Topics
2. Non-linear relationships
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AustraliaNSW
Year 12

2.01 Characteristics of parabolas

Lesson
Worksheet
Practice
Worksheet
Characteristics of parabolas
1

State whether the given graph is a parabola or not:

a
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
b
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
c
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
d
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
e
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
f
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
2

State whether the given equation is quadratic or not:

a

y = 7 - x

b

y = 8 x^{2}

c

y = 9 x^{2} - 5 x - 2

d

y = \dfrac{5}{x}

e

y = 6 x^{3} + 2 x^{2} - 8

f

y = 3 x + 6

g

y = 5 x^{2} - 6 x

h

y = x^{4} + x^{2} + 7

i

y = 1 + x^{3}

j

y = 4 + 8 x + 2 x^{2}

k

y = x^{2} + 2 + \dfrac{6}{x}

l

y = - 7 x^{2} + 3

3

Evaluate the following quadratic expressions for each of the given x-values:

a
x^{2}
i

x = 4

ii

x = - 4

b
2x^{2}
i

x = 3

ii

x = - 3

c
x^{2} + 3
i

x = 0

ii

x = 2

iii

x = - 2

d
x^{2} - 8
i

x = 0

ii

x = 8

iii

x = - 8

e
5x^{2} + 6
i

x = 0

ii

x = 1

iii

x = 2

f
4x^{2} - 5
i

x = 0

ii

x = -1

iii

x = -4

g
x^{2} - 2x + 6
i

x = 5

ii

x = -5

iii

x = -3

4

For each of the following parabolas, state whether:

i

It is concave up or concave down

ii

The y-value at the vertex is a minimum or a maximum

a
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
b
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
c
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
d
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
5

For each of the following graphs of quadratic equations:

i

State the coordinates of the turning point.

ii

State the equation of the axis of symmetry.

a
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
b
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
6

State the coordinates of the x-intercept(s) of each of the following parabolas:

a
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
b
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
7

For each of the parabolas defined by the following equations, state:

i

Whether the parabola is concave up or concave down.

ii

The y-intercept.

a
y = x^{2} + 4
b
y = - x^{2} + 3
c
y = x^{2} + 7
d
y = -x^{2} - 5
e
y = -x^{2} + 3x+9
f
y = x^{2} + 3x -4
g
y = 6x -x^{2}+ 11
h
y = 4x + 3x^{2} - 8
8

Consider the graph showing the height of a soccer ball after it is kicked:

a

What type of function is this?

b

State the y-intercept.

c

What does the y-intercept represent in this context?

d

State the x-intercept.

e

What does the x-intercept represent in this context?

1
2
3
4
5
6
\text{time (seconds)}
2
4
6
8
10
12
\text{height (m)}
9

Consider the graph of the quadratic equations y = x^{2} - 4, y = x^{2} + 2 and \\y = x^{2} + 6:

a

Which feature of the three parabola graphs is different?

b

Which part of the three quadratic equations is different?

-10
-8
-6
-4
-2
2
4
6
8
10
x
-6
-4
-2
2
4
6
8
10
12
y
10

Consider the graph of the two quadratic equations y = x^{2} - 4 x + 4 and \\y = - x^{2} - 3 x + 4:

a

Which feature of the two parabola graphs is the same?

b

State the y-intercept.

c

Which part of the two quadratic equations is the same?

d

Describe your findings from part (b) and part (c).

-10
-8
-6
-4
-2
2
4
6
8
10
x
-6
-4
-2
2
4
6
8
10
12
y
11

Consider the graph of the two quadratic equations y = x^{2} - 5 x + 4 and \\y = 2 x^{2} - 10 x + 8:

a

Which feature of the graphs of these two parabolas is different?

b

State whether the following statements are true or false in regards to the two equations:

i

Each term in the second equation is twice as big as the matching term in the first equation.

ii

The two equations are the same.

iii

The two equations have the same numbers, but each sign is different.

iv

The two equations are unrelated.

-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
12

Consider the graph of the two quadratic equations y = x^{2} - 2 x - 8 and \\y = - x^{2} + 2 x + 8.

a

Which feature of the graphs of these two parabolas is different?

b

Do these parabolas both have a maximum value, both have a minimum value, or one of each?

c

Determine whether each of the following statements is true or false in regards to the two equations:

i

The two equations are the same.

ii

The two equations have the same numbers, but each sign is different.

iii

The two equations are unrelated.

iv

The two equations have the same x^{2} term and the same x term but different constant terms.

-10
-8
-6
-4
-2
2
4
6
8
10
x
-20
-16
-12
-8
-4
4
8
12
16
20
y
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Outcomes

MS2-12-1

uses detailed algebraic and graphical techniques to critically evaluate and construct arguments in a range of familiar and unfamiliar contexts

MS2-12-6

solves problems by representing the relationships between changing quantities in algebraic and graphical forms

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