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Mathematics LibreTexts

4.3: Graphing Exponential Functions

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  • Page ID 126530

  • Katherine Skelton
  • Highline College

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Learning Objectives

  • Identify basic graphs of exponential functions and sketch their graphs.
  • Identify transformations of exponential functions.
  • Graph exponential functions.
  • Find equations of exponential functions that model graphical information.

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 form \(f(x)=b^x\) whose base is greater than one. We’ll use the function \(f(x)=2^x\). Observe how the output values in Table \(\PageIndex{1}\) change as the input increases by \(1\).

Each output value is the product of the previous output and the base, \(2\). We call the base \(2\) the constant ratio . In fact, for any exponential function with the form \(f(x)=ab^x\), \(b\) 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 \(a\).

Notice from the table that

  • the output values are positive for all values of \(x\);
  • as \(x\) increases, the output values increase without bound; and
  • as \(x\) decreases, the output values grow smaller, approaching zero.

Figure \(\PageIndex{1}\) shows the exponential growth function \(f(x)=2^x\).

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.

The domain of \(f(x)=2^x\) is all real numbers, the range is \((0,\infty)\), and the horizontal asymptote is \(y=0\).

To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form \(f(x)=b^x\) whose base is between zero and one. We’ll use the function \(g(x)={\left(\dfrac{1}{2}\right)}^x\). Observe how the output values in Table \(\PageIndex{2}\) change as the input increases by \(1\).

Again, because the input is increasing by \(1\), each output value is the product of the previous output and the base, or constant ratio \(\dfrac{1}{2}\).

  • the output values are positive for all values of \(x\);
  • as \(x\) increases, the output values grow smaller, approaching zero; and
  • as \(x\) decreases, the output values grow without bound.

Figure \(\PageIndex{2}\) shows the exponential decay function, \(g(x)={\left(\dfrac{1}{2}\right)}^x\).

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.

The domain of \(g(x)={\left(\dfrac{1}{2}\right)}^x\) is all real numbers, the range is \((0,\infty)\), and the horizontal asymptote is \(y=0\).

Characteristics of Basic Exponential Functions

An exponential function of the form \(f(x)=b^x\), \(b>0\), \(b≠1\), has the following characteristics:

  • one-to-one function
  • horizontal asymptote: \(y=0\)
  • domain: \((–\infty, \infty)\)
  • range: \((0,\infty)\)
  • x- intercept: none
  • y- intercept: \((0,1)\) 
  • point at \((1,b)\)
  • increasing if \(b>1\)
  • decreasing if \(b<1\)

Figure \(\PageIndex{3}\) compares the graphs of exponential growth and decay functions.

Graph of two functions where the first graph is of a function of f(x) = b^x when b>1 and the second graph is of the same function when b is 0<b<1. Both graphs have the points (0, 1) and (1, b) labeled.

Example \(\PageIndex{1}\)

Sketch the graph of \(f(x)=4^x\). State the domain, range, and asymptote.

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

  • Since \(b=4\) is greater than one, we know the function is increasing. The left tail of the graph will approach the horizontal asymptote, \(y=0\), and the right tail will increase without bound.
  • The \(y\)-intercept is \((0,1)\).
  • The point \((1,4)\) is on the graph.
  • We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

The domain is \((−\infty,\infty)\); the range is \((0,\infty)\); the horizontal asymptote is \(y=0\).

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

To get a better feeling for the effect of \(a\) and \(b\) on the graph of \(f(x)=ab^x\), examine the sets of graphs below. The first set shows various graphs, where \(a\) remains the same and we only change the value for \(b\).

屏幕快照 2019-06-24 下午8.01.01.png

Notice that the closer the value of \(b\) is to 1, the less steep the graph will be. The growth/decay is faster the further \(b\) is from 1.

In the next set of graphs, \(a\) is altered and our value for \(b\) remains the same.

屏幕快照 2019-06-24 下午8.02.55.png

Notice that changing the value for \(a\) changes the y-intercept. Since \(a\) is multiplying the \({b}^{x}\) term, \(a\) acts as a vertical stretch factor, not as a shift. Notice also that the end behavior for all of these functions is the same because the growth factor did not change and none of these \(a\) values introduced a vertical flip.

Example \(\PageIndex{2}\)

屏幕快照 2019-06-24 下午8.10.45.png

\[{f(x)=2(1.3)^{x} }\nonumber\]

\[{g(x)=2(1.8)^{x} }\nonumber\]

\[{h(x)=4(1.3)^{x} }\nonumber\]

\[{k(x)=4(0.7)^{x} }\nonumber\]

The graph of \(k(x)\) is the easiest to identify, since it is the only equation with a growth factor less than one, which will produce a decreasing graph. The graph of \(h(x)\) can be identified as the only growing exponential function with a vertical intercept at (0,4). The graphs of \(f(x)\) and \(g(x)\) both have a vertical intercept at (0,2), but since \(g(x)\) has a larger growth factor, we can identify it as the graph increasing faster.

