Transcript 18-SK-Power Equations

Section 4.4
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
x
• Exponential is of the form f(x) = ab
• y-intercept is (0, 3), so a = 3
• Input increases by 1, output multiplies by 2: b = 2
x
• f(x) = 3(2)
Section 4.4
Lehmann, Intermediate Algebra, 4ed
Slide 2
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
Lehmann, Intermediate Algebra, 4ed
Slide 3
Linear versus Exponential Functions
Using the Base Multiplier Property to Find Exponential Functions
Example
1. 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
x
1
• f  x   162 . 
 3
Section 4.4
Lehmann, Intermediate Algebra, 4ed
Slide 4
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
Lehmann, Intermediate Algebra, 4ed
Slide 5
Linear versus Exponential Functions
Solving Equations of the Form abn = k for b
Example
Find all real-number solutions.
1. b 2  25
2. b3  8
3. 2b  32
4
4.10b5  90 5. b 6  28
Solution
1.
• Solutions are 5 and –5
• Use the notation  5
Section 4.4
Lehmann, Intermediate Algebra, 4ed
Slide 6
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
Lehmann, Intermediate Algebra, 4ed
Slide 7
Linear versus Exponential Functions
Solving Equations of the Form abn = k for b
Solution
4.
Check that 1.55 approx. satisfies the equation.
6
5. The equation b = –28 has no real solution, since
an even exponent gives a positive number.
Section 4.4
Lehmann, Intermediate Algebra, 4ed
Slide 8
n
Solving Equations of the Form b = k for b
Solving Equations of the Form abn = k for b
Summary
n
To solve an equation of the form b = k for b,
1. If n is odd, the real-number solution is k 1 n
2. If n is even, and k ≥ 0, the real-number solutions
are  k 1 n.
3. If n is even and k < 0, there is no real number
solution.
Section 4.4
Lehmann, Intermediate Algebra, 4ed
Slide 9
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.
9
b
70
6
1. 5.42b – 3.19 = 43.74 2. 4 
3
b
Solution
Section 4.4
Lehmann, Intermediate Algebra, 4ed
Slide 10
One-Variable Equations Involving Exponents
Solving Equations of the Form abn = k for b
Solution Continued
2.
Section 4.4
Lehmann, Intermediate Algebra, 4ed
Slide 11
Finding Equations of an Exponential Function
U s i n g Tw o P o i n t s t o F i n d E q u a t i o n s o f E x p o n e n t i a l F u n c t i o n
Example
x
Find an approximate equation y = ab of the
exponential curve that contains the points (0, 3) and
(4, 70). Round the value of b to two decimal places.
Solution
x
• y-intercept is (0, 3): y = 3b
• Substitute (4, 70) and solve for b
Section 4.4
Lehmann, Intermediate Algebra, 4ed
Slide 12
Finding Equations of an Exponential Function
U s i n g Tw o P o i n t s t o F i n d E q u a t i o n s o f E x p o n e n t i a l F u n c t i o n
Solution Continued
x
• Our equation is y = 3(2.20)
• Graph contains (0, 3)
• b is rounded
• Doesn’t go through (0, 70), but it’s close
Section 4.4
Lehmann, Intermediate Algebra, 4ed
Slide 13