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3.02 Graphs of parabolas

Worksheet
Key features of parabolas
1

Consider the equation x = y^{2}.

a

Find the x-value of the point which has a y-coordinate of 4.

b

Find the x-value of the point which has a y-coordinate of - 4.

c

What is the distance between these two points?

2

Determine whether the following is true of the graph of x = y^{2} ?

a

There are always two points that contain the same x-coordinate, except for x = 0.

b

There are always two points that contain the same y-coordinate, except for y = 0.

3

Consider the equation x = y^{2}.

a

Make y the subject.

b

Explain why the graph of y = \pm \sqrt{x} has symmetry.

c

What is the equation of the axis of symmetry of x = y^{2} ?

4

Consider the graph of the relation x = - y^{2}.

a

State two functions that can be combined together to form the same graph as x = - y^{2}.

b

Over which values of x is the relation defined?

5

The point \left(k^{2} - 28 k - 35, 2 k-1\right) lies on the curve x = y^{2}. Find the values of k.

6

For each of the following parabolas:

i

State the coordinates of the vertex.

ii

Does the parabola open upward or downward?

a

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

b

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

c

\left(x - 3\right)^{2} = y + 2

d
y = - \left(x + 5\right)^{2} + 2
e
\left(x + 2\right)^{2} + 6 = y
7

Consider the parabola with equation \left(x + 1\right)^{2} = 8 y.

a

State the coordinates of the vertex of the parabola.

b

Determine the distance of the point \left(3, 2\right) from the vertex.

8

For each of the following parabolas:

i

State the coordinates of the vertex.

ii

Does the parabola open to the left or right?

a
\left(y - 5\right)^{2} - 4 = x
b
x = - \left(y + 6\right)^{2} + 3
9

What are the coordinates of the vertex of the parabola: x = \left(y + 3\right)^{2} - 4?

10

Consider the parabola with equation x = 2 \left(y - 4\right)^{2} - 1.

a

Is this a horizontal or vertical parabola?

b

Does the parabola open left or right?

c

What are the coordinates of the vertex?

11

For each of the following parabolas:

i

What are the coordinates of the vertex?

ii

In which direction does this parabola open?

a

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

b

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

c

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

d

x - 5 = - \left(y + 3\right)^{2}

12

Find the coordinates of the vertex of the parabola with equation x = - 3 y^{2} + 12 y + 11

Graph parabolas
13

Consider the parabola x = y^{2}.

a

State the range of values of x for which the relation is defined.

b

Complete the table for the values of x:

c

Plot the points from the table of values on a number plane.

d

Sketch the graph of the curve that passes through the plotted points.

x
y-3-2-10123
e

How many points on the graph correspond to any one particular value of x, for x > 0?

14

Consider the parabola x = - \dfrac{1}{4} y^{2}.

a

Complete the following table of coordinate pairs for the given equation:

b

Sketch the graph of the parabola.

x
y-2-101234
15

Consider the parabola x = y^{2} whose graph is shown:

a

How many points on the graph have an \\x-coordinate of 81?

b

State the values of y that correspond to an x-value of 81.

27
54
81
x
-9
-6
-3
3
6
9
y
16

Consider the parabola x = y^{2} + 4.

a

Complete the following table of coordinate pairs:

b

Sketch the graph of the parabola.

x
y-2-1012
17

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

a

Complete the following table of coordinate pairs:

b

Sketch the graph of the parabola.

x
y234567
18

Consider the parabola x = \dfrac{1}{4} y^{2}.

a

Sketch the graph of the parabola.

b

State the coordinates of the x-intercept.

19

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

a

Complete the following table of coordinate pairs:

b

Sketch the graph of the parabola.

c

State the coordinates of the x-intercept.

x
y-2-1012
20

For each of the following equations:

i

Sketch the graph of the parabola.

ii

State the domain of the function in interval notation.

iii

State the range of the function in interval notation.

a

y - 5 = x^{2}

b

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

c

x + 2 = y^{2}

d

x = \left(y - 3\right)^{2}

e

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

f

x - 1 = \left(y - 3\right)^{2}

g

y - 2 = - \dfrac{1}{3} \left(x - 1\right)^{2}

h

x - 2 = - 3 \left(y - 5\right)^{2}

i

y = x^{2} + 4 x + 3

j

x = y^{2} - 4 y + 3

k

y + 3 x^{2} - 24 x + 53 = 0

l

x + 3 y^{2} - 30 y + 78 = 0

m
x = \left(y - 3\right)^{2} - 5
n
y = \left(x + 2\right)^{2}-1
o
y = - 2 \left(x + 2\right)^{2} + 4
p
x = - \dfrac{1}{4} \left(y + 4\right)^{2} + 5
21

Consider the graphs of y= x^{2} and

x = y^{2} shown:

Describe the transformation required to transform y= x^{2} into x = y^{2}.

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

ACMMM021

recognise features of the graph of y^2=x including its parabolic shape and its axis of symmetry

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