屏幕快照 2019-06-24 下午8.13.10.png

Graphing Transformations of Exponential Functions

Transformations of exponential graphs behave similarly to those of other functions. Just as with other basic functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the basic function \(f(x)=b^x\) 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 Shift

The first transformation occurs when we add a constant \(d\) to the basic function \(f(x)=b^x\), giving us a vertical shiftd d units in the same direction as the sign. For example, if we begin by graphing a basic function, \(f(x)=2^x\), we can then graph two vertical shifts alongside it, using \(d=3\): the upward shift, \(g(x)=2^x+3\) and the downward shift, \(h(x)=2^x−3\). Both vertical shifts are shown in Figure \(\PageIndex{4}\).

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.

Observe the results of shifting \(f(x)=2^x\) vertically:

  • The domain, \((−\infty,\infty)\) remains unchanged.
  • The y- intercept shifts up \(3\) units to \((0,4)\).
  • The asymptote shifts up \(3\) units to \(y=3\).
  • The range becomes \((3,\infty)\).
  • The y- intercept shifts down \(3\) units to \((0,−2)\).
  • The asymptote also shifts down \(3\) units to \(y=−3\).
  • The range becomes \((−3,\infty)\).

Graphing a Horizontal Shift

The next transformation occurs when we add a constant \(c\) to the input of the basic function \(f(x)=b^x\), giving us a horizontal shift \(c\) units in the opposite direction of the sign. For example, if we begin by graphing the basic function \(f(x)=2^x\), we can then graph two horizontal shifts alongside it, using \(c=3\): the shift left, \(g(x)=2^{x+3}\), and the shift right, \(h(x)=2^{x−3}\). \(h(x)=2^{x−3}\). Both horizontal shifts are shown in Figure \(\PageIndex{5}\).

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.

Observe the results of shifting \(f(x)=2^x\) horizontally:

  • The domain, \((−\infty,\infty)\),remains unchanged.
  • The asymptote, \(y=0\),remains unchanged.
  • When the function is shifted left \(3\) units to \(g(x)=2^{x+3}\), the y-intercept of the basic function shifts to (-3,1). The y-intercept of \(g(x)\) is \((0,8)\) which can be found by evaluating the function at \(x=0\). 
  • When the function is shifted right \(3\) units to \(h(x)=2^{x−3}\), the y-intercept of the basic function shifts to (3,1). The y-intercept of \(h(x)\) is \((0,\dfrac{1}{8})\). 

Example \(\PageIndex{3}\)

Identify the transformations and use them to sketch the graph of \(f(x)=2^{x+1}−3\). State the domain, range, and asymptote.

We have an exponential equation of the form \(f(x)=b^{x+c}+d\), with \(b=2\), \(c=1\), and \(d=−3\).

The basic function is \(y=2^x\). The graph will shift left 1 unit and down 3 units.

Shifting left 1 unit and down 3 units results in the y-intercept of the basic graph shifting to \((−1,−2)\). The point \((1,2)\) on the basic graph shifts to \((0,-1)\). The horizontal asymptote \(y=−3\).

Using these key points, asymptote, and the shape of \(f(x)=2^x\), shift the whole graph left \(1\) 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).

The domain is \((−\infty,\infty)\); the range is \((−3,\infty)\); the horizontal asymptote is \(y=−3\).

You Try \(\PageIndex{1}\)

Identify the transformations and use them to sketch the graph of \(f(x)=2^{x−1}+3\).  State domain, range, and asymptote.

The basic function is \(y=2^x\). The graph will shift right 1 unit and up 3 units.

The domain is \((−\infty,\infty)\); the range is \((3,\infty)\); the horizontal asymptote is \(y=3\).

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).

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 basic function \(f(x)=b^x\) by \(−1\), we get a reflection about the x-axis. When we multiply the input by \(−1\), we get a reflection about the y-axis. The reflection about the x -axis, \(g(x)=−2^x\), is shown on the left side of Figure \(\PageIndex{7}\), and the reflection about the y-axis \(h(x)=2^{−x}\), is shown on the right side of Figure \(\PageIndex{7}\).

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.

Example \(\PageIndex{4}\)

Find and graph the equation for a function, \(g(x)\), that reflects \(f(x)={\left(\dfrac{1}{4}\right)}^x\) about the x-axis. State its domain, range, and asymptote.

Since we want to reflect the basic function \(f(x)={\left(\dfrac{1}{4}\right)}^x\) about the x-axis, we multiply \(f(x)\) by \(−1\) to get, \(g(x)=−{\left(\dfrac{1}{4}\right)}^x\). Next we create a table of points as in Table \(\PageIndex{3}\).

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).

The domain is \((−\infty,\infty)\); the range is \((−\infty,0)\); the horizontal asymptote is \(y=0\).

