1. Finding Equations of Exponential Function
Section 10.4
Finding Equations of Exponential Function

2. Finding an Equation of an Exponential Curve
Using the Base Multiplier Property to Find Exponential Functions
Example
An exponential curve contains the points listed in the table. Find an equation of the curve.
Solution
Exponential is of the form f(x) = abx
y-intercept is (0, 3), so a = 3
Input increases by 1, output multiplies by 2: b = 2
f(x) = 3(2)x

3. Verify results using graphing calculator 2nd  Tblset 2nd Table
Finding an Equation of an Exponential Curve
Using the Base Multiplier Property to Find Exponential Functions
Solution Continued
Verify results using graphing calculator
2nd  Tblset nd Table

4. Linear versus Exponential Functions
Using the Base Multiplier Property to Find Exponential Functions
Example
Find a possible equation of a function whose input – output pairs are listed in the table.
Solution
x increases by 1, y multiplies by 1/3: b = 1/3
y-intercept is (0, 162): a = 162

5. Linear versus Exponential Functions
Using the Base Multiplier Property to Find Exponential Functions
Example
2. Find a possible equation of a function whose input – output pairs are listed in the table.
Solution
x increases by 1, y subtracted by 4: Linear function
y-intercept is (0, 50)
y = -4x + 50

6. Find all real-number solutions.
Linear versus Exponential Functions
Solving Equations of the Form abn = k for b
Example
Find all real-number solutions.
Solution 1.
Solutions are 5 and –5
Use the notation 5

7. 2. 3. Check that both –2 and 2 satisfy the equation.
Linear versus Exponential Functions
Solving Equations of the Form abn = k for b
Solution 2. 3.
Check that both –2 and 2 satisfy the equation.

8. Linear versus Exponential Functions
Solving Equations of the Form abn = k for b
Solution 4.
Check that 1.55 approx. satisfies the equation.
5. The equation b6 = –28 has no real solution, since an even exponent gives a positive number.

9. Solving Equations of the Form bn = k for b
Solving Equations of the Form abn = k for b
Summary
To solve an equation of the form bn = k for b,
If n is odd, the real-number solution is
If n is even, and k ≥ 0, the real-number solutions are
If n is even and k < 0, there is no real number solution.

10. One-Variable Equations Involving Exponents
Solving Equations of the Form abn = k for b
Example
Find all real-number solutions. Round your answer to the second decimal place.
5.42b6 – 3.19 =
Solution

11. One-Variable Equations Involving Exponents
Solving Equations of the Form abn = k for b
Solution Continued 2.

12. Finding Equations of an Exponential Function
Using Two Points to Find Equations of Exponential Function
Example
Find an approximate equation y = abx of the exponential curve that contains the points (0, 3) and (4, 70). Round the value of b to two decimal places.
y-intercept is (0, 3): y = 3bx
Substitute (4, 70) and solve for b
Solution

13. Finding Equations of an Exponential Function
Using Two Points to Find Equations of Exponential Function
Solution Continued
Our equation is y = 3(2.20)x
Graph contains (0, 3)
b is rounded
Doesn’t go through (4, 70), but it’s close

14. Dividing Left Sides and Right Sides of Two Equations
Using Two Points to Find Equations of Exponential Function
Property
If a = b and c = d, then
In words: The quotient of the left sides of two equations is equal to the quotient of the right sides.
Find an approximate equation y = abx of the exponential curve that contains (2, 5) and (5, 63). Round the values of a and b to two decimal places.
Example

15. Finding an Equation of an Exponential Curve
Using Two Points to Find Equations of Exponential Function
Solution
Both points must satisfy the equation; so,
It’s easier to solve if we switch the equations
Divide left and right sides (a and b non zero)

16. Apply the property: Solve for b by finding the cube root of 63/5
Finding an Equation of an Exponential Curve
Using Two Points to Find Equations of Exponential Function
Solution Continued
Apply the property:
Solve for b by finding the cube root of 63/5

17. Finding an Equation of an Exponential Curve
Using Two Points to Find Equations of Exponential Function
Solution Continued
Substitute 2.33 for b
Substitute (2, 5) and solve for a
y = 0.92(2.33)x approx. passes (2, 5) and (5, 63)

18. Finding an Equation of an Exponential Curve
Using Two Points to Find Equations of Exponential Function
Solution Continued
Verify using a graphing calculator
We can find an equation of an exponential function using the base multiplier property or by using two points. Both methods give the same result.
Summary