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7.05 Graphs and transformations of exponential functions

Worksheet
Exponential graphs and transformations
1

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 are 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.

2

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

a

Complete the table of values:

x-5-4-3-2-1012345
y
b

Can the value of y ever be zero or positive? Explain your answer.

c

Is y = - \left(2^{x}\right) an increasing or decreasing function?

d

Describe the behaviour of the function as x increases.

e

State the domain.

f

State the range.

3

Find the missing coordinate in each ordered pair so that the pair is a solution of y = - 3^{x}:

a

\left(5, ⬚\right)

b

\left(⬚, - \dfrac{1}{27} \right)

c

\left( - 1 , ⬚\right)

d

\left(⬚, - 81 \right)

4

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}

5

Consider the function y = - 2.5 \times 4^{x}.

a

Is y = - 2.5 \times 4^{x} an increasing or decreasing function?

b

As x approaches -\infty, what value does y approach?

c

As x approaches \infty, y what value does y approach?

d

State the y-intercept of the graph.

6

Determine whether following are increasing or decreasing exponential functions:

a

y = 10\times\left(\dfrac{3}{5}\right)^{x}

b

y = 9 \times 3^{x}

c

y = 3 \times \left(0.5\right)^{x}

d

y = 0.2 \times 2^{x}

e

y = 2.2 \times 1.05^{x}

f

y = 50\times\left(\dfrac{7}{4}\right)^{x}

g

y = 500 \times 0.75^{x}

h

y = 80 \times 2^{-x}

i

y = -5 \times 2^{x}

j

y = -0.5 \times 0.8^{x}

k

y = -2\times\left(\dfrac{1}{4}\right)^{x}

l

y = -4 \times 3^{-x}

7

Which of the following graphs of exponential functions rises most steeply?

A

y = 2 \times \left(2.1\right)^{x}

B

y = 2 \times \left(2.2\right)^{x}

C

y = 2 \times \left(1.7\right)^{x}

D

y = 2 \times \left(1.2\right)^{x}

8

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

9

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.

10

For each of the following exponential functions:

i

Find the y-value of the y-intercept.

ii

Determine the horizontal asymptote.

iii

Sketch the graph of the function.

a
y = 3^{ - x }-1
b
y = - 3^{x} - 2
c
y = 4^{x - 2}
d
y = 3^{x - 4}
11

For each of the following exponential functions:

i

Find the y-value of the y-intercept.

ii

State the domain.

iii

State the range.

iv

Sketch the graph of the function.

a
y = 3^{x} - 5
b
y = 4^{x} + 3
c
y = 2^{x + 4}
d
y = 3^{x -1}
12

For each of the following exponential functions:

i

Find the y-value of the y-intercept.

ii

Create a table of values for -3 \leq x \leq 3.

iii

Determine the horizontal asymptote.

iv

Sketch the graph of the function.

iv

Is the function increasing or decreasing?

a
y = 2^{x - 2}
b
y = - 5^{x - 4}
c
y = - 3^{ - x }
d
y = 3^{ 2 x}
e
y = 3 \times 2^{x} + 2
13

Consider the function y = 8^{ - x } + 6.

a

What value is 8^{ - x } always greater than?

b

Hence, what value is 8^{ - x } + 6 always greater than?

c

How many x-intercepts does y = 8^{ - x } + 6 have?

d

State the equation of the asymptote of the curve y = 8^{ - x } + 6.

e

State the domain.

f

State the range.

14

Consider the function y = 2 \left(3^{x}\right).

a

Find the y-value of the y-intercept.

b

Can the function value ever be negative? Explain your answer.

c

State the domain.

d

State the range.

e

As x approaches positive infinity, what value does y approach?

f

Sketch the graph of the function.

15

Consider the function y = - 10^{x}.

a

Complete the following table:

x-2-10123
y
b

For any value of x, is y always negative for this equation? Explain your answer.

c

Sketch the graph of the function.

16

For each of the following functions, describe transformations that can be used to obtain the graph of f \left( x \right) from the graph of y=2^x:

a
y = 2^{x}+5
b
y = 2^{x-3}
c
y =3\times 2^{x}
d
y =-2^{x}+1
e
y = -2^{x+4}
f
y = 0.8 \times 2^{-x}
g
y = 2^{ 4- x}
h
y = 2^{ 3 x}
17

Consider a graph of y = 6^{ 3 x}.

a

A horizontal dilation by what factor takes the graph of y=6^{x} to y=6^{ 3 x}?

b

Determine whether the following functions are equivalent to y = 6^{ 3 x}:

i
y=216 \times 6^{x}
ii
y=\dfrac{1}{3} \times 6^{x}
iii
y=18^{x}
iv
y=216^{x}
c

Sketch the graph of y = 6^{ 3 x}.

