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1.01 Exponential functions

Worksheet
Exponential functions
1

Consider the equation y = 5 \times 4^{x}.

Find the value of y when:

a

x = 0

b

x = 2

c

x = - 1

2

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

a

Evaluate f \left( 4 \right).

b

Evaluate f \left( - 3 \right).

3

Consider the function f \left( x \right) = 3^{x} + 3^{ - x }.

a

Evaluate f \left( 3 \right).

b

Evaluate f \left( - 3 \right).

c

Does f \left( 3 \right) = f \left( - 3 \right)?

4

Would an exponential function generate the values shown in the following tables?

a
x123456
f(x)525125625312515\,625
b
x123456
f(x)4911.51513.511
5

What type of function would generate values as shown in the table?

x123456
f(x)234.87.611.818
A

approximately exponential

B

exactly exponential

C

not exponential

6

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}

7

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.

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

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?

9

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 }
10

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?

11

A linear function and exponential function have been graphed on the following number plane:

a

Over a 1 unit interval of x, by what constant amount does the linear function grow?

b

Over a 1 unit interval of x, by what constant ratio does the exponential function grow?

c

Would it be correct to state that the linear function always produces greater values than the exponential function? Explain your answer.

d

As x approaches infinity, which function increases more rapidly?

1
2
3
4
x
2
4
6
8
10
12
14
16
18
y
12

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.

-3
-2
-1
1
2
3
x
-1
1
2
3
4
5
y
13

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

14

Consider the function y = 6^{x}.

a

Can the value of y ever be negative?

b

As the values of x get larger and larger, what value does y approach?

c

As the values of x get smaller and smaller, what value does y approach?

d

Can the value of y ever be equal to 0?

e

Find the y-intercept of the curve.

f

How many x-intercepts does the curve have?

g

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

15

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.

-2
-1
1
2
x
2
4
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10
12
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18
y
16

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

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
-2
-1
1
2
3
4
5
y
b
-5
-4
-3
-2
-1
1
2
3
4
5
x
-2
-1
1
2
3
4
5
y
17

State whether the following are increasing or decreasing exponential functions:

a
y = \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}
18

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}
19

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).

Exponential decay
20

Consider the function y = 3^{ - x }.

a

Complete the table of values:

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

Is y = 3^{ - x } an increasing function or a decreasing function?

c

Describe the rate of decrease of the function.

d

Find the y-intercept of the curve.

e

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

21

Consider the function y = - 3^{x}.

a

Complete the table of values:

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

Can the function values ever be positive?

c

Can the function value ever be 0?

d

Is y = - \left(3^{x}\right) an increasing function or a decreasing function?

e

Describe the rate of decrease of the function.

f

Find the y-intercept of the curve.

g

Find the horizontal asymptote of the curve y = - 3^{x}.

h

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

22

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

a

Can the value of y ever be negative?

b

As the value of x gets larger and larger, what value does y approach?

c

As the value of x gets smaller and smaller, what value does y approach?

d

Can the value of y ever be equal to 0?

e

Find the y-intercept of the curve.

f

How many x-intercepts does the curve have?

g

Sketch the graph of y = 8^{ - x }.

23

Given the function y = - 10^{x}, what is the largest possible function value? Explain your answer.

24

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?

25

State the range of the following functions:

a
y = - 4 \left(2^{ - x }\right) + 5
b
y = - 2^{x} + 5
c
y = 5 + 2^{ - x }
d
y = - 4 \left(2^{x}\right)
e
y = - 2^{x} - 5
f
y = 2^{x} - 5
26

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

a

Find the y-intercept of the curve y = - 6^{x}.

b

Find the horizontal asymptote of the curve y = - 6^{x}.

c

Is the function y = - 6^{x}, an increasing or decreasing function?

d

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

27

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

a

Jenny thinks she has found a set of solutions for the equation as shown in the table:

x-2-10123
y- \dfrac{1}{100}- \dfrac{1}{10}-1-10-100-1000

She notices that all the y values are negative and concludes that for any value of x, y must always be negative. Is she correct? Explain your answer.

b

Sketch the curve of y = - 10^{x}.

c

Find the values of x for which y = 0.

