NAME ______________________________________________ DATE______________ PERIOD _____
9-1
Study Guide and Intervention Exponential Functions
Exponential Functions where a 0, b
An exponential function has the form y abx, 0, and b 1.
Properties of an Exponential Function
1. 2. 3. 4. 5.
Exponential Growth and Decay
If a If a
The The The The The
function is continuous and one-to-one. domain is the set of all real numbers. x-axis is the asymptote of the graph. range is the set of all positive numbers if a graph contains the point (0, a).
0 and all negative numbers if a ! 0.
0 and b 1, the function y abx represents exponential growth. 0 and 0 ! b ! 1, the function y abx represents exponential decay.
Example 1
Sketch the graph of y 0.1(4)x. Then state the function’s domain and range. Make a table of values. Connect the points to form a smooth curve. x
1
0
1
2
3
y
0.025
0.1
0.4
1.6
6.4
y
x
O
The domain of the function is all real numbers, while the range is the set of all positive real numbers. Example 2
Exercises Sketch the graph of each function. Then state the function’s domain and range.
1 14 2
1. y 3(2) x
2. y 2
y
x
y
y
O
O
3. y 0.25(5) x
x
x
O
x
Determine whether each function represents exponential growth, decay, or neither. 4. y 0.3(1.2) x Chapter 9
1 45 2
5. y 5
x
6
6. y 3(10)x Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Determine whether each function represents exponential growth, decay, or neither. a. y 0.5(2) x b. y 2.8(2) x c. y 1.1(0.5) x exponential growth, neither, since 2.8, exponential decay, since since the base, 2, is the value of a is less the base, 0.5, is between greater than 1 than 0. 0 and 1
NAME ______________________________________________ DATE______________ PERIOD _____
9-1
Study Guide and Intervention
(continued)
Exponential Functions
Property of Equality for Exponential Functions
If b is a positive number other than 1, then b x b y if and only if x y.
Property of Inequality for Exponential Functions
If b 1 then b x b y if and only if x y and b x ! b y if and only if x ! y.
Example 1
Solve 4 x 1
4x 1 2x 5 (22) x 1 2 x 5 2(x 1) x 5 2x 2 x 5 x7
1 125
Example 2
2 x 5.
Original equation
Solve 52x 1 .
Rewrite 4 as 22.
52x 1
1 Original inequality 125
Prop. of Inequality for Exponential Functions
52x 1
53
Distributive Property
2x 1 3 Prop. of Inequality for Exponential Functions 2x 2 Add 1 to each side. x 1 Divide each side by 2. The solution set is {x | x 1}.
Subtract x and add 2 to each side.
1
Rewrite as 53. 125
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises Simplify each expression. 1. (3Ïw2)Ïw2
2. 25Ïw2 125Ïw2
3. (xÏw2y3Ïw2)Ïw2
4. (xÏw6)(xÏw5)
5. (xÏw6)Ïw5
6. (2x )(5x3 )
Solve each equation or inequality. Check your solution. 7. 32x 1 3x 2
8. 23x 4x 2
10. 4x 1 82x 3
11. 8x 2
13. 4Ïwx 16Ïw5
14. xÏw3 36
1 27
1 16
15. xÏw2 81 Ï8w
2x 1
18. 52x ! 125x 5
2
21. 82x 5 ! 4x 8
3 4
17. 42x 2
19. 104x 1
20. 73x ! 49x
Chapter 9
12. 252x 125x 2
Îã
16. 3x 4 ! 100x 2
1 9
9. 32x 1
7
1
Glencoe Algebra 2
Lesson 9-1
Exponential Equations and Inequalities All the properties of rational exponents that you know also apply to real exponents. Remember that am an am n, (am)n amn, and am an am n.
