In this lesson we will be writing exponential functions of best fit given:
Let’s first look at writing an equation of the form \(y=a\cdot b^x\) given two points \(\left(x_1,y_1\right)\) and \(\left(x_2,y_2\right)\). You will be solving a system of equations for \(a\) and \(b\) using the substitution method, as follows:
1. Set up two equations in the form \(y=a\cdot b^x\) by substituting in the given \(\left(x,y\right)\) values of the coordinates.
2. Solve one of the equations for \(a\) (you can also solve for \(b\) but the algebra is less complicated if you solve for \(a\)).
3. Substitute the expression for \(a\) into the second equation.
4. Solve for \(b\).
5. Use the value of \(b\) to solve for \(a\).
6. Write the equation in \(y=a\cdot b^x\) form.
Example 1: Write the exponential function in the form \(y=a\cdot b^x\) that goes through the points \(\left(2,36\right)\) and \(\left(5,972\right)\).
\(\begin{align}&36=ab^2\;\;\;\;972=ab^5\ & \ \ \ \ \ &\text{1) Set up the system of equations.}\\
&a=\frac{36}{b^2}\ & \ \ \ \ \ &\text{2) Solve one equation for a.}\\
&\frac{36}{b^2}\cdot b^5=972\ & \ \ \ \ \ &\text{3) Substitute the expression for a into the second equation.}\\
&36b^3=972\ & \ \ \ \ \ &\text{4) Simplify the expression using rules of exponents.}\\
&b^3=27\ & \ \ \ \ \ &\text{5) Divide by the coefficient.}\\
&b=3\ & \ \ \ \ \ &\text{6) Solve for b.}\\
&36=a\ \cdot3^2\ & \ \ \ \ \ &\text{7) Substitute the value of b into either equation and solve for a.}\\
&a=4\ & \ \ \ \ \ &\text{8) Solve for a.}\\
&y=4 \cdot3^x\ & \ \ \ \ \ &\text{9) Write the equation in}\ \ y=a\cdot b^x\ \text{form.}\end{align}\)
Example 2: Write the exponential function that goes through the points \(\left(-1,15\right)\) and \(\left(2,\large\frac{5}{9}\right)\).
- Two points on an exponential graph (without a calculator)
- A set of data (regression with the calculator)
Let’s first look at writing an equation of the form \(y=a\cdot b^x\) given two points \(\left(x_1,y_1\right)\) and \(\left(x_2,y_2\right)\). You will be solving a system of equations for \(a\) and \(b\) using the substitution method, as follows:
1. Set up two equations in the form \(y=a\cdot b^x\) by substituting in the given \(\left(x,y\right)\) values of the coordinates.
2. Solve one of the equations for \(a\) (you can also solve for \(b\) but the algebra is less complicated if you solve for \(a\)).
3. Substitute the expression for \(a\) into the second equation.
4. Solve for \(b\).
5. Use the value of \(b\) to solve for \(a\).
6. Write the equation in \(y=a\cdot b^x\) form.
Example 1: Write the exponential function in the form \(y=a\cdot b^x\) that goes through the points \(\left(2,36\right)\) and \(\left(5,972\right)\).
\(\begin{align}&36=ab^2\;\;\;\;972=ab^5\ & \ \ \ \ \ &\text{1) Set up the system of equations.}\\
&a=\frac{36}{b^2}\ & \ \ \ \ \ &\text{2) Solve one equation for a.}\\
&\frac{36}{b^2}\cdot b^5=972\ & \ \ \ \ \ &\text{3) Substitute the expression for a into the second equation.}\\
&36b^3=972\ & \ \ \ \ \ &\text{4) Simplify the expression using rules of exponents.}\\
&b^3=27\ & \ \ \ \ \ &\text{5) Divide by the coefficient.}\\
&b=3\ & \ \ \ \ \ &\text{6) Solve for b.}\\
&36=a\ \cdot3^2\ & \ \ \ \ \ &\text{7) Substitute the value of b into either equation and solve for a.}\\
&a=4\ & \ \ \ \ \ &\text{8) Solve for a.}\\
&y=4 \cdot3^x\ & \ \ \ \ \ &\text{9) Write the equation in}\ \ y=a\cdot b^x\ \text{form.}\end{align}\)
Example 2: Write the exponential function that goes through the points \(\left(-1,15\right)\) and \(\left(2,\large\frac{5}{9}\right)\).
The second type of problem we will be solving involves regression on the calculator. We have done linear, quadratic and cubic regression in previous units. To refresh your memory on how to input the data and calculate the regression equation on the TI graphing calculator, review the steps and screenshots below.
The calculator defaults to List 1 and List 2, but you may use any list you want as long as you identify those lists in Step 4 above.
If you do not have List 1 or List 2, Select 5: SetUpEditor from Step 1 then ENTER
Clear all lists before entering new data by highlighting the list name in Step 2 then pressing CLEAR and ENTER.
To learn how to do regression on Desmos, click here.
Example 3: The population of Naperville has been increasing exponentially according to the data below. Let \(x\) represent the number of years since \(1940\) and \(y\) represent the population. Find the exponential model of best fit and predict the population of Naperville in \(2050\), assuming the growth remains constant.