Writing Exponential Equations Solutions

• Applying what you learned about systems of equations, you can write the equation to an exponential function from    two points   .

• Use repeated    substitution    to determine the values of   a and b   .

To write an exponential equation given two points:

1. Substitute $\underline{ \mathbf{(}{\mathit{x}}_{\mathbf{1}}\mathbf{,}\mathbf{ }{\mathit{y}}_{\mathbf{1}}\mathbf{)} }$ into$y=a{b}^x.$
2.    Isolate    a.
3. Substitute   the expression equal to a    and $\underline{ \mathbf{(}{\mathit{x}}_{\mathbf{2}}\mathbf{,}\mathbf{ }{\mathit{y}}_{\mathbf{2}}\mathbf{)} }$ into$y=a{b}^x.$
4. Solve for the   value of b   .
5. Solve for the   value of a   .
6. Write the function by substituting   the values of a and b    into$y=a{b}^x.$

Note

You can also isolate b first but for consistency, in this lesson, all problems isolate a first.

Example 5

Write an exponential equation that passes through the points $\left(–2, –8\right)$ and $(3, –\frac{1}{4}).$

Note

Use graphing technology to check your work. Enter the equation and make sure the given points are on the graph.