Example \(\PageIndex{5}\)

Identify the transformations and use them to sketch the graph of \(h(x)=3^{-x+2}-4\).  State domain, range, and asymptote.

The basic function is \(y=3^x\). Since \(x\) is negative, we must first factor so we can identify the transformations. \[\begin{align*} f(x) &=3^{-x+2}-4 \\&= 3^{-(x-2)}-4\end{align*}\]

There is a y-axis reflection, a horizontal shift to the right 2 units, and a vertical shift down 4 units. The domain is \((−\infty,\infty)\); the range is \((0,\infty)\); the horizontal asymptote is \(y=-4\).

Graph of Transformed Exponential Function h(x)

You Try \(\PageIndex{2}\)

Find and graph the equation for a function, \(g(x)\), that reflects \(f(x)={1.25}^x\) about the y-axis. State its domain, range, and asymptote.

The equation of the reflected graph is \(y=={1.25}^-x\); the domain is \((−\infty,\infty)\); the range is \((0,\infty)\); the horizontal asymptote is \(y=0\).

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).

Graphing a Stretch or Compression

While horizontal and vertical shifts involve adding constants to the input or to the function itself, a vertical stretch or compression occurs when we multiply the basic function \(f(x)=b^x\) by a constant \(|a|>0\). For example, if we begin by graphing the basic function \(f(x)=2^x\),we can then graph the stretch, using \(a=3\),to get \(g(x)=3{(2)}^x\) as shown on the left in Figure \(\PageIndex{10}\), and the compression, using \(a=\dfrac{1}{3}\),to get \(h(x)=\dfrac{1}{3}{(2)}^x\) as shown on the right in Figure \(\PageIndex{10}\).

Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.

Example \(\PageIndex{6}\)

Sketch a graph of \(f(x)=4{\left(\dfrac{1}{2}\right)}^x\). State the domain, range, and asymptote.

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

  • Since \(b=\dfrac{1}{2}\) this is a decreasing exponential function. The left end of the graph will increase without bound as \(x\) decreases, and the right tail will approach the x -axis as \(x\) increases.
  • Since \(a=4\), the graph of \(f(x)=4{\left(\dfrac{1}{2}\right)}^x\) will be stretched by a factor of \(4\).
  • Plot the y-intercept, \((0,4)\), along with two other points. We can use \((−1,8)\) and \((1,2)\). You may plot other points so your graph is more accurate also.

Draw a smooth curve connecting the points, as shown in Figure \(\PageIndex{11}\).

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).

The domain is \((−\infty,\infty)\); the range is \((0,\infty)\); the horizontal asymptote is \(y=0\).

You Try \(\PageIndex{3}\)

Sketch the graph of \(f(x)=\dfrac{1}{2}{(4)}^x\). State the domain, range, and asymptote.

The domain is \((−\infty,\infty)\); the range is \((0,\infty)\); the horizontal asymptote is \(y=0\). 

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).

Summary of Transformations of Exponential Functions

Now that we have worked with each type of transformation for the exponential function, we can summarize them to arrive at the general equation for translating exponential functions.

Transformations of Exponential Functions

A transformation of an exponential function has the form

\(f(x)=ab^{x+c}+d\)

where the basic function, \(y=b^x\), \(b>1\), is

  • shifted horizontally \(c\) units to the left.
  • stretched vertically by a factor of \(|a|\) if \(|a|>0\). 
  • compressed vertically by a factor of \(|a|\) if \(0<|a|<1\).
  • shifted vertically \(d\) units.
  • reflected about the x-axis when \(a<0\).

For \(f(x)=b^{-x}\), the graph of the basic function is reflected about the y-axis.

Remember that the order for transformations follow the order of operations. This means that you start with stretch/compressions and reflections first, then shifts.

You Try \(\PageIndex{4}\)

What is the domain, range, asymptote, and end behavior of \(y=8-e^{-x+7}\)?

The domain is \((-\infty,\infty)\), the range is \((-\infty,8)\), the vertical asymptote is at \(x=8\), as \(x \rightarrow -\infty\), \(y\rightarrow -\infty\), and as \(x \rightarrow \infty\), \(y\rightarrow 8\).

Finding Equations of Exponential Functions that Model Graphical Information

Example \(\pageindex{7}\).

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

\(f(x)=e^x\) is vertically stretched by a factor of \(2\) , reflected across the y-axis, and then shifted up \(4\) units.

We want to find an equation of the general form \(f(x)=ab^{x+c}+d\). We use the description provided to find a, b, c, and d.

  • We are given the basic function \(f(x)=e^x\), so \(b=e\).
  • The function is stretched by a factor of \(2\), so \(a=2\).
  • There is no horizontal shift so \(c=0\).
  • The function is reflected about the y -axis. We replace \(x\) with \(−x\) to get: \(e^{−x}\).
  • The graph is shifted vertically 4 units, so \(d=4\).

