same direction as the sign. For example, if we begin by graphing a parent function,f(x)=2x, we can then graph two vertical shifts alongside it, usingd=3:the upward shift,g(x)=2x+3and the downward shift,h(x)=2x3.Both vertical shifts are shown in (Figure).

Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions’ transformations are described in the text.
Figure 5.

Observe the results of shiftingf(x)=2xvertically:

  • The domain,(,)remains unchanged.
  • When the function is shifted up3units tog(x)=2x+3:
    • The y-intercept shifts up3units to(0,4).
    • The asymptote shifts up3units toy=3.
    • The range becomes(3,).
  • When the function is shifted down3units toh(x)=2x3:
    • The y-intercept shifts down3units to(0,2).
    • The asymptote also shifts down3units toy=3.
    • The range becomes(3,).

Graphing a Horizontal Shift

The next transformation occurs when we add a constantcto the input of the parent functionf(x)=bx, giving us a horizontal shiftcunits in the opposite direction of the sign. For example, if we begin by graphing the parent functionf(x)=2x, we can then graph two horizontal shifts alongside it, usingc=3:the shift left,g(x)=2x+3, and the shift right,h(x)=2x3.Both horizontal shifts are shown in (Figure).

Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions’ asymptotes are at y=0Note that each functions’ transformations are described in the text.
Figure 6.

Observe the results of shiftingf(x)=2xhorizontally:

  • The domain,(,),remains unchanged.
  • The asymptote,y=0,remains unchanged.
  • The y-intercept shifts such that:
    • When the function is shifted left3units tog(x)=2x+3,the y-intercept becomes(0,8).This is because2x+3=(8)2x,so the initial value of the function is8.
    • When the function is shifted right3units toh(x)=2x3,the y-intercept becomes(0,18).Again, see that2x3=(18)2x,so the initial value of the function is18.

Shifts of the Parent Function f(x) = bx

For any constantscandd,the functionf(x)=bx+c+dshifts the parent functionf(x)=bx

  • verticallydunits, in the same direction of the sign ofd.
  • horizontallycunits, in the opposite direction of the sign ofc.
  • The y-intercept becomes(0,bc+d).
  • The horizontal asymptote becomesy=d.
  • The range becomes(d,).
  • The domain,(,),remains unchanged.

How To

Given an exponential function with the formf(x)=bx+c+d,graph the translation.

  1. Draw the horizontal asymptotey=d.
  2. Identify the shift as(c,d).Shift the graph off(x)=bxleftcunits ifcis positive, and rightcunits ifcis negative.
  3. Shift the graph off(x)=bxupdunits ifdis positive, and downdunits ifdis negative.
  4. State the domain,(,),the range,(d,),and the horizontal asymptotey=d.

Graphing a Shift of an Exponential Function

Graphf(x)=2x+13.State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165135175234″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135175234″]

We have an exponential equation of the formf(x)=bx+c+d, withb=2,c=1, andd=3.

Draw the horizontal asymptotey=d, so drawy=3.

Identify the shift as(c,d), so the shift is(1,3).

Shift the graph off(x)=bxleft 1 units and down 3 units.

Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1).
Figure 7.

The domain is(,);the range is(3,);the horizontal asymptote isy=3.[/hidden-answer]

Try It

Graphf(x)=2x1+3.State domain, range, and asymptote.

[reveal-answer q=”fs-id1165137731918″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137731918″]

The domain is(,);the range is(3,);the horizontal asymptote isy=3.

Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).[/hidden-answer]

How To

Given an equation of the formf(x)=bx+c+dforx, use a graphing calculator to approximate the solution.

  • Press [Y=]. Enter the given exponential equation in the line headed “Y1=”.
  • Enter the given value forf(x)in the line headed “Y2=”.
  • Press [WINDOW]. Adjust the y-axis so that it includes the value entered for “Y2=”.
  • Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value off(x).
  • To find the value ofx,we compute the point of intersection. Press [2ND] then [CALC]. Select “intersect” and press [ENTER] three times. The point of intersection gives the value of x for the indicated value of the function.

Approximating the Solution of an Exponential Equation

Solve42=1.2(5)x+2.8graphically. Round to the nearest thousandth.

