Finding Equations of Exponential Function Section 10.4 Finding Equations of Exponential Function

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

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 2nd Table

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

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

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

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.

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.

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.

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 = 43.74 2. Solution

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

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

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

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

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)

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

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)

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