18

Consider a graph of y = 9^{\frac{x}{2}}.

a

A horizontal dilation by what factor takes the graph of y=9^{x} to y=9^{\frac{x}{2}}?

b

Determine whether the following functions are equivalent to y =9^{\frac{x}{2}}:

i
y=2 \times 9^{x}
ii
y=3 \times 9^{x}
iii
y=3^{x}
iv
y=\left(\dfrac{9}{2}\right)^{x}
c

Sketch the graph of y = 9^{ \frac{x}{2}}.

19

Consider the graph of y = 2^{x+5}.

a

Using index laws, rewrite 2^{x + 5} as A \times 2^{x}.

b

Hence, for the graph of y=2^x, a horizontal translation left by 5 units is equivalent to a vertical dilation by what factor?

c

Sketch the graph of y = 2^{x + 5}.

20

Consider the graph of y = 5^{x-1}.

a

Using index laws, rewrite 5^{x - 1} as A \times 5^{x}.

b

Hence, for the graph of y=5^x, a horizontal translation right by 1 unit is equivalent to a vertical dilation by what factor?

c

Sketch the graph of y = 5^{x - 1}.

21

Consider a graph of y = 3^{ 2 x + 3}.

a

Determine whether the following functions are equivalent to y = 3^{ 2 x + 3}:

i
y=8 \times 3^{x}
ii
y=27 \times 9^{x}
iii
y=9 \times 6^{x}
iv
y=\dfrac{1}{2} \times 9^{x}
b

State the domain.

c

State the range.

22

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

a

State the asymptote of y = - 3^{x}.

b

Hence, find the asymptote of y = 2 - 3^{x}.

c

How many x-intercepts would \\ y = 2 - 3^{x} have?

d

State the domain of y = 2 - 3^{x}.

e

State the range of y = 2 - 3^{x}.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-7
-6
-5
-4
-3
-2
-1
1
y
23

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

a

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

b

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

-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
1
2
3
4
5
6
7
y
24

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}.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
25

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.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
26

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 - 2}.

b

Skecth the graph of y = 3^{x - 2}.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
1
2
3
4
5
6
7
8
9
y
27

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 - 3}.

b

Sketch the graph of y = 5^{x - 3}.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
1
2
3
4
5
6
7
8
9
y
28

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 } + 2.

b

Sketch the graph of y = 3^{ - x } + 2.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-2
-1
1
2
3
4
5
6
7
8
9
10
11
y
29

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

a

Describe a transformation of the graph of y = 4^{ - x } that would obtain y = 4^{ - \left( x + 3 \right)}.

b

Sketch the graph of y = 4^{ - \left( x + 3 \right)}.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-2
2
4
6
8
10
12
14
16
y
30

Determine whether each of the following exponential functions has a range of y < 5:

a

y = -4 \left(2^{ - x }\right) + 5

b

y = 5 + 2^{ - x }

c

y = - 4 \left(2^{x-5}\right)

d

y = - 2^{x} + 5

e

y = - 2^{x} - 5

f

y = 2^{x} - 5

31

Find the equation of the new function that results from the following transformations:

a

The graph of y = 2^{x} is translated down by 7 units.

b

The graph of y = 11^{x} is moved to the left by 10 units.

32

For each of the following, write the equation of the new function that results from y = 7^{x} after the following translations:

a
Translated three units to the right and then five units downward.
b
A reflection in the x-axis, then translated five units upward and three units to the left.
c
Translated five units upward, three units to the left and then reflected in the x-axis.
33

Consider the function y = 4^{x}. Find the equation of the new function that results from the following transformations:

a

The function is first reflected about the x-axis.

b

This new function is then multiplied by - 2.

34

Consider the function y = 6^{x}.

a

Find the equation of the new function that results from the following transformations:

i

The function is dilated by a factor of 5 vertically.

ii

This new function is then translated 3 units up.

b

Find the horizontal asymptote of the new function.

c

Find the y-value of the y-intercept of the new function.

d

Hence, sketch the graph of the new function.

35

Consider the function y = 5^{x}.

a

Find the equation of the new function that results from y = 5^{x} being translated 3 units to the right and 5 units downwards.

b

Find the horizontal asymptote of the new function.

c

Find the y-intercept of the new function.

d

Find the x-intercept of the new function.

e

Hence, sketch the graph of the new function.

36

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.

37

The graph of the exponential function P, given by y = 4^{ - x } is shown on the right:

a

Sketch the graph of Q given by the equation y = 8 x + 12 on the same set of axes as P.

b

How many times do P and Q intersect?

c

Which function has the higher function value at the y-intercept?

d

Which function has the higher function value at x = 1?