[reveal-answer q=”fs-id1165137653309″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137653309″]

Press [Y=] and enter1.2(5)x+2.8next to Y1=. Then enter 42 next to Y2=. For a window, use the values –3 to 3 forxand –5 to 55 fory.Press [GRAPH]. The graphs should intersect somewhere nearx=2.

For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The x-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for Guess?) To the nearest thousandth,x2.166.

[/hidden-answer]

Try It

Solve4=7.85(1.15)x2.27graphically. Round to the nearest thousandth.

[reveal-answer q=”fs-id1165137854192″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137854192″]

x1.608

[/hidden-answer]

Graphing a Stretch or Compression

While 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 functionf(x)=bxby a constant|a|>0.For example, if we begin by graphing the parent functionf(x)=2x,we can then graph the stretch, usinga=3,to getg(x)=3(2)xas shown on the left in (Figure), and the compression, usinga=13,to geth(x)=13(2)xas shown on the right in (Figure).

Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.
Figure 8. (a)g(x)=3(2)xstretches the graph off(x)=2xvertically by a factor of3.(b)h(x)=13(2)xcompresses the graph off(x)=2xvertically by a factor of13.

Stretches and Compressions of the Parent Function f(x) = bx

For any factora>0,the functionf(x)=a(b)x

  • is stretched vertically by a factor ofaif|a|>1.
  • is compressed vertically by a factor ofaif|a|<1.
  • has a y-intercept of(0,a).
  • has a horizontal asymptote aty=0, a range of(0,), and a domain of(,),which are unchanged from the parent function.

Graphing the Stretch of an Exponential Function

Sketch a graph off(x)=4(12)x.State the domain, range, and asymptote.

[reveal-answer q=”274741″]Show Solution[/reveal-answer]
[hidden-answer a=”274741″]

Before graphing, identify the behavior and key points on the graph.

  • Sinceb=12is between zero and one, the left tail of the graph will increase without bound asxdecreases, and the right tail will approach the x-axis asxincreases.
  • Sincea=4,the graph off(x)=(12)xwill be stretched by a factor of4.
  • Create a table of points as shown in (Figure).
    x 3 2 1 0 1 2 3
    f(x)=4(12)x 32 16 8 4 2 1 0.5
  • Plot the y-intercept,(0,4),along with two other points. We can use(1,8)and(1,2).

Draw a smooth curve connecting the points, as shown in (Figure).

Graph of the function, f(x) = 4(1/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2).
Figure 9.

The domain is(,);the range is(0,);the horizontal asymptote isy=0.

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Try It

Sketch the graph off(x)=12(4)x.State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137694067″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137694067″]

The domain is(,);the range is(0,);the horizontal asymptote isy=0.Graph of the function, f(x) = (1/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).[/hidden-answer]

Graphing Reflections

In 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 functionf(x)=bxby1,we get a reflection about the x-axis. When we multiply the input by1,we get a reflection about the y-axis. For example, if we begin by graphing the parent functionf(x)=2x, we can then graph the two reflections alongside it. The reflection about the x-axis,g(x)=2x,is shown on the left side of (Figure), and the reflection about the y-axish(x)=2x, is shown on the right side of (Figure).

Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis.
Figure 10. (a)g(x)=2xreflects the graph off(x)=2xabout the x-axis. (b)g(x)=2xreflects the graph off(x)=2xabout the y-axis.

Reflections of the Parent Function f(x) = bx

The functionf(x)=bx

  • reflects the parent functionf(x)=bxabout the x-axis.
  • has a y-intercept of(0,1).
  • has a range of(,0)
  • has a horizontal asymptote aty=0and domain of(,),which are unchanged from the parent function.

The functionf(x)=bx

  • reflects the parent functionf(x)=bxabout the y-axis.
  • has a y-intercept of(0,1), a horizontal asymptote aty=0, a range of(0,), and a domain of(,), which are unchanged from the parent function.

Writing and Graphing the Reflection of an Exponential Function

Find and graph the equation for a function,g(x),that reflectsf(x)=(14)xabout the x-axis. State its domain, range, and asymptote.

[reveal-answer q=”fs-id1165137937537″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137937537″]

Since we want to reflect the parent functionf(x)=(14)xabout the x-axis, we multiplyf(x)by1to get,g(x)=(14)x.Next we create a table of points as in (Figure).

x 3 2 1 0 1 2 3
g(x)=(14)x 64 16 4 1 0.25 0.0625 0.0156

Plot the y-intercept,(0,1),along with two other points. We can use(1,4)and(1,0.25).

