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iGCSE (2021 Edition)

6.05 Graphs of logarithmic and 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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b
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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 }

x
y
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
y
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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Graphs of logarithmic functions
31

Consider the function y = \log_{4} x and its given graph:

a

Complete the following table of values:

x\dfrac{1}{16}\dfrac{1}{4}416256
y
b

Find the x-intercept.

c

How many y-intercepts does the function have?

d

Find the x-value for which \log_{4} x = 1.

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32

Consider the functions graphed below:

Which of these graphs represents a logarithmic function of the form y = \log_{a} \left(x\right)?

A
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B
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33

Consider the function y = \log_{2} x.

a

Complete the following table of values:

x\dfrac{1}{2}12416
y
b

Sketch a graph of the function.

c

State the equation of the vertical asymptote.

34

Sketch the graph of y = \log_{5} x.

35

Consider the function y = \log_{4} x.

a

Complete the table of values.

x\dfrac{1}{1024}\dfrac{1}{4}1416256
y
b

Is \log_{4} x an increasing or decreasing function?

c

Describe the behaviour of \log_{4} x as x approaches 0.

d

State the value of y when x = 0.

36

Consider the function y = \log_{a} x, where a is a value greater than 1.

a

For which of the following values of x will \log_{a} x be negative?

A

x = - 9

B

x = \dfrac{1}{9}

C

x = 9

D

\log_{a} x is never negative

b

For which of the following values of x will \log_{a} x be positive?

A

x = 5

B

x = - 5

C

x = \dfrac{1}{5}

D

\log_{a} x will never be positive

c

Is there a value that \log_{a} x will always be greater than?

d

Is there a value that \log_{a} x will always be less than?

37

Consider the given graph of the logarithmic function y = \log_{a} x:

a

Is \log_{a} x an increasing or decreasing function?

b

Which is a possible value for a,\dfrac{2}{3} or \dfrac{3}{2} ?

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38

Consider the functions y = \log_{2} x and y = \log_{3} x.

a

Sketch the two functions on the same set of axes.

b

Describe how the size of the base relates to the steepness of the graph.

39

Consider the given graph of f \left( x \right) = \log_{k} x:

a

Determine the value of the base k.

b

Hence, state the equation of f \left( x \right).

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Transformations of logarithmic functions
40

Consider the functions f\left(x\right) = \log_{2} x and g\left(x\right) = \log_{2} x + 2.

a

Complete the table of values below:

x\dfrac{1}{2}1248
f\left(x\right)=\log_2 x
g\left(x\right)=\log_2 x + 2
b

Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.

c

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

d

Determine whether each of the following features of the graph will remain unchanged after the given transformation:

i

The vertical asymptote.

ii

The general shape of the graph.

iii

The x-intercept.

iv

The range.

41

Consider the functions f\left(x\right) = \log_{2} \left( - x \right) and g\left(x\right) = \log_{2} \left( - x \right) - 3.

a

Complete the table of values below:

x-8-4-2-1-\dfrac{1}{2}
f\left(x\right)=\log_2 \left( - x \right)
g\left(x\right)=\log_2 \left( - x \right) - 3
b

Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.

c

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

d

Determine whether each of the following features of the graph will remain unchanged after the given transformation:

i

The vertical asymptote.

ii

The general shape of the graph.

iii

The x-intercept.

iv

The domain.

42

Sketch the graph of the following functions:

a
y = \log_{3} x translated 2 units up.
b

y = \log_{3} x translated 4 units down.

c

y= \log_{2} x + 4.

43

The graph of y = \log_{6} x is transformed to create the graph of y = \log_{6} x + 4. Describe a tranformation that could achieve this.

44

For each of the following functions:

i

State the equation of the function after it has been translated.

ii

Sketch the translated graph.

a

y = \log_{5} x translated downwards by 2 units.

b

y = \log_{3} \left( - x \right) translated upwards by 2 units.

45

Consider the graph of y = \log_{6} x which has a vertical asymptote at x = 0. This graph is transformed to give each of the new functions below. State the equation of the asymptote for each new graph:

a
y = \log_{6} x - 7
b
y = \log_{6} x +2
c
y = 3\log_{6} x
d
y = \log_{6} \left(x - 2\right)
46

Given the graph of y = \log_{8} \left( - x \right), sketch the graph of y = 3 \log_{8} \left( - x \right).

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47

Given the graph of y = \log_{2} x, sketch the graph of the following functions:

a
y = \dfrac{1}{3} \log_{2} x
b
y = - \dfrac{1}{2} \log_{2} x
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48

Find the equation of the following functions, given it is of the stated form:

a

y = k \log_{2} x

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b

y = 4 \log_{b} x

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c

y = \log_{4} x + c

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49

The function graphed has an equation of the form y = k \log_{2} x + c and passes through points A\left(4,11\right) and B\left(8,15\right):

a

Use the given points to form two equations relating c and k.

b

Hence, find the values of c and k.

c

State the equation of the function.

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Outcomes

0606C7.1A

Know simple properties and graphs of the logarithmic function including lnx and graphs of k ln(ax + b) where n, k, a and b are integers.

0606C7.1B

Know simple properties and graphs of the exponential function including e^x and graphs of ke^(nx) + a where n and k are integers.

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