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Name Class Date

Explore Exploring Graphs of Exponential Functions Exponential functions follow the general shape y = a b x .

Graph the exponential functions on a graphing calculator, and match the graph to the correct function rule.

1. y = 3 (2) x

2. y = 0.5 (2) x

3. y = 3 (0.5) x

4. y = -3 (2) x

a.

c.

b.

d.

B In all the functions 1–4 above, the base b > 0.

Use the graphs to make a conjecture: State the domain and range of y = a b x if a > 0.

In all the functions 1–4 above, the base b > 0.

Use the graphs to make a conjecture: State the domain and range of y = a b x if a < 0.

What is the y-intercept of ƒ (x) = 0.5 (2) x ?

Resource Locker

Module 10 445 Lesson 2

10 . 2 Graphing Exponential Functions

Essential Question: How do you graph an exponential function of the form f (x) = ab x ?

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x 0 4

4

2

2

-4 -2

-4

-2

(0, 1) (1, 2)

(2, 4)

f(x) = 2x

(-1, 0.5)

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Note the similarities between the y-intercept and a. What is their relationship?

Reflect

1. Discussion What is the domain for any exponential function y = a b x ?

2. Discussion Describe the values of b for all functions y = a b x .

Explain 1 Graphing Increasing Positive Exponential Functions The symbol ∞ represents infinity. We can describe the end behavior of a function by describing what happens to the function values as x approaches positive infinity (x → ∞) and as x approaches negative infinity (x → -∞) .

Example 1 Graph each exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph.

ƒ (x) = 2 x

Choose several values of x and generate ordered pairs.

x f (x) = 2 x -1 0.5 0 1

1 2

2 4

Graph the ordered pairs and connect them with a smooth curve.

a = 1

b = 2

y-intercept: (0, 1)

End Behavior: As x-values approach positive infinity (x → ∞) , y-values approach positive infinity (y → ∞) . As x-values approach negative infinity (x → -∞) , y-values approach zero (y → 0) .

Using symbols only, we say: As x → ∞, y → ∞, and as x → -∞, y → 0.

Module 10 446 Lesson 2

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-4 -2 -15

y

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B ƒ (x) = 3 (4) x

Choose several values of x and generate ordered pairs.

x f (x) = 3 (4) x -1 0

1

2

Graph the ordered pairs and connect them with a smooth curve.

a =

b =

y-intercept: ( , ) End Behavior: As x → ∞, y → and as x → -∞, y → .

Reflect

3. If a > 0 and b > 1, what is the end behavior of the graph?

4. Describe the y-intercept of the exponential function ƒ (x) = a b x in terms of a and b.

Your Turn

5. Graph the exponential function ƒ (x) = 2 (2) x After graphing, identify a and b, the y-intercept, and the end behavior of the graph.

x f (x) = 2 (2) x

Module 10 447 Lesson 2

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y

x

42

6

-4 -2

-12

-18

-6

f(x) = -2(3)x (0, -2) (1, -6)

(2, -18)

y

x 0 4

40

2

20

-4 -2

-40

-60

-20

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Explain 2 Graphing Decreasing Negative Exponential Functions

Example 2 Graph each exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph.

ƒ (x) = -2 (3) x

Choose several values of x and generate ordered pairs.

x f (x) = -2 (3) x

-1 -0.7 0 -2 1 -6 2 -18

Graph the ordered pairs and connect them with a smooth curve.

a = -2

b = 3

y-intercept: (0, -2) End Behavior: As x → ∞, y → -∞ and as x → -∞, y → 0.

B ƒ (x) = -3 (4) x

Choose several values of x and generate ordered pairs.

x f (x) = -3 (4) x

-1 0

1

2

Graph the ordered pairs and connect them with a smooth curve.

a =

b =

y-intercept: ( , ) End Behavior: As x → ∞, y → and as x → -∞, y → .

Module 10 448 Lesson 2

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y

x 0 4

20

2

10

-4 -2

-20

-30

-10

y

x 0 4

4

2

2

-4 -2

-4

-2

(0, 1) (2, 0.25)

(1, 0.5)

f(x) = 0.5x

(-1, 2)

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Reflect

6. If a < 0 and b > 1, what is the end behavior of the graph?

Your Turn

7. Graph the exponential function. ƒ (x) = -3 (3) x After graphing, identify a and b, the y-intercept, and the end behavior of the graph.

Explain 3 Graphing Decreasing Positive Exponential Functions

Example 3 Graph each exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph. Use inequalities to discuss the behavior of the graph.

ƒ (x) = (0.5) x

Choose several values of x and generate ordered pairs.

x f (x) = (0.5) x -1 2 0 1

1 0.5

2 0.25

Graph the ordered pairs and connect them with a smooth curve.

a = 1

b = 0.5

y-intercept: (0, 1)

End Behavior: As x → ∞, y → 0 and as x → -∞, y → ∞.

x f (x) = -3 (3) x

Module 10 449 Lesson 2

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-4 -2

-4

-2

y

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-4 -2

-4

-2

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B ƒ (x) = 2 (0.4) x

Choose several values of x and generate ordered pairs.

x f (x) = 2 (0.4) x -1 0

1

2

Graph the ordered pairs and connect them with a smooth curve.

a =

b =

y-intercept: ( , ) End Behavior: As x → ∞, y → and as x → -∞, y → .

Reflect

8. If a > 0 and 0 < b < 1, what is the end behavior of the graph?

Your Turn

9. Graph the exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph.

ƒ (x) = 3 (0.5) x

x f (x) = 3 (0.5) x

Module 10 450 Lesson 2

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y

x 0

2

1

1

-2 -1

-2

-3

-1 f(x) = -0.5x

(-1, -2)

(0, -1) (1, -0.5)

(2, -0.25)

y

x 0 2

6

1

3

-2 -1

-6

-9

-3

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Explain 4 Graphing Increasing Negative Exponential Functions

Example 4 Graph each exponential function. After graphing, identify a and b, the y-intercept, and the end behavior of the graph.

ƒ (x) = -0. 5 x

Choose several values of x and generate ordered pairs.

x f (x) = -0. 5 x -1 -2 0 -1 1 -0.5 2 -0.25

Graph the ordered pairs and connect them with a smooth curve.

a = -1

b = 0.5

y-intercept: (0, -1)

End Behavior: As x → ∞, y → 0 and as x → -∞, y → -∞.

B ƒ (x) = -3 (0.4) x

Choose several values of x and generate ordered pairs.

x f (x) = -3 (0.4) x -1 0

1

2

Graph the ordered pairs and connect them with a smooth curve.

a =

b =

y-intercept: ( , ) End Behavior: As x → ∞, y → and as x → -∞, y → .

Module 10 451 Lesson 2

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y

x 0 2

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1

2

-2 -1

-4

-6

-2

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