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Mathematics LibreTexts

4.2: Graphs of Exponential Functions

  • Page ID
    34903
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    Learning Objectives

    • Graph exponential functions and their transformations.
    • Review Laws of Exponents

    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. Seeing their graphs gives us another layer of insight for predicting future events.

    Graphing Exponential Functions

    Exponential growth is modelled by functions of the form \(f(x)=b^x\) where the base is greater than one. Exponential decay occurs when the base is between zero and one. We’ll use the functions \(f(x)=2^x\) and  \(g(x)={\left(\tfrac{1}{2}\right)}^x\) to get some insight into the behaviour of graphs that model exponential growth and decay. In each table of values below, observe how the output values change as the input increases by \(1\).

    Exponential Growth: \(f(x)=2^x\).

    Table \(\PageIndex{1}\): Exponential Growth
    \(x\) \(−3\) \(−2\) \(−1\) \(0\) \(1\) \(2\) \(3\)
    \(f(x)=2^x\) \(\dfrac{1}{8}\) \(\dfrac{1}{4}\) \(\dfrac{1}{2}\) \(1\) \(2\) \(4\) \(8\)

    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 \(f(x+1)\) will be the product of the base and the previous output, \(b f(x)=b \cdot ab^x=ab^{x+1}=f(x+1)\), 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.

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

    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 \(\PageIndex{1}\): Graph of the exponential growth function \(f(x)=2^x\).
    Notice that the graph gets close to the x-axis, but never touches it.

    Exponential Decay: \(g(x)={\left(\frac{1}{2}\right)}^x\).

    Table \(\PageIndex{2}\): Exponential Decay
    \(x\) \(−3\) \(−2\) \(−1\) \(0\) \(1\) \(2\) \(3\)
    \(g(x)=\big( \dfrac{1}{2} \big) \) \(8\) \(4\) \(2\) \(1\) \(\dfrac{1}{2}\) \(\dfrac{1}{4}\) \(\dfrac{1}{8}\)

    Again, as the input is increases by \(1\), each output value is the product of the previous output and the base, \(\tfrac{1}{2}\), so the constant ratio of the function is \(\tfrac{1}{2}\). Notice from the table that

    • 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.

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

    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 \(\PageIndex{2}\): Graph of the exponential decay function, \(g(x)={\left(\frac{1}{2}\right)}^x\).

    Notice that \(g(x)={\left(\frac{1}{2}\right)}^x = (2^{-1})^x = f(-x)\). Thus the graph of \(g\) is simply a reflection over the \(y\)-axis of the graph of \(f\).

    CHARACTERISTICS OF THE GRAPH OF THE PARENT FUNCTION \(f(x) = b^x\)

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

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

    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.

    Figure \(\PageIndex{3}\): Graphs of exponential growth and decay functions.

     How to: Given an exponential function of the form \(f(x)=b^x\), graph the function

    1. Create a table of points.
    2. Plot at least \(3\) points from the table. Include the (y\)-intercept \((0,1)\) and the horizontal asymptote,  \(y=0\).
    3. Draw a smooth curve through the points that approaches the asymptote.
    4. State the domain \((−\infty,\infty)\), and the range \((0,\infty)\).

    Example \(\PageIndex{1}\): Sketching the Graph of an Exponential Function of the Form \(f(x) = b^x\)

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

    Solution

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

    Table \(\PageIndex{3}\)
    \(x\) \(−3\) \(−2\) \(−1\) \(0\) \(1\) \(2\) \(3\)
    \(f(x)={0.25}^x\) \(64\) \(16\) \(4\) \(1\) \(0.25\) \(0.0625\0 \(0.015625\)
    • Since \(b=0.25\) 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 \(y=0\).
    • Create a table of points as in Table \(\PageIndex{3}\).
    • Plot the asymptote, and 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 shown below.

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

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

    Try It \(\PageIndex{1}\)

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

    Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).The domain is \((−\infty,\infty)\); the range is \((0,\infty)\); the horizontal asymptote is \(y=0\).


     

    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 \(f(x)=b^x\) without loss of general 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

    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.The first transformation occurs when we add a constant \(d\) to the parent function \(f(x)=b^x\), giving us a vertical shift \(d\) units in the same direction as the sign.  For example, if we begin by graphing a parent 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 shifts are shown in the figure to the right.

