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Graphs of Exponential Functions

Graph exponential functions.

Graphs of Exponential Functions

Graph exponential functions (IA 10.2.1).
Function transformations (exponential) (CA 3.5.1-3.5.5).

Example

Graph exponential functions.

Try it.

On the same coordinate system graph \(f(x)={2}^{x}\) and \(g(x)={2}^{x+1}.\)

Solution

We will use point plotting to graph the functions.

Looking at the graphs of the functions \(f(x)={2}^{x}\) and \(g(x)={2}^{x+1}\) above, we see that adding one in the exponent caused a horizontal shift of one unit to the left. We can use this pattern to graph other functions using horizontal shifts.

Try it.

On the same coordinate system graph \(f(x)={3}^{x}\) and \(g(x)={3}^{x}-2.\)

Solution

We will use point plotting to graph the functions.

Looking at the graphs of the functions \(f(x)={3}^{x}\) and \(g(x)={3}^{x}-2\), we see that subtracting 2 caused vertical shift of down two units. Notice that the horizontal asymptote also shifted down 2 units. We can use this pattern to help graph other functions with a vertical shift.

Try it.

On the same coordinate system graph \(f(x)={3}^{x}\) and \(g(x)={3}^{x-1}.\)

Try it.

On the same coordinate system graph \(f(x)={3}^{x}\) and \(g(x)={3}^{x}+1.\)

Condensed — the full section is in OpenStax Precalculus 2e.

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 change as the input increases by \(1.\)

\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(f(x)={2}^{x}\)	\(\frac{1}{8}\)	\(\frac{1}{4}\)	\(\frac{1}{2}\)	\(1\)	\(2\)	\(4\)	\(8\)

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 \(f(x)=a{b}^{x},\) \(b\) 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.

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

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 \(g(x)={(\frac{1}{2})}^{x}.\) Observe how the output values in change as the input increases by \(1.\)

\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(g(x)=(\frac{1}{2}{)}^{x}\)	\(8\)	\(4\)	\(2\)	\(1\)	\(\frac{1}{2}\)	\(\frac{1}{4}\)	\(\frac{1}{8}\)

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

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.

Condensed — the full section is in OpenStax Precalculus 2e.

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 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.

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 vertical shifts are shown in .

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 \((-3,\infty ).\)

Condensed — the full section is in OpenStax Precalculus 2e.

Key Concepts

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 \(y=0.\) See .
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
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 \(f(x)={b}^{x}.\) See .
Approximate solutions of the equation \(f(x)={b}^{x+c}+d\) can be found using a graphing calculator. See .
The equation \(f(x)=a{b}^{x},\) where \(a>0,\) represents a vertical stretch if \(|a|>1\) or compression if \(0<|a|<1\) of the parent function \(f(x)={b}^{x}.\) See .
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. See .
All translations of the exponential function can be summarized by the general equation \(f(x)=a{b}^{x+c}+d.\) See .
Using the general equation \(f(x)=a{b}^{x+c}+d,\) we can write the equation of a function given its description. See .

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

On the same coordinate system graph \(f(x)={2}^{x}\) and \(g(x)={2}^{x+1}.\)

Otkrij odgovor

We will use point plotting to graph the functions.

Looking at the graphs of the functions \(f(x)={2}^{x}\) and \(g(x)={2}^{x+1}\) above, we see that adding one in the exponent caused a horizontal shift of one unit to the left. We can use this pattern to graph other functions using horizontal shifts.
On the same coordinate system graph \(f(x)={3}^{x}\) and \(g(x)={3}^{x}-2.\)

Otkrij odgovor

We will use point plotting to graph the functions.