Substituting in the general form we get,

\(=2e^{−x+0}+4\)

\(=2e^{−x}+4\)

The domain is \((−\infty,\infty)\); the range is \((4,\infty)\); the horizontal asymptote is \(y=4\).

You Try \(\PageIndex{5}\)

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

\(f(x)=e^x\) is compressed vertically by a factor of \(\dfrac{1}{3}\), reflected across the x -axis and then shifted down \(2\) units.

\(f(x)=−\dfrac{1}{3}e^{x}−2\); the domain is \((−\infty,\infty)\); the range is \((−\infty,2)\); the horizontal asymptote is \(y=2\).

Example \(\PageIndex{8}\)

Find an exponential function that passes through the points \((−2,6)\) and \((2,1)\). Round to the nearest fourth decimal place.

Because we are not given the base for the function, we substitute both points into an equation of the form \(f(x)=ab^x\), and then solve the system for \(a\) and \(b\).

  • Substituting \((−2,6)\) gives \(6=ab^{−2}\)
  • Substituting \((2,1)\) gives \(1=ab^2\)

Use the first equation to solve for \(a\) in terms of \(b\):

\[\begin{align*} 6&= ab^{-2}\\ 6&=\dfrac{a}{b^2} \qquad \text{Use properties of exponents to rewrite with positive expoents}\\ 6b^2&= a \qquad \text{Multiply by common denominator} \end{align*}\]

Substitute a in the second equation, and solve for \(b\):

\[\begin{align*} 1&= ab^{2}\\ 1&= 6b^2 b^2 \qquad \text{Substitute a} \\ 1&= 6b^4 \qquad \text{Now, solve for b} \\ \dfrac{1}{6} &= b^4 \\ \left(\dfrac{1}{6}\right )^\frac{1}{4} &= \left(b^4\right)^\frac{1}{4} \\ b&= \left (\dfrac{1}{6} \right )^{\frac{1}{4}} \qquad \text{Round 4 decimal places rewrite the denominator}\\ b&\approx 0.6389 \end{align*}\]

Use the value of \(b\) in the first equation to solve for the value of \(a\):

\[\begin{align*} a&= 6b^{2}\\ &\approx 6(0.6389)^2 \\ &\approx 2.4492 \end{align*}\]

Thus, the equation is \(f(x)=2.4492{(0.6389)}^x\).

We can graph our model to check our work. Notice that the graph in Figure \(\PageIndex{12}\) passes through the initial points given in the problem, \((−2, 6)\) and \((2, 1)\). The graph is an example of an exponential decay function.

Graph of the exponential function, f(x)=2.4492(0.6389)^x, with labeled points at (-2, 6) and (2, 1).

You Try \(\PageIndex{6}\)

Given the two points \((1,3)\) and \((2,4.5)\), find the equation of the exponential function that passes through these two points.

\(f(x)=2{(1.5)}^x\)

6.2 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 form f ( x ) = b x f ( x ) = b x whose base is greater than one. We’ll use the function f ( x ) = 2 x . f ( x ) = 2 x . Observe how the output values in Table 1 change as the input increases by 1. 1.

Each output value is the product of the previous output and the base, 2. 2. We call the base 2 2 the constant ratio . In fact, for any exponential function with the form f ( x ) = a b x , f ( x ) = a b x , b b 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 a . a .

Notice from the table that

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

Figure 1 shows the exponential growth function f ( x ) = 2 x . f ( x ) = 2 x .

The domain of f ( x ) = 2 x f ( x ) = 2 x is all real numbers, the range is ( 0 , ∞ ) , ( 0 , ∞ ) , and the horizontal asymptote is y = 0. y = 0.

To get a sense of the behavior of exponential decay , we can create a table of values for a function of the form f ( x ) = b x f ( x ) = b x whose base is between zero and one. We’ll use the function g ( x ) = ( 1 2 ) x . g ( x ) = ( 1 2 ) x . Observe how the output values in Table 2 change as the input increases by 1. 1.

Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio 1 2 . 1 2 .

  • as x x increases, the output values grow smaller, approaching zero; and
  • as x x decreases, the output values grow without bound.

Figure 2 shows the exponential decay function, g ( x ) = ( 1 2 ) x . g ( x ) = ( 1 2 ) x .

The domain of g ( x ) = ( 1 2 ) x g ( x ) = ( 1 2 ) x is all real numbers, the range is ( 0 , ∞ ) , ( 0 , ∞ ) , and the horizontal asymptote is y = 0. y = 0.

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

An exponential function with the form f ( x ) = b x , f ( x ) = b x , b > 0 , b > 0 , b ≠ 1 , b ≠ 1 , has these characteristics:

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

Figure 3 compares the graphs of exponential growth and decay functions.

Given an exponential function of the form f ( x ) = b x , f ( x ) = b x , graph the function.

