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9.05 Graphs of exponential functions

Worksheet
Graphs of exponential functions
1

For each of the following functions:

i

Complete the following table of values:

x-5-4-3-2-101234510
y
ii

State whether the function is an increasing or decreasing function.

iii

Describe the rate of change of the function.

iv

State the y-intercept of the curve.

a
y = 3^{x}
b
y = 3^{ - x }
2

Consider the graph of the equation y = 4^{x}:

a

Is each y-value of the function positive or negative?

b

State the value of y the graph approaches but does not reach.

c

State the equation and name of the horizontal line, which y = 4^{x} gets closer and closer to but never intersects.

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3

Do either of the functions y = 9^{x} or y = 9^{ - x } have x-intercepts? Explain your answer.

4

Consider the expression 3^{x}.

a

Evaluate the expression when x = - 4.

b

Evaluate the expression when x = 0.

c

Evaluate the expression when x = 4.

d

What happens to the value of 3^{x} as x gets larger?

e

What happens to the value of 3^{x} as x gets smaller?

5

Consider the expression 2^{ - x }.

a

Evaluate the expression when x = 2.

b

Evaluate the expression when x = - 2.

c

What happens to the value of 2^{ - x } as x gets larger?

d

What happens to the value of 2^{ - x } as x gets smaller?

6

Consider the graphs of the functions y = 4^{x} and y = 4^{ - x } below. Describe the rate of change for each function.

a
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7

Consider the two functions y = 4^{x} and y = 5^{x}. Which one increases more rapidly for x > 0?

8

Determine the y-intercept of all exponential functions of the form:

a

y = a^{x}

b

y = a^{ - x }

c

y = - a^{x}

d

y = - a^{-x}

9

Consider the given graphs of the two exponential functions P and Q:

State whether the following pairs of equations could be the equations of the graphs P and Q:

a

P: \, y = 2^{x} \\ Q: \, y = 2^{ - x }

b

P: \, y = \left(3.5\right)^{x} \\ Q: \, y = 6^{ - x }

c

P: \, y = 2^{x} \\ Q: \, y = 5^{ - x }

d

P: \, y = 5^{x} \\ Q: \, y = 2^{ - x }

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10

The points \left(3, n\right), \left(k, 16\right) and \left(m, \dfrac{1}{4}\right) all lie on the curve with equation y = 2^{x}. Find the value of:

a

n

b

k

c

m

11

Consider the functions y = 2^{x}, y = 3^{x} and y = 5^{x}.

a
Sketch the three functions on the same set of axes.
b

Determine whether each of the following statements is true:

i

None of the curves cross the x-axis.

ii

They all have the same y-intercept.

iii

All of the curves pass through the point \left(1, 2\right).

iv

All of the curves have a maximum value.

c

State the y-intercept of each curve?

12

Consider the functions y = 2^{-x}, y = 3^{-x} and y = 5^{-x}.

a
Sketch the three functions on the same set of axes.
b
State the y-intercept of each curve.
c

Describe the nature of these functions for large values of x.

13

Consider the graph of the following functions y = 3^{x} and y = 3^{ - x }:

a

State the coordinates of the point of intersection of the two curves.

b

Describe the behaviour of both these functions for large values of x.

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Transformations of exponential functions
14

Consider the functions y = 2^{x} and y = 2^{x} - 2.

a

Find the y-intercept of y = 2^{x}.

b

Hence, determine the y-intercept of y = 2^{x} - 2.

c

State the horizontal asymptote of y = 2^{x}.

d

Hence, determine the horizontal asymptote of y = 2^{x} - 2.

15

Consider the given graph of y = 5^{x}.

a

Describe a transformation of the graph of y = 5^{x} that would obtain y = - 5^{x}.

b

Sketch the graph of y = - 5^{x}.

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16

Consider the given graph of y = 3^{x}.

a

Describe a transformation of the graph of y = 3^{x} that would obtain y = 3^{x} - 4.

b

Sketch the graph of y = 3^{x} - 4.

