topic badge
AustraliaVIC
Level 10

3.04 Transforming parabolas

Worksheet
Transformations of parabolas
1

For the following equations:

i

Complete the table of values:

ii

Plot the graph of the parabola.

x- 3- 2- 10123
y
a

y = 3 x^{2} + 2

b

y = 3 x^{2} - 3

c

y = - 3 x^{2} + 4

d

y = - 2 x^{2} - 4

2

For the following equations:

i

Complete the given table of values.

ii

Plot the graph of the parabola.

iii

Find the maximum y -value.

iv

Find the coordinates of the vertex.

a

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

x12345
y
b

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

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

What feature is at the point \left(h, k\right) on the parabola defined by the equation: y = a \left(x - h\right)^{2} + k

4

Consider the following parabola:

a

Find the coordinates of the vertex.

b

Given that the graph has equation of the form y = a \left(x - h\right)^{2} + k, find the equation of the parabola.

1
2
3
4
5
6
x
2
4
6
8
10
12
14
16
y
5

For the following sentences:

i

Write the equations in the form y=\left(x + a\right)^{2} + b.

ii

Find the coordinates of the vertex.

iii

Sketch the graph of the parabola.

a
y=x^{2} + 6 x
b
y = x^{2} + 6 x + 1
c
y=x^{2} - 6 x + 1
d
y = x^{2} + x + 3
6

For the following equations:

i

Describe its successive transformations from y = x^{2}.

ii

Find the coordinates of the vertex.

iii

Sketch the graph of the parabola.

iv

Find the axis of symmetry.

a

y = \left(x - 3\right)^{2} - 4

b

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

7

For the following equations:

i

Find the x-intercepts.

ii

Find the y-intercept.

iii

Find the coordinates of the vertex.

iv

Sketch the graph of the parabola.

a

y = \left(x - 2\right)^{2} - 16

b

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

c

y = - \left(x + 4\right)^{2} + 9

d

y = 4 - \left(x - 1\right)^{2}

8

For the following equations:

i

Find the y-intercept.

ii

Is the graph concave up or down?

iii

Find the minimum y-value.

iv

Find the coordinates of the vertex.

v

Sketch the graph of the parabola.

vi

Find the axis of symmetry.

a

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

b

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

9

Consider the parabola described by the equation:y = 2 \left(x - 1\right)^{2} - 3

a

Is the parabola concave up or down?

b

Is the parabola more or less steep than the parabola y = x^{2}?

c

Find the coordinates of the vertex of the parabola.

d

Sketch the graph of this function.

10

Consider the parabola described by the equation:y = - \dfrac{1}{3} \left(x - 2\right)^{2} + 2

a

Is the parabola concave up or down?

b

Is the parabola more or less steep than the parabola y = - x^{2}?

c

Find the coordinates of the vertex of the parabola.

d

Sketch the graph of this function.

11

Consider the parabola described by the equation:y = - 3 \left(x + 5\right)^{2} - 4

a

Find the coordinates of the vertex of this parabola.

b

Find the axis of symmetry of this parabola.

c

Find the y-coordinate of the graph at x = - 4.

d

Sketch the graph of this function.

e

Sketch the axis of symmetry of the parabola on the same number plane.

12

For the following transformations:

i

Find the equation of the resulting parabola.

ii

Find the minimum y-value.

iii

Find the x-value that results in the minimum y-value.

iv

Find the axis of symmetry.

v

Sketch the graph of the resulting parabola.

a

The graph of y = \left(x + 6\right)^{2} is translated 6 units up.

b

The graph of y = \left(x - 3\right)^{2} is translated 3 units down.

13

The graph of y = - \left(x + 3\right)^{2} is translated 2 units up.

a

Find the equation of the resulting parabola.

b

Find the maximum x and y-values.

c

Find the axis of symmetry.

d

Find the coordinates of the vertex.

e

Sketch the graph of the resulting parabola.

14

Sketch the graph of y = \left(x - 4\right)^{2} and its transformation y = 2 \left(x - 4\right)^{2} - 4 on the same number plane.

15

Sketch the graph of the function f\left(x\right) = x^{2} and its transformation g\left(x\right) = - 3 \left(x + 2\right)^{2} + 5 on the same number plane.

16

A parabola has the equation: y = x^{2} + 4 x-1

a

Express the equation in the form y = \left(x - h\right)^{2} + k .

b

Find the y-intercept of the curve.

c

Find the coordinates of the vertex.

d

Is the parabola concave up or down?

e

Sketch the graph of the function.

17

Consider the quadratic function:y = x^{2} - 12 x + 32

a

Find the zeros of the quadratic function by completing the square.

b

Express the equation in the form y = a \left(x - h\right)^{2} + k by completing the square.

c

Find the coordinates of the vertex.

d

Sketch the graph of the function.

18

The following parabola is symmetrical about the line x = 2, and its vertex lies 6 units below the x-axis. It has the form:y = \left(x - h\right)^{2} + k

a

Find the equation of the parabola.

b

Sketch the graph of the prabola.

19

A parabola has x-intercepts at \left(1, 0\right) and \left( - 5 , 0\right) and is of the form:y = \left(x - h\right)^{2} + k

a

Find the axis of symmetry.

b

Find the equation of the parabola.

20

Over the summer, Susana and her friends build a bike ramp to launch themselves into the local lake. Susana decides that the shape of the ramp will be parabolic, and reckons that the parabola is given by the equation: y = \dfrac{1}{4} \left(x + 2\right)^{2} + 2where y is the height in metres above the ground, and x is the horizontal distance in metres from the edge of the lake.

a

If the ramp starts 6 \text{ m} back from the edge of the lake, how high is the start of the ramp?

b

At what height will the rider leave the ramp?

c

At what other distance x is the rider also at a height of 3 \text{ m}?

d

Graph the shape of the ramp on a number plane.

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

Outcomes

VCMNA339

Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital technology as appropriate

What is Mathspace

About Mathspace