  • Create a table of points.
  • Plot at least 3 3 point from the table, including the y -intercept ( 0 , 1 ) . ( 0 , 1 ) .
  • Draw a smooth curve through the points.
  • State the domain, ( − ∞ , ∞ ) , ( − ∞ , ∞ ) , the range, ( 0 , ∞ ) , ( 0 , ∞ ) , and the horizontal asymptote, y = 0. y = 0.

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

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

Before graphing, identify the behavior and create a table of points for the graph.

  • Since b = 0.25 b = 0.25 is 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 asymptote y = 0. y = 0.
  • Plot the y -intercept, ( 0 , 1 ) , ( 0 , 1 ) , along with two other points. We can use ( − 1 , 4 ) ( − 1 , 4 ) and ( 1 , 0.25 ) . ( 1 , 0.25 ) .

Draw a smooth curve connecting the points as in Figure 4 .

The domain is ( − ∞ , ∞ ) ; ( − ∞ , ∞ ) ; the range is ( 0 , ∞ ) ; ( 0 , ∞ ) ; the horizontal asymptote is y = 0. y = 0.

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

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 function f ( x ) = b x f ( x ) = b x 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 Shift

The first transformation occurs when we add a constant d d to the parent function f ( x ) = b x , f ( x ) = b x , giving us a vertical shift d d units in the same direction as the sign. For example, if we begin by graphing a parent function, f ( x ) = 2 x , f ( x ) = 2 x , we can then graph two vertical shifts alongside it, using d = 3 : d = 3 : the upward shift, g ( x ) = 2 x + 3 g ( x ) = 2 x + 3 and the downward shift, h ( x ) = 2 x − 3. h ( x ) = 2 x − 3. Both vertical shifts are shown in Figure 5 .

Observe the results of shifting f ( x ) = 2 x f ( x ) = 2 x vertically:

  • The domain, ( − ∞ , ∞ ) ( − ∞ , ∞ ) remains unchanged.
  • The y- intercept shifts up 3 3 units to ( 0 , 4 ) . ( 0 , 4 ) .
  • The asymptote shifts up 3 3 units to y = 3. y = 3.
  • The range becomes ( 3 , ∞ ) . ( 3 , ∞ ) .
  • The y- intercept shifts down 3 3 units to ( 0 , − 2 ) . ( 0 , − 2 ) .
  • The asymptote also shifts down 3 3 units to y = − 3. y = − 3.
  • The range becomes ( − 3 , ∞ ) . ( − 3 , ∞ ) .

Graphing a Horizontal Shift

The next transformation occurs when we add a constant c c to the input of the parent function f ( x ) = b x , f ( x ) = b x , giving us a horizontal shift c c units in the opposite direction of the sign. For example, if we begin by graphing the parent function f ( x ) = 2 x , f ( x ) = 2 x , we can then graph two horizontal shifts alongside it, using c = 3 : c = 3 : the shift left, g ( x ) = 2 x + 3 , g ( x ) = 2 x + 3 , and the shift right, h ( x ) = 2 x − 3 . h ( x ) = 2 x − 3 . Both horizontal shifts are shown in Figure 6 .

Observe the results of shifting f ( x ) = 2 x f ( x ) = 2 x horizontally:

  • The domain, ( − ∞ , ∞ ) , ( − ∞ , ∞ ) , remains unchanged.
  • The asymptote, y = 0 , y = 0 , remains unchanged.
  • When the function is shifted left 3 3 units to g ( x ) = 2 x + 3 , g ( x ) = 2 x + 3 , the y -intercept becomes ( 0 , 8 ) . ( 0 , 8 ) . This is because 2 x + 3 = ( 8 ) 2 x , 2 x + 3 = ( 8 ) 2 x , so the initial value of the function is 8. 8.
  • When the function is shifted right 3 3 units to h ( x ) = 2 x − 3 , h ( x ) = 2 x − 3 , the y -intercept becomes ( 0 , 1 8 ) . ( 0 , 1 8 ) . Again, see that 2 x − 3 = ( 1 8 ) 2 x , 2 x − 3 = ( 1 8 ) 2 x , so the initial value of the function is 1 8 . 1 8 .

Shifts of the Parent Function f ( x ) = b x

For any constants c c and d , d , the function f ( x ) = b x + c + d f ( x ) = b x + c + d shifts the parent function f ( x ) = b x f ( x ) = b x

  • vertically d d units, in the same direction of the sign of d . d .
  • horizontally c c units, in the opposite direction of the sign of c . c .
  • The y -intercept becomes ( 0 , b c + d ) . ( 0 , b c + d ) .
  • The horizontal asymptote becomes y = d . y = d .
  • The range becomes ( d , ∞ ) . ( d , ∞ ) .

Given an exponential function with the form f ( x ) = b x + c + d , f ( x ) = b x + c + d , graph the translation.