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17

The graph of y = 2^{x} is translated down by 7 units, state its new equation.

18

Consider the original graph y = 3^{x}. The function values of the graph are multiplied by 2 to form a new graph.

a

For each point on the original graph, find the point on the new graph:

Point on original graphPoint on new graph
\left(-1,\dfrac{1}{3}\right)(-1,⬚)
(0,1)(0,⬚)
(1,3)(1,⬚)
(2,9)(2,⬚)
b

State the equation of the new graph.

c

Graph the original and new graph on the same set of axes.

d

Describe the postion of new graph in relation to the original graph.

19

Of the two functions y = 2^{x} and y = 3 \times 2^{x}, which is increasing more rapidly for x > 0?

20

Consider the graphs of the two exponential functions R and S:

a

One of the graphs is of y = 4^{x} and the other graph is of y = 6^{x}.

Which is the graph of y = 6^{x}?

b

For x < 0, is the graph of y = 6^{x} above or below the graph of y = 4^{x}. Explain your answer.

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21

Consider the function y = - 5^{x}.

a

State the equation of the asymptote of y = - 5^{x}.

b

Hence, determine the equation of the asymptote of y = 2 - 5^{x}.

c

How many x-intercepts would the graph of y = 2 - 5^{x} have?

22

Consider the function y = 4 \left(2^{x}\right).

a

Find the y-intercept of the curve.

b

Is the function value ever negative?

c

As x approaches infinity, what value does y approach?

d

Sketch the graph of y = 4 \left(2^{x}\right).

23

Consider the function y = 2 - 4^{ - x }.

a

Find the y-intercept of the curve.

b

Is this an increasing or decreasing function?

c

As x approaches infinity, what value does y approach?

d

Sketch the graph of y = 2 - 4^{ - x }.

24

Consider the function y = 4^{x} + 3.

a

Find the y-intercept of the curve.

b

State the domain of the function.

c

State the range of the function.

d

Sketch the graph of y = 4^{x} + 3.

25

Consider the function y = \left(\dfrac{1}{2}\right)^{x}

a

Determine whether the following functions are equivalent to y = \left(\dfrac{1}{2}\right)^{x}:

i

y = \dfrac{1}{2^{x}}

ii

y = 2^{ - x }

iii

y = - 2^{x}

iv

y = - 2^{ - x }

b

Hence, describe a trasformation that would obtain the graph of y = \left(\dfrac{1}{2}\right)^{x} from the graph of y =2^{x}.

c

Graph the functions y = 2^{x} and y = \left(\dfrac{1}{2}\right)^{x} on the same set of axes.

26

Consider the equation y = \left(\dfrac{1}{3}\right)^{x}.

a

Rewrite the equation in the form y = k^{ - x }.

b

Describe a trasformation that would obtain the graph of y = \left(\dfrac{1}{3}\right)^{x} from the graph of y =3^{x}.

c

Graph the functions y = 3^{x} and y = \left(\dfrac{1}{3}\right)^{x} on the same set of axes.

27

For each of the following functions:

i

Find the y-intercept of the curve.

ii

State the equation of the horizontal asymptote.

iii

Sketch a graph of the function.

a

y = 3^{x} + 2.

b

y = 2^{x} - 2

c

y = - 3^{x} + 2

d

y = 3^{ - x }-1

28

Consider the function y = 2^{x - 2}.

a

Find the y-intercept of the curve.

b

Complete table of values for y = 2^{x - 2}.

x-3-2-10123
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c

State the horizontal asymptote of the curve.

d

Sketch a graph of the function.

29

Sketch a graph of each of the following functions:

a
y = 2^{x + 5}
b
y = 3^{x-1}
30

Consider the given graphs of f \left(x\right) = 3^{x} and g \left(x\right):

a

Describe a transformation that can be used to obtain g \left(x\right) from f \left(x\right).

b

State the equation of g \left(x\right).

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