Looking at the graphs of the functions \(f(x)={3}^{x}\) and \(g(x)={3}^{x}-2\), we see that subtracting 2 caused vertical shift of down two units. Notice that the horizontal asymptote also shifted down 2 units. We can use this pattern to help graph other functions with a vertical shift.
On the same coordinate system graph \(f(x)={3}^{x}\) and \(g(x)={3}^{x-1}.\)
On the same coordinate system graph \(f(x)={3}^{x}\) and \(g(x)={3}^{x}+1.\)
Graph \(f(x)={3}^{x+2}-3\)
Otkrij odgovor
1. Make a table for \(f(x)={3}^{x}\)
2. Add a column on the left for \(x+2\) , by subtracting 2 from all the input values
3. Add a column on the right by subtracting 3 from all the y-value
4. Two outside columns have the points for the new graph
Graph \(f(x)={2}^{x-3}-1\)
- ⓐ Given \(f(x)={3}^{x}\) , reflect it about y-axis and write an equation of a new function below.
- ⓑ Given \(f(x)={3}^{x}\) , reflect it about x-axis and write an equation of a new function below.
- ⓒ Given \(f(x)={3}^{x}\) , shift the graph up 4 units and write an equation of a new function below.
- ⓓ Graph the equations found in parts a, b, and c on the coordinate system provided and check your work using a graphing utility.
Sketch a graph of \(f(x)={0.25}^{x}.\) State the domain, range, and asymptote.
Otkrij odgovor
Before graphing, identify the behavior and create a table of points for the graph.
- 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 .
  \(x\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(3\)
  \(f(x)={0.25}^{x}\) \(64\) \(16\) \(4\) \(1\) \(0.25\) \(0.0625\) \(0.015625\)
- 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 .
The domain is \((-\infty ,\infty );\) the range is \((0,\infty );\) the horizontal asymptote is \(y=0.\)
Sketch the graph of \(f(x)={4}^{x}.\) State the domain, range, and asymptote.

Otkrij odgovor

The domain is \((-\infty ,\infty );\) the range is \((0,\infty );\) the horizontal asymptote is \(y=0.\)
Graph \(f(x)={2}^{x+1}-3.\) State the domain, range, and asymptote.

Otkrij odgovor

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

Otkrij odgovor

The domain is \((-\infty ,\infty );\) the range is \((3,\infty );\) the horizontal asymptote is \(y=3.\)
Solve \(42=1.2{(5)}^{x}+2.8\) graphically. Round to the nearest thousandth.

Otkrij odgovor

Press [Y=] and enter \(1.2{(5)}^{x}+2.8\) next to Y₁=. Then enter 42 next to Y2=. For a window, use the values –3 to 3 for \(x\) and –5 to 55 for \(y.\) Press [GRAPH]. The graphs should intersect somewhere near \(x=2.\)
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, \(x\approx 2.166.\)
Solve \(4=7.85{(1.15)}^{x}-2.27\) graphically. Round to the nearest thousandth.

Otkrij odgovor

\(x\approx -1.608\)
Sketch a graph of \(f(x)=4{(\frac{1}{2})}^{x}.\) State the domain, range, and asymptote.
Otkrij odgovor
Before graphing, identify the behavior and key points on the graph.
- Since \(b=\frac{1}{2}\) 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 \(f(x)={(\frac{1}{2})}^{x}\) will be stretched by a factor of \(4.\)
- Create a table of points as shown in .
  \(x\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(3\)
  \[f(x)=4(\frac{1}{2}{)}^{x}\] \(32\) \(16\) \(8\) \(4\) \(2\) \(1\) \(0.5\)
- 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 .
The domain is \((-\infty ,\infty );\) the range is \((0,\infty );\) the horizontal asymptote is \(y=0.\)
Sketch the graph of \(f(x)=\frac{1}{2}{(4)}^{x}.\) State the domain, range, and asymptote.

Otkrij odgovor

The domain is \((-\infty ,\infty );\) the range is \((0,\infty );\) the horizontal asymptote is \(y=0.\)
Find and graph the equation for a function, \(g(x),\) that reflects \(f(x)={(\frac{1}{4})}^{x}\) about the x-axis. State its domain, range, and asymptote.

Otkrij odgovor

Since we want to reflect the parent function \(f(x)={(\frac{1}{4})}^{x}\) about the x-axis, we multiply \(f(x)\) by \(-1\) to get, \(g(x)=-{(\frac{1}{4})}^{x}.\) Next we create a table of points as in .
\(x\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(3\)
\[g(x)=-(\frac{1}{4}{)}^{x}\] \(-64\) \(-16\) \(-4\) \(-1\) \(-0.25\) \(-0.0625\) \(-0.0156\)
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:
The domain is \((-\infty ,\infty );\) the range is \((-\infty ,0);\) the horizontal asymptote is \(y=0.\)
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.