  • Draw the horizontal asymptote y = d . y = d .
  • Identify the shift as ( − c , d ) . ( − c , d ) . Shift the graph of f ( x ) = b x f ( x ) = b x left c c units if c c is positive, and right c c units if c c is negative.
  • Shift the graph of f ( x ) = b x f ( x ) = b x up d d units if d d is positive, and down d d units if d d is negative.
  • State the domain, ( − ∞ , ∞ ) , ( − ∞ , ∞ ) , the range, ( d , ∞ ) , ( d , ∞ ) , and the horizontal asymptote y = d . y = d .

Graphing a Shift of an Exponential Function

Graph f ( x ) = 2 x + 1 − 3. f ( x ) = 2 x + 1 − 3. State the domain, range, and asymptote.

We have an exponential equation of the form f ( x ) = b x + c + d , f ( x ) = b x + c + d , with b = 2 , b = 2 , c = 1 , c = 1 , and d = − 3. d = − 3.

Draw the horizontal asymptote y = d y = d , so draw y = −3. y = −3.

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

Shift the graph of f ( x ) = b x f ( x ) = b x left 1 units and down 3 units.

The domain is ( − ∞ , ∞ ) ; ( − ∞ , ∞ ) ; the range is ( − 3 , ∞ ) ; ( − 3 , ∞ ) ; the horizontal asymptote is y = −3. y = −3.

Graph f ( x ) = 2 x − 1 + 3. f ( x ) = 2 x − 1 + 3. State domain, range, and asymptote.

Given an equation of the form f ( x ) = b x + c + d f ( x ) = b x + c + d for x , x , use a graphing calculator to approximate the solution.

  • Press [Y=] . Enter the given exponential equation in the line headed “ Y 1 = ”.
  • Enter the given value for f ( x ) f ( x ) in the line headed “ Y 2 = ”.
  • Press [WINDOW] . Adjust the y -axis so that it includes the value entered for “ Y 2 = ”.
  • Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value of f ( x ) . f ( x ) .
  • To find the value of x , x , 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

Solve 42 = 1.2 ( 5 ) x + 2.8 42 = 1.2 ( 5 ) x + 2.8 graphically. Round to the nearest thousandth.

Press [Y=] and enter 1.2 ( 5 ) x + 2.8 1.2 ( 5 ) x + 2.8 next to Y 1 =. Then enter 42 next to Y2= . For a window, use the values –3 to 3 for x x and –5 to 55 for y . y . Press [GRAPH] . The graphs should intersect somewhere near x = 2. x = 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, x ≈ 2.166. x ≈ 2.166.

Solve 4 = 7.85 ( 1.15 ) x − 2.27 4 = 7.85 ( 1.15 ) x − 2.27 graphically. Round to the nearest thousandth.

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 function f ( x ) = b x f ( x ) = b x by a constant | a | > 0. | a | > 0. For example, if we begin by graphing the parent function f ( x ) = 2 x , f ( x ) = 2 x , we can then graph the stretch, using a = 3 , a = 3 , to get g ( x ) = 3 ( 2 ) x g ( x ) = 3 ( 2 ) x as shown on the left in Figure 8 , and the compression, using a = 1 3 , a = 1 3 , to get h ( x ) = 1 3 ( 2 ) x h ( x ) = 1 3 ( 2 ) x as shown on the right in Figure 8 .

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

For any factor a > 0 , a > 0 , the function f ( x ) = a ( b ) x f ( x ) = a ( b ) x

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

Graphing the Stretch of an Exponential Function

Sketch a graph of f ( x ) = 4 ( 1 2 ) x . f ( x ) = 4 ( 1 2 ) x . State the domain, range, and asymptote.

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

  • Since b = 1 2 b = 1 2 is between zero and one, the left tail of the graph will increase without bound as x x decreases, and the right tail will approach the x -axis as x x increases.
  • Since a = 4 , a = 4 , the graph of f ( x ) = ( 1 2 ) x f ( x ) = ( 1 2 ) x will be stretched by a factor of 4. 4.
  • Plot the y- intercept, ( 0 , 4 ) , ( 0 , 4 ) , along with two other points. We can use ( − 1 , 8 ) ( − 1 , 8 ) and ( 1 , 2 ) . ( 1 , 2 ) .

Draw a smooth curve connecting the points, as shown in Figure 9 .

Sketch the graph of f ( x ) = 1 2 ( 4 ) x . f ( x ) = 1 2 ( 4 ) x . State the domain, range, and asymptote.

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 function f ( x ) = b x f ( x ) = b x by −1 , −1 , we get a reflection about the x -axis. When we multiply the input by −1 , −1 , we get a reflection about the y -axis. For example, if we begin by graphing the parent function f ( x ) = 2 x , f ( x ) = 2 x , we can then graph the two reflections alongside it. The reflection about the x -axis, g ( x ) = −2 x , g ( x ) = −2 x , is shown on the left side of Figure 10 , and the reflection about the y -axis h ( x ) = 2 − x , h ( x ) = 2 − x , is shown on the right side of Figure 10 .