Draw a smooth curve connecting the points:

Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25).
Figure 11.

The domain is(,);the range is(,0);the horizontal asymptote isy=0.[/hidden-answer]

Try It

Find and graph the equation for a function,g(x), that reflectsf(x)=1.25xabout the y-axis. State its domain, range, and asymptote.

[reveal-answer q=”fs-id1165135368458″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135368458″]

The domain is(,);the range is(0,);the horizontal asymptote isy=0.

Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).[/hidden-answer]

Summarizing Translations of the Exponential Function

Now 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 the Parent Function f(x)=bx
Translation Form
Shift

  • Horizontallycunits to the left
  • Verticallydunits up
f(x)=bx+c+d
Stretch and Compress

  • Stretch if|a|>1
  • Compression if0<|a|<1
f(x)=abx
Reflect about the x-axis f(x)=bx
Reflect about the y-axis f(x)=bx=(1b)x
General equation for all translations f(x)=abx+c+d

Translations of Exponential Functions

A translation of an exponential function has the form

f(x)=abx+c+d

Where the parent function,y=bx,b>1,is

  • shifted horizontallycunits to the left.
  • stretched vertically by a factor of|a|if|a|>0.
  • compressed vertically by a factor of|a|if0<|a|<1.
  • shifted verticallydunits.
  • reflected about the x-axis whena<0.

Note the order of the shifts, transformations, and reflections follow the order of operations.

Writing a Function from a Description

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

  • f(x)=exis vertically stretched by a factor of2, reflected across the y-axis, and then shifted up4units.
[reveal-answer q=”fs-id1165135532412″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135532412″]

We want to find an equation of the general formf(x)=abx+c+d.We use the description provided to finda, b, c, and d.

  • We are given the parent functionf(x)=ex, sob=e.
  • The function is stretched by a factor of2, soa=2.
  • The function is reflected about the y-axis. We replacexwithxto get:ex.
  • The graph is shifted vertically 4 units, sod=4.

Substituting in the general form we get,

f(x)=abx+c+d=2ex+0+4=2ex+4

The domain is(,);the range is(4,);the horizontal asymptote isy=4.[/hidden-answer]

Try It

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.

  • f(x)=exis compressed vertically by a factor of13, reflected across the x-axis and then shifted down 2 units.
[reveal-answer q=”fs-id1165137724110″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137724110″]

f(x)=13ex2;the domain is(,);the range is(,2);the horizontal asymptote isy=2.

[/hidden-answer]

Access this online resource for additional instruction and practice with graphing exponential functions.

Key Equations

General Form for the Translation of the Parent Function f(x)=bx f(x)=abx+c+d

Key Concepts

  • The graph of the functionf(x)=bxhas a y-intercept at(0,1),domain(,),range(0,), and horizontal asymptotey=0.See (Figure).
  • Ifb>1,the function is increasing. The left tail of the graph will approach the asymptotey=0, and the right tail will increase without bound.
  • If[latex]\,0
  • The equationf(x)=bx+drepresents a vertical shift of the parent functionf(x)=bx.
  • The equationf(x)=bx+crepresents a horizontal shift of the parent functionf(x)=bx.See (Figure).
  • Approximate solutions of the equationf(x)=bx+c+dcan be found using a graphing calculator. See (Figure).
  • The equationf(x)=abx, wherea>0, represents a vertical stretch if|a|>1or compression if0<|a|<1of the parent functionf(x)=bx.See (Figure).
  • When the parent functionf(x)=bxis multiplied by1,the result,f(x)=bx, is a reflection about the x-axis. When the input is multiplied by1,the result,f(x)=bx, is a reflection about the y-axis. See (Figure).
  • All translations of the exponential function can be summarized by the general equationf(x)=abx+c+d.See (Figure).
  • Using the general equationf(x)=abx+c+d, we can write the equation of a function given its description. See (Figure).

Section Exercises

Verbal

What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

[reveal-answer q=”fs-id1165135386464″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135386464″]

An asymptote is a line that the graph of a function approaches, asxeither increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.