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

    • The domain, \((−\infty,\infty)\) remains unchanged.
    • When the function is shifted up \(3\) units to \(g(x)=2^x+3\):
      • The \(y\)-intercept shifts up \(3\) units to \((0,4)\).
      • The asymptote shifts up \(3\) units to \(y=3\).
      • The range becomes \((3,\infty)\).
    • When the function is shifted down \(3\) units to \(h(x)=2^x−3\):
      • 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

    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.The next transformation occurs when we add a constant \(c\) to the input of the parent 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 parent 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, \(h(x)=2^{x−3}\).  Both horizontal shifts are shown in the figure to the right.

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

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

    SHIFTS OF THE PARENT FUNCTION \(f(x) = b^x\)

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

    • vertically \(d\) units, in the same direction of the sign of \(d\).
    • horizontally \(c\) units, in the opposite direction of the sign of \(c\).
    • The y-intercept, at \(f(0),\) becomes \((0,b^c+d)\).
    • The horizontal asymptote becomes \(y=d\), and occurs when the exponential part of the function, \(b^{x+c}\) approaches zero.
    • The range becomes \((d,\infty)\).
    • The domain, \((−\infty,\infty)\), remains unchanged.

     How to: Given an exponential function with the form \(f(x)=b^{x+c}+d\), graph the translation

    1. Draw the horizontal asymptote \(y=d\).
    2. Identify the shift as \((−c,d)\). Shift the graph of \(f(x)=b^x\) left \(c\) units if \(c\) is positive, and right \(c\) units if \(c\) is negative.
    3. Shift the graph of \(f(x)=b^x\) up \(d\) units if \(d\) is positive, and down \(d\) units if \(d\) is negative.
    4. State the domain, \((−\infty,\infty)\), the range, \((d,\infty)\), and the horizontal asymptote \(y=d\).

    Example \(\PageIndex{2}\): Graphing a Shift of an Exponential Function

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

     

    Solution

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

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

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

    Shift the graph of \(f(x)=b^x\) left \(1\) units and down \(3\) 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, -2), (0, -1), and (1, 1).

    Try It \(\PageIndex{2}\)

    Graph \(f(x)=2^{x−1}+3\).  State domain, range, and asymptote.
    Answer

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


     

    4.2reflectionExample.png 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.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 \(f(x)=b^x\)  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. For example, if we begin by graphing the parent function \(f(x)=2^x\), we can then graph the two reflections alongside it. The reflection about the \(x\)-axis, \(g(x)=−2^x\), is illustrated in the graph on the left, and the reflection about the \(y\)-axis \(h(x)=2^{−x}\), is shown in the graph on the right.

     

     

    REFLECTIONS OF THE PARENT FUNCTION \(f(x) = b^x\)

    The function \(f(x)=−b^x\)

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

    The function \(f(x)=b^{−x}\)

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

    Example \(\PageIndex{3}\): Writing and Graphing the Reflection of an Exponential Function

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

    Solution

    Since we want to reflect the parent function \(f(x)=( \tfrac{1}{4} )^x\) about the \(x\)-axis, we multiply \(f(x)\) by \(−1\) to get, \(g(x)=−f(x)=−( \tfrac{1}{4} )^x\).  Next we create a table of points below.

    \(x\) \(−2\) \(−1\) \(0\) \(1\) \(2\)

    \(g(x)=−( \tfrac{1}{4} )^x\)

    \(−16\)

    \(−4\)

    \(−1\)

    \(−0.25\)

    \(−0.0625\)

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

    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 \(\PageIndex{11}\). Graph of  \(g(x)=−( \tfrac{1}{4} )^x\).

    Try It \(\PageIndex{3}\)

    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.

    Answer

    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).  \(g(x) = f(-x) = 1.25^{-x}\)

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

    Included in the graph is the horizontal asymptote \(y=0\), and the points for \(g(-1) = 1.25\), \(g(0)=1\), and \(g(1) = .8\).

     

     

    4.2 VerticalStretch Example.png - Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.Graphing a Vertical 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 \(f(x)=b^x\) by a constant \(|a|>0\). For example, if we begin by graphing the parent function \(f(x)=2^x\), we can then graph the stretch, using \(a=3\),to get \(g(x)=3{(2)}^x\) as shown in Figure (a), and the compression, using \(a=\dfrac{1}{3}\), to get \(h(x)=\dfrac{1}{3}{(2)}^x\) as shown on the right in Figure (b).