Transformations of exponential functions
28

State whether the following statements are true or false for the graph of y = - 2.5 \times 4^{x}:

a

As x approaches -\infty, y approaches 0.

b

The graph is decreasing.

c

As x approaches \infty, y approaches 0.

d

The graph has a y-intercept of \left(0, - 4 \right).

e

The graph has a y-intercept of \left(0, - 2.5 \right).

29

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

30

For each of the following exponential functions:

i
Find the y-intercept.
ii
State the horizontal asymptote of the curve.
iii
Sketch the curve on a number plane.
a
y = 5^{x} + 3
b
y = 3^{x} - 5
c
y = 2^{x - 2}
d
y = 2^{x + 4}
e
y = 2^{x} + 4
f
y = - 3^{ - x }
g
y = 3^{x} + 2
h
y = 2^{x} - 2
i
y = - 3^{x} + 2
j
y = - 3^{x} - 2
k
y = 3^{ - x }-1
l
y = 3^{ - x } + 1
m
y = 4^{x - 2}
n
y = - 5^{x - 4}
31

Consider the function y = 3^{ - x } + 2.

a

What value is 3^{ - x } always greater than?

b

Hence, what value is 3^{ - x } + 2 always greater than?

c

Hence how many x-intercepts does y = 3^{ - x } + 2 have?

d

State the equation of the asymptote of the curve y = 3^{ - x } + 2.

32

Consider the function y = 9^{x} + 5.

a

What value is 9^{x} always greater than?

b

Hence, what value is 9^{x} + 5 always greater than?

c

How many x-intercepts does y = 9^{x} + 5 have?

d

State the equation of the asymptote of the curve y = 9^{x} + 5.

33

Consider the linear function f \left(x\right) = 5 x + 2 and the exponential function g \left(x\right) = 5 \left(3\right)^{x}.

a

Find the value of f\left(5\right) - f\left(4\right).

b

Find the value of \dfrac{g \left( 5 \right)}{g \left( 4 \right)}.

c

Simplify f \left(k + 1\right) - f \left(k\right).

d

What does the result of part (c) demonstrate?

e

Simplify \dfrac{g \left(k + 1\right)}{g \left(k\right)}.

f

What does the result of part (e) demonstrate?

34

Consider the function y = - 3^{ - x }.

a

Find the y-intercept of the curve y = - 3^{ - x }.

b

Find the horizontal asymptote of the curve y = - 3^{ - x }.

c

Hence sketch the curve y = - 3^{ - x }.

d

Is the function y = - 3^{ - x }, an increasing or decreasing function?

35

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 }.

36

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
37

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.

38

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
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5
x
-5
-4
-3
-2
-1
1
2
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5
y
39

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

a

What transformation must be done to obtain the graph of y = 3^{ - x } + 2 from y = 3^{ - x }?

b

Hence sketch the graph of y = 3^{ - x } + 2 on a number plane.

-4
-3
-2
-1
1
2
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x
-4
-3
-2
-1
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y
40

Write the equation of the resulting function if the graph of y = 7^{x} is translated three units to the right and then five units downward.

41

Describe how the graph of y = 3^{4 - x} could be obtained from the graph of y = 3^{x}.

42

Describe how the graph of y = 3 \times 2^{x} - 1 could be obtained from the graph of y = 2^{x}.

43

Consider the function y = 5^{x - 3}.

a

Describe how the graph of y = 5^{x - 3} could be obtained from the graph of y = 5^{x}?

b

Given the graph of y = 5^{x}, sketch the graph of y = 5^{x - 3}.