[/hidden-answer]

What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

Algebraic

The graph off(x)=3xis reflected about the y-axis and stretched vertically by a factor of4.What is the equation of the new function,g(x)?State its y-intercept, domain, and range.

[reveal-answer q=”fs-id1165135369259″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135369259″]

g(x)=4(3)x;y-intercept:(0,4);Domain: all real numbers; Range: all real numbers greater than0.

[/hidden-answer]

The graph off(x)=(12)xis reflected about the y-axis and compressed vertically by a factor of15.What is the equation of the new function,g(x)?State its y-intercept, domain, and range.

The graph off(x)=10xis reflected about the x-axis and shifted upward7units. What is the equation of the new function,g(x)?State its y-intercept, domain, and range.

[reveal-answer q=”fs-id1165137851416″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137851416″]

g(x)=10x+7;y-intercept:(0,6);Domain: all real numbers; Range: all real numbers less than7.

[/hidden-answer]

The graph off(x)=(1.68)xis shifted right3units, stretched vertically by a factor of2,reflected about the x-axis, and then shifted downward3units. What is the equation of the new function,g(x)?State its y-intercept (to the nearest thousandth), domain, and range.

The graph off(x)=2(14)x20 is shifted left2units, stretched vertically by a factor of4,reflected about the x-axis, and then shifted downward4units. What is the equation of the new function,g(x)?State its y-intercept, domain, and range.

[reveal-answer q=”fs-id1165137724981″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137724981″]

g(x)=2(14)x; y-intercept:(0, 2);Domain: all real numbers; Range: all real numbers greater than0.

[/hidden-answer]

Graphical

For the following exercises, graph the function and its reflection about the y-axis on the same axes, and give the y-intercept.

f(x)=3(12)x

g(x)=2(0.25)x

[reveal-answer q=”159818″]Show Solution[/reveal-answer]
[hidden-answer a=”159818″]Graph of two functions, g(-x)=-2(0.25)^(-x) in blue and g(x)=-2(0.25)^x in orange.

y-intercept:(0,2)

[/hidden-answer]

h(x)=6(1.75)x

For the following exercises, graph each set of functions on the same axes.

f(x)=3(14)x,g(x)=3(2)x,andh(x)=3(4)x

[reveal-answer q=”fs-id1165137639767″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137639767″]Graph of three functions, g(x)=3(2)^(x) in blue, h(x)=3(4)^(x) in green, and f(x)=3(1/4)^(x) in orange.[/hidden-answer]

f(x)=14(3)x,g(x)=2(3)x,andh(x)=4(3)x

For the following exercises, match each function with one of the graphs in (Figure).

Graph of six exponential functions.
Figure 12.

f(x)=2(0.69)x

[reveal-answer q=”fs-id1165137722409″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137722409″]

B

[/hidden-answer]

f(x)=2(1.28)x

f(x)=2(0.81)x

[reveal-answer q=”fs-id1165137767451″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137767451″]

A

[/hidden-answer]

f(x)=4(1.28)x

f(x)=2(1.59)x

[reveal-answer q=”fs-id1165135541572″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135541572″]

E

[/hidden-answer]

f(x)=4(0.69)x

For the following exercises, use the graphs shown in (Figure). All have the formf(x)=abx.

Graph of six exponential functions.
Figure 13.

Which graph has the largest value forb?

[reveal-answer q=”fs-id1165134040575″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165134040575″]

D

[/hidden-answer]

Which graph has the smallest value forb?

Which graph has the largest value fora?

[reveal-answer q=”fs-id1165137836509″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137836509″]

C

[/hidden-answer]

Which graph has the smallest value fora?

For the following exercises, graph the function and its reflection about the x-axis on the same axes.

f(x)=12(4)x

[reveal-answer q=”fs-id1165135581221″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135581221″]Graph of two functions, f(x)=(1/2)(4)^(x) in blue and -f(x)=(-1/2)(4)^x in orange.[/hidden-answer]

f(x)=3(0.75)x1

f(x)=4(2)x+2

[reveal-answer q=”fs-id1165135187279″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135187279″]Graph of two functions, -f(x)=(4)(2)^(x)-2 in blue and f(x)=(-4)(2)^x+1 in orange.[/hidden-answer]

For the following exercises, graph the transformation off(x)=2x.Give the horizontal asymptote, the domain, and the range.

f(x)=2x

h(x)=2x+3

[reveal-answer q=”979418″]Show Solution[/reveal-answer]
[hidden-answer a=”979418″]

Graph of h(x)=2^(x)+3.

