 4.04 Transforming parabolas

Worksheet
Transformations of parabolas
1

For the following equations:

i

Complete the table of values:

ii

Plot the graph of the parabola.

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}

b

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

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.

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.