1. Finding Equations of Exponential Function
Section 4.4
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
Section 4.4
Slide 2

3. Verify results using graphing calculator
Finding an Equation of an Exponential Curve
Using the Base Multiplier Property to Find Exponential Functions
Solution Continued
Verify results using graphing calculator
Section 4.4
Slide 3

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 .
Section 4.4
Slide 4

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
Section 4.4
Slide 5

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
Section 4.4
Slide 6

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.
Section 4.4
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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.
Section 4.4
Slide 8

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.
Section 4.4
Slide 9

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
Section 4.4
Slide 10

11. One-Variable Equations Involving Exponents
Solving Equations of the Form abn = k for b
Solution Continued 2.
Section 4.4
Slide 11

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
Section 4.4
Slide 12

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 (0, 70), but it’s close
Section 4.4
Slide 13