Otkrij odgovor

The domain is \((-\infty ,\infty );\) the range is \((0,\infty );\) the horizontal asymptote is \(y=0.\)
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.
Otkrij odgovor
We want to find an equation of the general form \(\ f(x)=a{b}^{x+c}+d.\) 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 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,
\[\begin{array}{ll}f(x) & =a{b}^{x+c}+d \\ & =2{e}^{-x+0}+4 \\ & =2{e}^{-x}+4\end{array}\]
The domain is \((-\infty ,\infty );\) the range is \((4,\infty );\) the horizontal asymptote is \(y=4.\)
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 \(\frac{1}{3},\) reflected across the x-axis and then shifted down \(2\) units.
Otkrij odgovor

\(f(x)=-\frac{1}{3}{e}^{x}-2;\) the domain is \((-\infty ,\infty );\) the range is \((-\infty ,-2);\) the horizontal asymptote is \(y=-2.\)
What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

Otkrij odgovor

An asymptote is a line that the graph of a function approaches, as \(x\) either 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.
What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?
The graph of \(f(x)={3}^{x}\) is reflected about the y-axis and stretched vertically by a factor of \(4.\) What is the equation of the new function, \(g(x)?\) State its y-intercept, domain, and range.

Otkrij odgovor

\(g(x)=4{(3)}^{-x};\) y-intercept: \((0,4);\) Domain: all real numbers; Range: all real numbers greater than \(0.\)
The graph of \(f(x)={(\frac{1}{2})}^{-x}\) is reflected about the y-axis and compressed vertically by a factor of \(\frac{1}{5}.\) What is the equation of the new function, \(g(x)?\) State its y-intercept, domain, and range.
The graph of \(f(x)={10}^{x}\) is reflected about the x-axis and shifted upward \(7\) units. What is the equation of the new function, \(g(x)?\) State its y-intercept, domain, and range.

Otkrij odgovor

\(g(x)=-{10}^{x}+7;\) y-intercept: \((0,6);\) Domain: all real numbers; Range: all real numbers less than \(7.\)
The graph of \(f(x)={(1.68)}^{x}\) is shifted right \(3\) units, stretched vertically by a factor of \(2,\) reflected about the x-axis, and then shifted downward \(3\) units. What is the equation of the new function, \(g(x)?\) State its y-intercept (to the nearest thousandth), domain, and range.
The graph of \(f\left(x\right)=-\frac{1}{2}{(\frac{1}{4})}^{x-2}+4\) is shifted downward \(4\) units, and then shifted left \(2\) units, stretched vertically by a factor of \(4,\) and reflected about the x-axis. What is the equation of the new function, \(g(x)?\) State its y-intercept, domain, and range.

Otkrij odgovor

\(g(x)=2{(\frac{1}{4})}^{x};\) y-intercept: \((0,\ \text{2});\) Domain: all real numbers; Range: all real numbers greater than \(0.\)
\(f(x)=3{(\frac{1}{2})}^{x}\)
\(g(x)=-2{(0.25)}^{x}\)

Otkrij odgovor

y-intercept: \((0,-2)\)
\(h(x)=6{(1.75)}^{-x}\)
\(f(x)=3{(\frac{1}{4})}^{x},\) \(g(x)=3{(2)}^{x},\) and \(h(x)=3{(4)}^{x}\)
\(f(x)=\frac{1}{4}{(3)}^{x},\) \(g(x)=2{(3)}^{x},\) and \(h(x)=4{(3)}^{x}\)
\(f(x)=2{(0.69)}^{x}\)

Otkrij odgovor

B
\(f(x)=2{(1.28)}^{x}\)
\(f(x)=2{(0.81)}^{x}\)

Otkrij odgovor

A
\(f(x)=4{(1.28)}^{x}\)
\(f(x)=2{(1.59)}^{x}\)

Otkrij odgovor

E
\(f(x)=4{(0.69)}^{x}\)
Which graph has the largest value for \(b?\)

Otkrij odgovor

D
Which graph has the smallest value for \(b?\)
Which graph has the largest value for \(a?\)

Otkrij odgovor

C

Symbols used here

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\neq

not equal
The two sides are different.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

\binom{n}{k}

binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

Euler's number
2.71828…, the base whose exponential is its own derivative.

imaginary unit
i² = −1.

\theta

theta
The usual name for an angle.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbf{v},\ \vec{v}

vector
A quantity with magnitude and direction; a column of numbers.

How to: Graphs of Exponential Functions

Graph exponential functions.
Graph exponential functions using transformations.
Graph exponential functions (IA 10.2.1).
Function transformations (exponential) (CA 3.5.1-3.5.5).
Identify the vertical and horizontal shifts from the formula.
The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.
The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.
Note the order of the shifts, transformations, and reflections follow the order of operations.

Questions people ask

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

Pokušaj sam.

Parts of this page are adapted from OpenStax Precalculus 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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