    VERTICAL STRETCHES AND COMPRESSIONS OF THE PARENT FUNCTION \(f(x) = b^x\)

    For any factor \(a>0\), the function \(f(x)=a{(b)}^x\)

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

    Example \(\PageIndex{4}\): Graphing the Vertical Stretch of an Exponential Function

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

    Solution

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

    • Since \(b=\dfrac{1}{2}\) is between zero and one, the left tail 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 the parent function \(y={\Big(\dfrac{1}{2}\Big)}^x\) will be stretched vertically by a factor of \(4\).
    • Create a table of points of the parent function as shown in Table \(\PageIndex{4}\). Then multiply all the \(y\) coordinates by 4.
      Table \(\PageIndex{4}\)
      \(x\) \(−3\) \(−2\) \(−1\) \(0\) \(1\) \(2\) \(3\)
      \( y={\Big(\dfrac{1}{2}\Big)}^x\)  \(8\) \(4\) \(2\) \(1\) \(\dfrac{1}{2}\) \(\dfrac{1}{4}\) \(\dfrac{1}{8}\)

      \(f(x)=4{\Big(\dfrac{1}{2}\Big)}^x\)

      \(32\) \(16\) \(8\) \(4\) \(2\) \(1\) \(0.5\)
    • 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 \(\PageIndex{9}\).

     

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

    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 \(\PageIndex{9}\)

    Try It \(\PageIndex{4}\)

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

    Answer

    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).

    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 \(\PageIndex{6}\) to arrive at the general equation for translating exponential functions.

    Table \(\PageIndex{6}\): Translations of the Parent Function \(f(x)=b^x\)
    Translation Form
    Shift
    • Horizontally \(c\) units to the left
    • Vertically \(d\) units up

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

    Reflect about the \(x\)-axis

    \(f(x)=−b^x\)

    Reflect about the \(y\)-axis

    \(f(x)=b^{−x}={\left(\dfrac{1}{b}\right)}^x\)

    Stretch and Compress
    • Stretch if \(|a|>1\)
    • Compression if \(0<|a|<1\)

    \(f(x)=ab^x\)

    General equation for all translations

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

     

     How to: Graph TRANSLATIONS OF EXPONENTIAL FUNCTIONS

    A translation of an exponential function has the form \(f(x)=ab^{x+c}+d\) or  \(f(x)=ab^{-x+c}+d\)

    The translations to the parent function, \(y=b^x\), \(b>1\), needed to obtain \(f\) are given below. The order these are done is important. 

    1. FIRST. Shift horizontally \(c\) units. ( \( x \longrightarrow (x - c) \) ).
      • To the left if \(c > 0\)  
      • To the right if \(c < 0\).  
    2. Reflect over the \(y\)-axis if the coefficient in front of \(x\) is negative. ( \( x \longrightarrow -x )\).
    3. Reflect over the \(x\)-axis if \(a<0\). ( \( y \longrightarrow -y )\).
    4. Vertically stretch or shrink by a factor of \(|a|\) . ( \( y \longrightarrow |a|y )\).
      • If \(a>0\) the graph is stretched vertically. 
      • If \(0<a<1\) the graph is compressed vertically.
    5. LAST. Shift vertically \(d\) units. ( \( y \longrightarrow y + d )\).
      • Up if \(d > 0\)
      • Down if \(d < 0\)

    If the exponent \((-x+c)\) is rewritten \(-(x-c)\) then replace steps 1 and 2 with: reflect over \(y\)-axis first then go right \(c\) units

    Construct an Exponential Equation from a Description 

    Example \(\PageIndex{5}\): Writing a Function from a Description

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

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

    Solution

    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 parent function \(f(x)=e^x\), so \(b=e\).
    • The function is vertically stretched by a factor of \(2\), so \(a=2\).
    • 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,

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

    \(=2e^{−x+0}+4\)

    \(=2e^{−x}+4\)

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

    Try It \(\PageIndex{5}\)

    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.

    Answer

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

    Exponent Properties

    When learning about exponents, we are typically given expressions that have a whole number exponent and a base that is a variable, like \(x^2\). These type of expressions are technically called power functions. In contrast, the base can be a constant and the exponent can have a variable in it, like \(2^x\). Expressions in this form are called exponential functions. When simplifying expressions with the variable in the exponent, we often use the Laws of Exponents "backwards".  A review of properties of exponents from a different perspective is briefly discussed here. Competency with exponents is a very useful skill to have. The table below summarizes these properties.