-5
-4
-3
-2
-1
1
2
3
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5
x
-25
-20
-15
-10
-5
5
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y
44

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

a

Describe how to shift the graph of \\ y = 4^{ - x } to get the graph of y = 4^{ - \left( x + 3 \right)}.

b

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

-3
-2
-1
1
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x
-56
-48
-40
-32
-24
-16
-8
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40
48
56
y
45

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

a

Beginning with the above function, what equation do we obtain if we:

i

Horizontally dilate the graph by a factor of 2 from the x-axis and then translate the function by 5 units to the right?

ii

Translate the graph by 5 units to the right and then horizontally dilate the graph by a factor of 2 from the x-axis?

b

Did the order of the transformations affect the final equation in part (a)?

46

Consider the equation y = 5^{x}.

a

The above function is translated 3 units to the right and 5 units downwards. What is the equation of the new function?

b

What is the horizontal asymptote of the new function?

c

What is the y-intercept of the new function?

d

What is the x-intercept of the new function?

e

Sketch the graph of the new function.

47

Consider the equation y = 6^{x}.

a

The above function is dilated by a factor of 5 vertically, and then translated by 3 units upwards. Find the equation of the new function.

b

What is the horizontal asymptote of the new function?

c

What is the y-intercept of the new function?

d

Sketch the graph of the new function.

48

Consider the function y = 2^{x}. By translating this function horizontally 5 units to the left we get the function y = 2^{x + 5}.

a

Using index laws, rewrite 2^{x + 5} as A \times 2^{x}, where A is a positive constant.

b

Hence, 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}.

49

Consider the graph of y = 5^{x}. By translating this function horizontally 4 units to the right we get the function y = 5^{x - 4}.

a

Using index laws, rewrite 5^{x - 4} as A \times 5^{x}, where A is a positive constant.

b

Hence, a horizontal translation right by 4 units is equivalent to a vertical dilation of what factor?

c

Sketch the graph of y = 5^{x - 4}.

50

Consider the function y = 3^{ 2 x + 3}.

a

Which function below is equivalent?

A

y=8 \times 3^{x}

B

y=27 \times 9^{x}

C

y=9 \times 6^{x}

D

y=\dfrac{1}{2} \times 9^{x}

b

What is the domain of y = 3^{ 2 x + 3}?

c

What is the range of y = 3^{ 2 x + 3}?

51

Find the equation corresponding to the graph of y = 7^{x} after having been translated 5 units upward, 3 units to the left then reflected across the x-axis.

52

For each of the following exponential functions:

i
Find the y-intercept.
ii
State the horizontal asymptote.
iii
Sketch the curve
a
y = 3 \times 2^{x}
b
y = 3 \times 2^{x} + 2
c
y = 5 \times 2^{x - 3}
53

The function y = 4^{x} is dilated by a factor of 4 vertically, and then translated 3 unit downwards.

a

Find the equation of the new function.

b

What is the equation horizontal asymptote of the new function?

c

What is the value of the y-intercept of the new function?

d

Hence, sketch the graph of the new function.

54

The function y = 2^{x} is translated 4 units to the left and 4 units up.

a

What is the equation of the new function?

b

What is the equation of the horizontal asymptote of the new function?

c

What is the value of the y-intercept of the new function?

d

Does the new function have an x-intercept?

e

Hence, sketch the graph of the new function.

55

The function y = 4^{x} is translated 3 units to the right and is then dilated graphically by a factor of 2 horizontally from the y-axis.

a

What is the equation of the new function?

b

What is the equation of the horizontal asymptote of the new function?

c

What is the value of the y-intercept of the new function?

d

Hence, sketch the graph of the new function.

56

The function y = 4^{ - x } is dilated graphically by a factor of 2 horizontally from the y-axis and then translated 1 unit downwards.

a

What is the equation of the new function?

b

What is the equation of the horizontal asymptote of the new function?

c

What is the value of the y-intercept of the new function?

d

Hence, sketch the graph of the new function.

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