Find Exponential Functions from Their Graphs

Exponential Functions

To find a possible equation of an exponential function from its graph, we generally consider the form:

\[ f(x) = a \cdot b^{x-c} + d \]

It is recommended to review the tutorial on graphs of exponential functions to understand the properties of these graphs before attempting to find the function from a graph.

Examples with Detailed Solutions

Example 1

Find the exponential function of the form \( y = b^x \) whose graph is shown below.

graph of exponential function for example 1

Solution to Example 1

From the graph, for \( x = 1 \), we have \( y = 4 \). Substituting into the equation \( y = b^x \):

\[ b^1 = 4 \implies b = 4 \]

Hence, the function is:

\[ y = 4^x \]

Example 2

Find the exponential function of the form \( y = b^{x-c} \) whose graph is shown below.

graph of exponential function for example 2

Solution to Example 2

From the graph: \( x = 2 \), \( y = 1 \) and \( x = 3 \), \( y = 2 \). Substituting into \( y = b^{x-c} \):

\[ b^{2 - c} = 1 \quad \text{(equation 1)}, \quad b^{3 - c} = 2 \quad \text{(equation 2)} \]

Equation (1) gives \( b^{2-c} = b^0 \implies 2-c = 0 \implies c = 2 \). Substituting \( c = 2 \) into equation (2):

\[ b^{3-2} = 2 \implies b = 2 \]

Hence, the function is:

\[ y = 2^{x-2} \]

Example 3

Find the exponential function of the form \( y = -b^x + d \) whose graph is shown below.

graph of exponential function for example 3

Solution to Example 3

From the graph: \( x = 0 \), \( y = 0 \) and \( x = 1 \), \( y = -2 \). Substituting into \( y = -b^x + d \):

\[ -b^0 + d = 0 \implies -1 + d = 0 \implies d = 1 \]

\[ -b^1 + 1 = -2 \implies b = 3 \]

Hence, the function is:

\[ y = -3^x + 1 \]

Example 4

Find the exponential function of the form \( y = a \cdot b^x + d \) with horizontal asymptote \( y = 1 \).

graph and asymptote of exponential function for example 4

Solution to Example 4

Horizontal asymptote: \( y = d \implies d = 1 \). From the graph: \( x = 0 \), \( y = 4 \):

\[ a \cdot b^0 + 1 = 4 \implies a = 3 \]

Point \( (1,7) \) on the graph gives:

\[ 3 \cdot b^1 + 1 = 7 \implies b = 2 \]

Hence, the function is:

\[ y = 3 \cdot 2^x + 1 \]

Example 5

Find the exponential function of the form \( y = a \cdot e^{x-1} + d \) with horizontal asymptote \( y = -2 \).

graph and asymptote of exponential function for example 5

Solution to Example 5

Horizontal asymptote: \( d = -2 \). For \( x = 1 \), \( y = -4 \):

\[ a \cdot e^{1-1} - 2 = -4 \implies a \cdot 1 - 2 = -4 \implies a = -2 \]

Hence, the function is:

\[ y = -2 \cdot e^{x-1} - 2 \]

Example 6

Find the exponential function of the form \( y = a \cdot b^{x-1} + d \) with horizontal asymptote \( y = -1 \).

graph and asymptote of exponential function for example 6

Solution to Example 6

Horizontal asymptote: \( d = -1 \). From the graph: \( (1,2) \) and \( (2,5) \):

\[ a \cdot b^{1-1} - 1 = 2 \implies a = 3 \]

\[ 3 \cdot b^{2-1} - 1 = 5 \implies b = 2 \]

Hence, the function is:

\[ y = 3 \cdot 2^{x-1} - 1 \]

Exercises with Answers

Find the exponential function for each graph below. The broken line represents the asymptote.

graphs of exponential functions for exercises

Answers

  1. (A): \( y = e^{x-1} \)
  2. (B): \( y = -2 \cdot 2^x - 2 \)
  3. (C): \( y = -\left(\frac{1}{2}\right)^{x+3}+1 \)

References & Further Reading