Reflections of the Parent Function f ( x ) = b x f ( x ) = b x

The function f ( x ) = − b x f ( x ) = − b x

  • reflects the parent function f ( x ) = b x f ( x ) = b x about the x -axis.
  • has a y -intercept of ( 0 , − 1 ) . ( 0 , − 1 ) .
  • has a range of ( − ∞ , 0 ) . ( − ∞ , 0 ) .
  • has a horizontal asymptote at y = 0 y = 0 and domain of ( − ∞ , ∞ ) , ( − ∞ , ∞ ) , which are unchanged from the parent function.

The function f ( x ) = b − x f ( x ) = b − x

  • reflects the parent function f ( x ) = b x f ( x ) = b x about the y -axis.
  • has a y -intercept of ( 0 , 1 ) , ( 0 , 1 ) , a horizontal asymptote at y = 0 , y = 0 , a range of ( 0 , ∞ ) , ( 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 ) , g ( x ) , that reflects f ( x ) = ( 1 4 ) x f ( x ) = ( 1 4 ) x about the x -axis. State its domain, range, and asymptote.

Since we want to reflect the parent function f ( x ) = ( 1 4 ) x f ( x ) = ( 1 4 ) x about the x- axis, we multiply f ( x ) f ( x ) by − 1 − 1 to get, g ( x ) = − ( 1 4 ) x . g ( x ) = − ( 1 4 ) x . Next we create a table of points as in Table 5 .

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

Draw a smooth curve connecting the points:

The domain is ( − ∞ , ∞ ) ; ( − ∞ , ∞ ) ; the range is ( − ∞ , 0 ) ; ( − ∞ , 0 ) ; the horizontal asymptote is y = 0. y = 0.

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

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 Table 6 to arrive at the general equation for translating exponential functions.

Translations of Exponential Functions

A translation of an exponential function has the form

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

  • shifted horizontally c c units to the left.
  • stretched vertically by a factor of | a | | a | if | a | > 0. | a | > 0.
  • compressed vertically by a factor of | a | | a | if 0 < | a | < 1. 0 < | a | < 1.
  • shifted vertically d d units.
  • reflected about the x- axis when a < 0. a < 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 ) = e x f ( x ) = e x is vertically stretched by a factor of 2 2 , reflected across the y -axis, and then shifted up 4 4 units.

We want to find an equation of the general form   f ( x ) = a b x + c + d .   f ( x ) = a b x + c + d . We use the description provided to find a , a , b , b , c , c , and d . d .

  • We are given the parent function f ( x ) = e x , f ( x ) = e x , so b = e . b = e .
  • The function is stretched by a factor of 2 2 , so a = 2. a = 2.
  • The function is reflected about the y -axis. We replace x x with − x − x to get: e − x . e − x .
  • The graph is shifted vertically 4 units, so d = 4. d = 4.

Substituting in the general form we get,

The domain is ( − ∞ , ∞ ) ; ( − ∞ , ∞ ) ; the range is ( 4 , ∞ ) ; ( 4 , ∞ ) ; the horizontal asymptote is y = 4. y = 4.

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

  • f ( x ) = e x f ( x ) = e x is compressed vertically by a factor of 1 3 , 1 3 , reflected across the x -axis and then shifted down 2 2 units.

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

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  1. Graphing Exponential Functions Example #1

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COMMENTS

  1. 03.04 Graphing Exponential Functions Dailan

    Write an exponential function to represent the spread of Carter's social media posts. In exponential function form, the spread of Carter's social media posts would be f(x) = 10(2)^x; Graph each function using at least three points for each curve. All graphs should be placed together on the same coordinate plane, so be sure to label each curve.

  2. 03.04 Graphing Exponential Functions

    2. Write an exponential function to represent the spread of Carter's social media post. In exponential function form, the spread of Carter's social media posts would be (x) = 10(2)^x 3. Graph each function using at least three points for each curve. All graphs should be placed together on the same coordinate plane, so be sure to label ...

  3. Graphing Exponential Functions Assignment Flashcards

    Answer 2: A. Slide 7 of 9 (answers read left to right then up and down) B. Slide 8 of 9 (answers read left to right then up and down) C. Slide 9 of 9 (answers read left to right then up and down) Sketch the graph of. Reflect the graph across the y-axis to show the function. Stretch the graph vertically by a factor of 3 to show the function.

  4. 4.3: Graphs of Exponential Functions

    Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value off(x). f(x). To find the value ofx, x,we compute the point of intersection. Press [2ND] then [CALC]. Select "intersect" and press [ENTER] three times.

  5. Graphing Exponential Functions Assignment Flashcards

    Study with Quizlet and memorize flashcards containing terms like How does the graph of g(x) = 10x - 8 compare to the graph of f(x) = 10x?, This graph shows transformations between f(x) = 10x and g(x) = a · 10x. Identify the following functions., Which situation best describes the transformation between f(x) = 10x and g(x) = -2 · 10x? and more.