Horizontal asymptote:h(x)=3; Domain: all real numbers; Range: all real numbers strictly greater than3.

[/hidden-answer]

f(x)=2x2

For the following exercises, describe the end behavior of the graphs of the functions.

f(x)=5(4)x1

[reveal-answer q=”fs-id1165135160376″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135160376″]As x,
f(x);[/hidden-answer]

f(x)=3(12)x2

f(x)=3(4)x+2

[reveal-answer q=”fs-id1165135543075″]Show Solution[/reveal-answer]

[hidden-answer a=”fs-id1165135543075″]

As x,
f(x)2;[/hidden-answer]

For the following exercises, start with the graph off(x)=4x.Then write a function that results from the given transformation.

Shift f(x) 4 units upward

Shiftf(x)3 units downward

[reveal-answer q=”fs-id1165137731311″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137731311″]

f(x)=4x3

[/hidden-answer]

Shiftf(x)2 units left

Shiftf(x)5 units right

[reveal-answer q=”fs-id1165137572737″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137572737″]

f(x)=4x5

[/hidden-answer]

Reflectf(x)about the x-axis

Reflectf(x)about the y-axis

[reveal-answer q=”fs-id1165135209553″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135209553″]
f(x)=4x[/hidden-answer]

For the following exercises, each graph is a transformation ofy=2x.Write an equation describing the transformation.

Graph of f(x)=2^(x) with the following translations: vertical stretch of 4, a reflection about the x-axis, and a shift up by 1.

Graph of f(x)=2^(x) with the following translations: a reflection about the x-axis, and a shift up by 3.

[reveal-answer q=”fs-id1165135536375″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135536375″]

y=2x+3

[/hidden-answer]

Graph of f(x)=2^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis and y-axis, and a shift up by 3.

For the following exercises, find an exponential equation for the graph.

Graph of f(x)=3^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis, and a shift up by 7.

[reveal-answer q=”fs-id1165135199467″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135199467″]

y=2(3)x+7

[/hidden-answer]

Graph of f(x)=(1/2)^(x) with the following translations: vertical stretch of 2, and a shift down by 4.

Numeric

For the following exercises, evaluate the exponential functions for the indicated value ofx.

g(x)=13(7)x2forg(6).

[reveal-answer q=”fs-id1165135175180″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135175180″]

g(6)=800+13800.3333

[/hidden-answer]

f(x)=4(2)x12forf(5).

h(x)=12(12)x+6forh(7).

[reveal-answer q=”fs-id1165134044680″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165134044680″]

h(7)=58

[/hidden-answer]

Technology

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.

50=(12)x

116=14(18)x
[reveal-answer q=”fs-id1165135309920″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135309920″]

x2.953

[/hidden-answer]

12=2(3)x+1

5=3(12)x12

[reveal-answer q=”fs-id1165137605832″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137605832″]

x0.222

[/hidden-answer]

30=4(2)x+2+2

Extensions

Explore and discuss the graphs ofF(x)=(b)xandG(x)=(1b)x.Then make a conjecture about the relationship between the graphs of the functionsbxand(1b)xfor any real numberb>0.

[reveal-answer q=”fs-id1165137635181″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137635181″]

The graph ofG(x)=(1b)xis the refelction about the y-axis of the graph ofF(x)=bx;For any real numberb>0and functionf(x)=bx,the graph of(1b)xis the the reflection about the y-axis,F(x).

[/hidden-answer]

Prove the conjecture made in the previous exercise.

Explore and discuss the graphs off(x)=4x,g(x)=4x2,andh(x)=(116)4x.Then make a conjecture about the relationship between the graphs of the functionsbxand(1bn)bxfor any real number n and real numberb>0.

[reveal-answer q=”fs-id1165135693738″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135693738″]

The graphs ofg(x)andh(x)are the same and are a horizontal shift to the right of the graph off(x);For any real number n, real numberb>0, and functionf(x)=bx, the graph of(1bn)bxis the horizontal shiftf(xn).

[/hidden-answer]

Prove the conjecture made in the previous exercise.

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Algebra and Trigonometry OpenStax Copyright © 2015 by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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