topic badge
AustraliaNSW
Stage 5.1

3.04 Graphs of parabolas

Worksheet
Graphs of parabolas
1

Consider the equation y = x^{2}.

a

Complete the table of values:

x- 3- 2- 10123
y
b

Use the table of values to sketch the graph of the function on a number plane.

c

Are the y-values ever negative?

d

Find the equation of the axis of symmetry.

e

Find the minimum y-value.

f

For every y-value greater than 0, how many corresponding x-values are there?

2

Consider the equation y = - x^{2}.

a

Complete the table of values:

x- 3- 2- 10123
y
b

Use the table of values to sketch the graph of the function on a number plane.

c

Are the y-values ever positive?

d

Find the equation of the axis of symmetry.

e

Find the maximum y-value.

3

For each of the following equations:

i

Complete the table of values:

x- 3- 2- 10123
y
ii

Sketch the graph of the function on a number plane.

a

y = 3 x^{2}

b

y = - 2 x^{2}

c
y = 3 x^{2} + 3
d
y = - 2 x^{2} + 5
e
y = 3 x^{2} - 3
f
y = - 2 x^{2} - 2
g

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

h

y = - \dfrac{1}{2} x^{2}

Transformations of parabolas
4

For each of the following equations:

i

Complete the table of values:

x12345
y
ii

Sketch the graph of the function on a number plane.

iii

Find the minimum y-value.

iv

Find the x-value that corresponds to the minimum y-value.

v

Find the coordinates of the vertex.

a
y = \left(x - 3\right)^{2}
b
y = - \left(x - 3\right)^{2}
c
y = \left(x - 1\right)^{2}
d
y = \left(x - 4\right)^{2}
5

For each of the following equations:

i

Complete the table of values:

x- 3- 2-101
y
ii

Sketch the graph of the function on a number plane.

iii

Find the minimum y-value.

iv

Find the x-value that corresponds to the minimum y-value.

v

Find the coordinates of the vertex.

a
y = \left(x + 1\right)^{2}
b
y = - \left(x + 1\right)^{2}
c
y=\left(x + 2\right)^{2}
d
y=-\left(x + 3\right)^{2}
6

For each of the following equations:

i

State the coordinates of the vertex.

ii

Solve for the equation of the axis of symmetry.

iii

Sketch the graph of the function on a number plane.

iv

Plot the axis of symmetry.

a

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

b

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

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

Describe the transformation required to transform the parabola y = -x^{2} into the parabola y = -\left(x+4\right)^{2}.

8

State whether the following parabolas will be concave up or concave down:

a
y=\left(x-4\right)^{2}
b
y=-\left(x+2\right)^{2}
c
y=-\left(x-8\right)^{2}
d
y=\left(x+7\right)^{2}
9

For each of the following equations:

i

Complete the table of values:

x- 2- 1012
y
ii

Sketch the graph of the function on a number plane.

iii

State the coordinates of the vertex.

a
y = 2x^{2}
b
y = 3x^{2}
c
y = -4x^{2}
d
y = \dfrac{1}{2}x^{2}
10

For each of the following equations:

i

Solve for the equation of the axis of symmetry.

ii

Sketch the graph of the function and its axis of symmetry on a number plane.

a
y=4x^{2}
b

y = -3x^{2}

c

y = \dfrac{1}{4}x^{2}

d
y= -\dfrac{1}{2}x^{2}
11

Describe the transformation required to transform the parabola y = x^{2} into the parabola y =5x^{2}.

12

Rewrite the equation, y=x^{2}, after the following transformations take place:

a
Vertically translated by 3 units.
b
Vertically translated by -2 units.
c
Horizontally translated by 5 units.
d
Horizontally translated by -4 units.
e
Vertically scaled by 2 units.
f
Vertically scaled by -3 units.
g
Vertically scaled by \dfrac{1}{2} units.
h
Vertically reflected about the x-axis.
13

Graph the equation, y=x^{2}, after the following transformations take place:

a

Vertically translated by 4 units.

b

Horizontally translated by -5 units.

c

Vertically translated about the x axis.

d

Vertically scaled by 2 units.

Applications
14

William is training for a remote control plane aerobatics competition. He wants to fly the plane along the path of a parabola so he has chosen the equation: y = 3 x^{2} where y is the height in metres of the plane from the ground, and x is the horizontal distance in metres of the plane from its starting point.

a

Complete the table of values:

x \text{ (m)}01234
y \text{ (m)}
b

Sketch the shape of the path on a number plane.

c

Find the lowest height of the plane.

d

Find the x-value that corresponds to the minimum y-value.

e

Find the coordinates of the vertex.

15

On Jupiter the equation, d = 12.5 t^{2}, can be used to approximate the distance in metres, d, that an object falls in t seconds, if air resistance is ignored.

a

Complete the table of values:

\text{time }(t)0246
\text{distance } (d)
b

Sketch the shape of the path on a number plane.

c

Use the equation to determine the number of seconds, t, that it would take an object to fall 84.4\text{ m}. Round the value of t to the nearest second.

Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

MA5.1-7NA

graphs simple non-linear relationships

What is Mathspace

About Mathspace