    Rules of Exponents
    For nonzero real numbers a and b and integers m and n
      Product Rule Quotient Rule Power Rule
    Like Bases Rule \(a^m⋅a^n=a^{m+n}\) \(
    \dfrac{a^m}{a^n}=a^{m−n}\)
    \((a^m)^n=a^{m⋅n}\)
    Reverse of Like Bases Rule \(a^{m+n}=a^m⋅a^n \) \(a^{m−n}=
    \dfrac{a^m}{a^n}\)
    \(a^{m⋅n}=(a^m)^n\)
    Like Exponents Rule \(a^n⋅b^n =  (a⋅b)^n  \) \( \dfrac{a^n}{b^n} =  \left(\dfrac{a}{b}\right)^n\) \( \dfrac{1}{b^n} =  \left(\dfrac{1}{b}\right)^n\)

    Example \(\PageIndex{6}\)

    Rewrite the following exponential expressions such that the result uses only one exponent, \(x\). Simplify.

    1. \(2^{x+3}\) 2. \(3^{x-2}\) 3. \(5^{3-x}\) 4. \(2^x \cdot 11^x\) 5. \(5^{3+2x} \) 6. \( \dfrac{4^x}{12^x} \) 7. \( \dfrac{6^{3x}}{9^{2x}} \) 8. \( \dfrac{8^{x-1}}{2^{2x-5}} \)
    Solution

    1. \(2^{x+3} = 2^x 2^3 = 2^x \cdot 8 = 8(2^x)\)

    2. \(3^{x-2} = 3^x3^{-2} = \dfrac{3^x}{3^2} = \dfrac{3^x}{9} \)

    3. \(5^{3-x} = 5^3 5^{-x} = \dfrac{5^3}{5^x} = \dfrac{125}{5^x} = 125 \left(  \dfrac{1}{5}  \right)^x \)

    4. \(2^x \cdot 11^x = (2 \cdot 11)^x = 22^x \)

    5. \(5^{3+2x} = 5^3 5^{2x} = 125 (5^2)^x = 125(25^x) \)

    6. \( \dfrac{4^x}{12^x} = \left( \dfrac{4}{12} \right)^x = \left( \dfrac{1}{3} \right)^x \)

    7. \( \dfrac{6^{3x}}{9^{2x}} = \dfrac{(6^3)^x}{(9^2)^x} = \left( \dfrac{6^3}{9^2} \right)^x = \left( \dfrac{8}{3} \right)^x \)

    8. \( \dfrac{8^{x-1}}{2^{2x-5}} \)\(= \dfrac{8^x 8^{-1}}{2^{2x} 2^{-5}} =  \dfrac{8^x}{2^{2x}} \cdot \dfrac{2^5}{8} \)\(=  \dfrac{2^5}{8} \cdot \dfrac{8^x}{4^x}= \dfrac{32}{8} \cdot  \left( \dfrac{8}{4} \right)^x = 4 (2^x) \)

    \(\PageIndex{6}\)

    Rewrite the following exponential expressions such that the result uses only one exponent, \(x\). Simplify.

    \(1.\quad 3^{x+1}\cdot 4^{-2x} \) \(2. \quad \dfrac{2^{3x+4}}{7^{2x+1}} \)
    Answer

    \( 1.\quad 3 \left( \dfrac{3}{16} \right) ^x \qquad 2. \quad\dfrac{16}{7} \left( \dfrac{8}{49} \right) ^x   \)

    Key Concepts

    General Form for the Translation of the Parent Function \(f(x)=b^x\) \(f(x)=ab^{x+c}+d\)   or   \(f(x)=ab^{-x+c}+d\)
    • The graph of the function \(f(x)=b^x\) has a y-intercept at \((0, 1)\),domain \((−\infty, \infty)\),range \((0, \infty)\), and horizontal asymptote \(y=0\).
    • If \(b>1\),the function is increasing. The left tail of the graph will approach the asymptote \(y=0\), and the right tail will increase without bound.
    • If \(0<b<1\), the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote \(y=0\).
    • The equation \(f(x)=b^x+d\) represents a vertical shift of the parent function \(f(x)=b^x\).
    • The equation \(f(x)=b^{x+c}\) represents a horizontal shift of the parent function \(f(x)=b^x\).
    • The equation \(f(x)=ab^x\), where \(a>0\), represents a vertical stretch if \(|a|>1\)  or compression if \(0<|a|<1\) of the parent function \(f(x)=b^x\). 
    • When the parent function \(f(x)=b^x\) is multiplied by \(−1\),the result, \(f(x)=−b^x\), is a reflection about the x-axis. When the input is multiplied by \(−1\),the result, \(f(x)=b^{−x}\), is a reflection about the y-axis.
    • All translations of the exponential function can be summarized by the general equation \(f(x)=ab^{x+c}+d\). 
    • Using the general equation \(f(x)=ab^{x+c}+d\), we can write the equation of a function given its description.