Finding Equations of Exponential Function

Section 4.4

Lehmann, Intermediate Algebra, 4edSection 4.4

An exponential curve contains the points listed in the table. Find an equation of the curve.

Slide 2

Finding an Equation of an Exponential CurveUsing the Base Multiplier Property to Find Exponential Functions

Example

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

Lehmann, Intermediate Algebra, 4edSection 4.4

• Verify results using graphing calculator

Slide 3

Finding an Equation of an Exponential CurveUsing the Base Multiplier Property to Find Exponential Functions

Solution Continued

Lehmann, Intermediate Algebra, 4edSection 4.4

1. Find a possible equation of a function whose input – output pairs are listed in the table.

Slide 4

Linear versus Exponential FunctionsUsing the Base Multiplier Property to Find Exponential Functions

Example

Solution• x increases by 1, y multiplies by 1/3: b = 1/3• y-intercept is (0, 162): a = 162

• . 1162

3

x

f x

Lehmann, Intermediate Algebra, 4edSection 4.4

2. Find a possible equation of a function whose input – output pairs are listed in the table.

Slide 5

Linear versus Exponential FunctionsUsing the Base Multiplier Property to Find Exponential Functions

Example

Solution• x increases by 1, y subtracted by 4: Linear function• y-intercept is (0, 50)

• y = 4x + 50

Lehmann, Intermediate Algebra, 4edSection 4.4

Find all real-number solutions.

Slide 6

Linear versus Exponential FunctionsSolving Equations of the Form abn = k for b

Example

Solution1.

• Solutions are 5 and –5• Use the notation 5

2 3 4

5 6

1. 25 2. 8 3. 2 32

4.10 90 5. 28

b b b

b b

Lehmann, Intermediate Algebra, 4edSection 4.4

2.

3.

Check that both –2 and 2 satisfy the equation.

Slide 7

Linear versus Exponential FunctionsSolving Equations of the Form abn = k for b

Solution

Lehmann, Intermediate Algebra, 4edSection 4.4

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.

Slide 8

Linear versus Exponential FunctionsSolving Equations of the Form abn = k for b

Solution

Lehmann, Intermediate Algebra, 4edSection 4.4

To solve an equation of the form bn = k for b,

1. If n is odd, the real-number solution is

2. If n is even, and k ≥ 0, the real-number solutions are .

3. If n is even and k < 0, there is no real number solution.

Slide 9

Solving Equations of the Form bn = k for bSolving Equations of the Form abn = k for b

Summary

1 nk

1 nk

Lehmann, Intermediate Algebra, 4edSection 4.4

Find all real-number solutions. Round your answer to the second decimal place.

1.5.42b6 – 3.19 = 43.74 2.

Slide 10

One-Variable Equations Involving ExponentsSolving Equations of the Form abn = k for b

Example

Solution

9

4

703

b

b

Lehmann, Intermediate Algebra, 4edSection 4.4

2.

Slide 11

One-Variable Equations Involving ExponentsSolving Equations of the Form abn = k for b

Solution Continued

Lehmann, Intermediate Algebra, 4edSection 4.4

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

Slide 12

Finding Equations of an Exponential FunctionUsing Two Points to Find Equations of Exponential Function

Example

Solution

Lehmann, Intermediate Algebra, 4edSection 4.4

•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

Slide 13

Finding Equations of an Exponential FunctionUsing Two Points to Find Equations of Exponential Function

Solution Continued