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3.03 Graphs of square root functions

Worksheet
Key features
1

Consider the graph of y = \sqrt{ - x }:

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

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2

Consider the graph of y = \sqrt{x}:

a

Describe the rate of increase of the function as x increases.

b

State the axes intercepts.

c

Does the function have an asymptote?

d

Does the function have a limiting value?

e

As x increases, what does y approach?

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3

Consider the function y = - \sqrt{x}.

a

Complete the table of values. Round any values to two decimal places if necessary.

b

Can the function values ever be positive?

x0123459
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c

The graph of the function y = - \sqrt{x} is shown. Is y = - \sqrt{x} an increasing function or a decreasing function?

d

Describe the rate of decrease of the function as x increases.

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4

Consider the function y = \sqrt{ - x }.

a

Complete the table of values. Round any values to two decimal places if necessary.

x-5-4-3-2-10
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b

The graph of y = \sqrt{ - x } is given.

Is y = \sqrt{ - x } an increasing function or a decreasing function?

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5

Consider the function y = \sqrt{x} + 3.

a

Can y ever be negative?

b

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

c

Determine the y-intercept of the curve.

d

How many x-intercepts does it have?

e

Sketch the graph.

6

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

a

Is the function increasing or decreasing from left to right?

b

Is the function more or less steep than y = \sqrt{x} ?

c

What are the coordinates of the vertex?

d

Sketch the graph.

7

Consider the function y = - \dfrac{1}{2} \sqrt{x} + 2.

a

Is the function increasing or decreasing from left to right?

b

Is the function more or less steep than y = \sqrt{x} ?

c

What are the coordinates of the vertex?

d

Sketch the graph.

Domain and range
8

Consider the function y = \sqrt{ - x }.

a

State the domain of the function.

b

State the range of the function.

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9

Consider the function y = \sqrt{x}.

a

Complete the table of values. Round any values to two decimal places if necessary.

b

State the domain of the function.

c

State the range of the function.

x0123459
y
d

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

e

Sketch the graph of y = \sqrt{x}.

10

Consider the function y = - \sqrt{x}.

a

State the domain of the function.

b

State the range of the function.

11

The function y = \sqrt{x} has domain x \geq 0 and range y \geq 0.

What is the domain and range of y = \sqrt{x} - 2 ?

12

Consider the function y = \sqrt{x - 5}.

a

State the domain of the function.

b

State the range of the function.

c

Do the functions y = \sqrt{x} and y = \sqrt{x - 5} increase at the same rate?

13

Consider the function y = \sqrt{ - x } + 6.

a

What is the smallest possible function value?

b

State the domain of the function.

c

State the range of the function.

14

A square root function has a range of y \leq 0 and a domain of x \geq 0. Determine whether the following could be the equation of the function:

a

y = \sqrt{x}

b

y = - \sqrt{ - x }

c

y = - \sqrt{x}

d

y = 5 \sqrt{x}

e

y = - 5 \sqrt{x}

f

y = \sqrt{ - x }

15

For each of the following functions:

i

Sketch the graph

ii

State the domain

iii

State the range

a

f \left( x \right) = - \sqrt{x + 1}

b

f \left( x \right) = - 2 \sqrt{x + 5}

c

f \left( x \right) = 3 \sqrt{\left( \dfrac{1}{3} x\right)}

d

f \left( x \right) = - \dfrac{\sqrt{x}}{2} - 2

e

f \left( x \right) = \dfrac{\sqrt{x - 1}}{2} + 2

16

For each of the following functions:

i

State the domain of the function.

ii

State the range of the function.

iii

Sketch the graph.

a
y = - \sqrt{x} + 5
b
y = - 5 \sqrt{x}
c
y = - \sqrt{ - x }
d
y = \sqrt{x - 3} + 2
17

For which values of x do the following expressions evaluate to a real number?

a

\sqrt{ 7 x}

b

\sqrt{x - 2}

c

\sqrt{3 - x}

d

\sqrt{15 - 5 x}

e

\sqrt{x^{2} + 6}

18

Consider the function f \left( x \right) = \sqrt{x - 2} + 3. State the domain of the function using interval notation.

Transformations
19

The graph of y = \sqrt{x} has a vertex at \left(0, 0\right). By considering the transformations that have taken place, state the coordinates of the vertex of y = - \sqrt{x} + 3.

20

The graph of y = \sqrt{x} has been translated to the graph of y = \sqrt{x} - 4.

a

Describe the transformation that has occured on the original function.

b

Hence, sketch the graph of y = \sqrt{x} - 4.

21

Consider the graph of y = \sqrt{x} shown:

a
i

Sketch the curve after y = \sqrt{x} has been reflected about the y-axis.

ii

What is the equation of this new graph?

b
i

Sketch the curve after y = \sqrt{x} has been reflected about the x-axis.

ii

What is the equation of this new graph?

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22

Consider the function y = \sqrt{x}:

a

Describe how we can transform the graph of y = \sqrt{x} to get the graph of y = \sqrt{x - 4} + 3.

b

Hence, sketch the graph of y = \sqrt{x - 4} + 3.

23

Sketch the curve y = 3 \sqrt{x - 2} + 3.

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Outcomes

ACMMM014

recognise features of the graphs of y=x^n for n∈N, n=−1 and n=½, including shape, and behaviour as x→∞ and x→−∞

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