-4
-3
-2
-1
1
2
3
4
x
-4
4
8
12
16
y
38

The graph of the exponential function P, given by y = - 4^{x} is shown on the right:

a

Sketch the graph of Q given by the equation y = - \dfrac{4 \left(x - 3\right) \left(x - 7\right)}{21} on the same set of axes as P.

b

How many times do P and Q intersect?

c

Which function has the higher function value at the y-intercept?

d

Describe the behaviour of functions P and Q on the domain 0 < x < 5.

-10
-8
-6
-4
-2
2
4
6
8
10
x
-12
-8
-4
4
y
39

Consider the graph of exponential function P given by y = 2^{x} + 2 and a table of values for parabola Q given by \\ y = - 3 \left(x + 5\right) \left(x + 2\right) + 2.

a

Sketch the graphs of Q and P on the same set of axes.

b

Describe the behaviour of function P and function Q as x approaches infinity.

c

Using the graph, determine how many solutions are there to the equation \\2^{x} + 2 + 3 \left(x + 5\right) \left(x + 2\right) - 2 = 0.

Parabola Q

x-5-4-3-2-1
y2882-10

Function P

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
40

Consider the exponential functions P given by y = 9 \left(3^{ - x }\right) and Q given by y = 5 \left(3^{ - x }\right).

a

Are the exponential functions P and Q increasing or decreasing?

b

Sketch the two functions on the same set of axes.

c

As x tends to \infty, what value does each function approach?

d

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

i
P
ii
Q
Function from a graph
41

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 }

x
y
42

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).

-12
-10
-8
-6
-4
-2
2
4
6
8
10
12
x
-2
-1
1
2
3
4
5
y
43

The graph of f \left(x\right) = 9^{x} and another exponential function, g \left(x\right) is shown:

g(x) increasing at exactly the same rate as f \left(x\right), but has a different y-intercept. Write down the equation of function g \left(x\right).

-2
-1
1
2
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
44

For each of the following graphs of exponential functions in the form y = a^{x}, state the equation of the function:

a
-5
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-3
-2
-1
1
2
3
4
5
x
1
2
3
4
5
6
7
8
9
10
y
b
-3
-2
-1
1
2
3
x
1
2
3
4
5
6
7
8
9
10
y
c
-3
-2
-1
1
2
3
x
4
8
12
16
20
y
d
-3
-2
-1
1
2
3
x
5
10
15
20
25
30
35
40
y
e
-2
-1
1
2
x
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
y
f
-3
-2
-1
1
2
3
x
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
y
45

For each of the following graphs of exponential functions in the form y = a^{x} + k:

i

State the equation for the horizontal asymptote.

ii

State the equation for the exponential function.

a
-3
-2
-1
1
2
3
x
5
10
15
20
y
b
-3
-2
-1
1
2
3
x
-2
2
4
6
8
10
y
c
-3
-2
-1
1
2
3
x
1
2
3
4
5
6
7
y
d
-3
-2
-1
1
2
3
x
-7
-6
-5
-4
-3
-2
-1
y
46

For each of the following graphs of exponential functions in the form y = a^{x - h}:

i

Find the value of h for this function.

ii

State the equation of the function.

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
1
2
3
4
5
y
b
-5
-4
-3
-2
-1
1
2
3
4
5
x
1
2
3
4
5
y
c
-5
-4
-3
-2
-1
1
2
3
4
5
x
1
2
3
4
5
y
d
1
2
3
4
5
6
7
x
1
2
3
4
5
6
7
y
47

For each of the following graphs of exponential functions in the form y = a^{x - h} + k:

i

Find the value of k for this function.

ii

Find the value of h for this function.

ii

State the equation of the function.

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
1
2
3
4
5
6
7
y
b
-5
-4
-3
-2
-1
1
2
3
4
5
x
1
2
3
4
5
6
7
8
9
y
c
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
d
-5
-4
-3
-2
-1
1
2
3
4
5
x
1
2
3
4
5
6
7
8
9
10
y
48

For each of the given pair of points, find the equation of the exponential function of the form y =A\times a^{x} that passes through the points:

a

P\left(0,3\right) and Q\left(1,6\right)

b

P\left(1,30\right) and Q\left(2,90\right)

c

P\left(1,20\right) and Q\left(2,5\right)

d

P\left(1,30\right) and Q\left(2,10\right)

e

P\left(2,20\right) and Q\left(3,40\right)

f

P\left(2,-18\right) and Q\left(5,-486\right)

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Outcomes

2.1.5

recognise the qualitative features of the graph of 𝑦 = 𝑎^𝑥 (𝑎>0), including asymptotes, and of its translations 𝑦 = 𝑎^𝑥+𝑏 and 𝑦=𝑎^(𝑥-𝑐)

1.1.27

examine dilations and the graphs of y=c f(x) and y=f(dx)

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