  6. 4.3: Graphing Exponential Functions

    We have an exponential equation of the form f(x) = bx + c + d, with b = 2, c = 1, and d = − 3. The basic function is y = 2x. The graph will shift left 1 unit and down 3 units. Shifting left 1 unit and down 3 units results in the y-intercept of the basic graph shifting to ( − 1, − 2).

  7. Graphing Exponential Functions Flashcards

    Study with Quizlet and memorize flashcards containing terms like y=3ⁿ, y=2ⁿ+4, y=-3ⁿ and more.

  8. 6.2 Graphs of Exponential Functions

    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 form f (x) = b x f (x) = b x whose base is greater than one. We'll use the function f (x) = 2 x. f (x) = 2 x.

  9. PDF Section 4.3 Graphing Exponential Functions

    Section 4.3 Lehmann, Intermediate Algebra, 4ed For an exponential function of the form y = abx, if the value of the independent variable increases by 1, the value of the dependent variable is multiplied by b. •For the function , as the value of x increases by 1, the value of y is multiplied by 3 •For the function , as the value of x

  10. Graphing Exponential Functions

    The graph for the function f (x) = 3x is given below. Match the given function, g(x), to its graph. g(x) = −3x+2 + 1. Learn Graphing Exponential Functions with free step-by-step video explanations and practice problems by experienced tutors.

  11. 0304-graphing-exponential-functions-dailan (pdf)

    Write an exponential function to represent the spread of Carter's social media posts. In exponential function form, the spread of Carter's social media posts would be f (x) = 10 (2)^x 3. Graph each function using at least three points for each curve. All graphs should be placed together on the same coordinate plane, so be sure to label each curve.

  12. Exponential graph: e^x

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  13. 03.04 Graphing Exponential Functions

    03.04 GRAPHING EXPONENTIAL FUNCTIONS - Free download as PDF File (.pdf), Text File (.txt) or read online for free. The document discusses how three students - Amber, Ben, and Carter - shared social media posts and tracked how many shares their posts received over time. Amber shared her post with 3 people who continued sharing it, increasing the number of shares daily as shown by the function f ...

  14. 03.04 Graphing Exponential Functions Assignment.docx

    3. Graph each function using at least three points for each curve. All graphs should be placed together on the same coordinate plane, so be sure to label each curve. You may graph your equation by hand on a piece of paper and scan your work, or you may use graphing technology. Carter is blue Amber is green Ben is red 4. Using the functions for each student, predict how many shares each student ...

  15. 03.04 Graphing Exponential Functions .pdf

    together on the same coordinate plane, so be sure to label each curve. You may graph your equation by hand on a piece of paper and scan your work, or you may use graphing technology. 4. Using the functions for each student, predict how many shares each student's post will be received on Day 3 and then on Day 10. Justify your answers.

  16. 03.04 Graphing Exponential Functions WORK

    03.04 Graphing Exponential Functions WORK - Free download as Word Doc (.doc / .docx), PDF File (.pdf), Text File (.txt) or read online for free. The document defines exponential functions to model the spread of social media posts by three students, Ben, Carter, and Amber. It then uses the functions to predict and compare the number of shares each post would receive on day 3 and day 10.

  17. 03.04 Graphing Exponential Functions Project Option 1

    All graphs should be placed together on the same coordinate plane, so be sure to label each curve. You may graph your equation by hand on a piece of paper and scan your work, or you may use graphing technology. Green is Amber's exponential function. Purple is Ben's exponential function. Black is Carter's exponential function. 4.

  18. 03.04 Graphing Exponential Functions Assignement

    Exponential Functions 1) Ben's social media post: f (x) = 2 (3)^x 2) Carter's social media post: f (x) = 10 (2)^x 3) Graph each function using at least 3 points for each curve. 4) Predictions Predict how many shares each student will have after 3 days and after 10 days. Amber 3 days: 192 10 days: 3145728 Ben 3 days: 54 10 days: 118098 Carter 3 ...

  19. 03.04 GRAPHING EXPONENTIAL FUNCTIONS.odt

    View 03.04 GRAPHING EXPONENTIAL FUNCTIONS.odt from AA 1How much do you share on social media? Do you have accounts linked to your computer, phone, and tablet? The average teen spends around five ... View Algebra 1- 3.04 assignment.rtf from ALGEBRA 1 at Online High School. Assignment In... Lesson 11 Quiz First Attempt_ Math 110 WC Spring 2020.pdf.

  20. 03.04 Graphing Exponential Functions .docx

    Write an exponential function to represent the spread of Ben's social media post. y=2(3)x 2. Write an exponential function to represent the spread of Carter's social media post. y=10(2)x 3. Graph each function using at least three points for each curve. All graphs should be placed together on the same coordinate plane, so be sure to label each ...