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3.02 Quadratic functions

Worksheet
Key features
1

Is the function y = - 2 \left(x - 3\right)^{2} - 1 a one-to-one function?

2

Consider the quadratic equation y = - x^{2} + 10 x + 16.

Find the y-intercept of the graph given by the equation.

3

Consider the graph of the given function y = f \left( x \right):

a

What is the minimum value of the graph?

b

What is the range of the function?

c

What is the domain of this function?

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4

Consider the graph of the quadratic equation.

a

State the y-intercept.

b

State the x-intercepts.

c

State the equation of the axis of symmetry.

d

State the coordinates of the maximum point.

e

Determine the values of x for which the gradient of the curve is negative.

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5

Find the maximum value of y for the quadratic function y = - x^{2} + 10 x - 25.

6

The minimum value of the function y = 9 x^{2} + 108 x + m is y = 6.

a

Find the value of x at the minimum point.

b

Find the value of m.

7

Determine the values of x for which the quadratic function is decreasing.

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8

Consider the parabola defined by the equation y = x^{2} + 5.

a

Is the parabola concave up or concave down?

b

What is the y-intercept of the parabola?

c

What is the minimum y-value of the parabola?

d

Hence determine the range of the parabola.

9

For each of the following equations of parabolas:

i

Determine if the parabola is concave up or concave down.

ii

Find the y-intercept of the parabola.

iii

Find the number of solutions of y=0.

iv

Find the number of x-intercepts.

v

Find the minimum y-value of the parabola.

a

y = x^{2} + 3

b

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

10

Consider the equation y = 25 - \left(x + 2\right)^{2}.

What is the maximum value of y?

11

State whether the following parabolas have any x-intercepts:

a

y = \left(x - 7\right)^{2} + 4

b

y = - \left(x - 7\right)^{2} + 4

c

y = - \left(x - 7\right)^{2} - 4

d

y = \left(x - 7\right)^{2} - 4

12

What is the axis of symmetry of the parabola y = k \left(x - 7\right) \left(x + 7\right) for any value of k?

13

Consider the following graph:

a

Find the x-intercepts.

b

Find the zeros of the function.

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14

Consider the graph of the function f \left( x \right) = - x^{2} - x + 6.

Using the graph, write down the solutions to the equation - x^{2} - x + 6 = 0.

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15

Use the quadratic formula to find the x-intercepts of y = 3 x^{2} + 3 x - 7 in exact form.

16

The parabola y = 2 x^{2} + b x + 1 has its axis of symmetry at x = - 1. Find the value of b.

17

For the graph y = x^{2}, find the two x-values that correspond to a y-value of 81.

18

Determine the value of c if the parabola y = x^{2} + 4 x + c has exactly one x-intercept.

19

Consider the given graph of the parabola:

a

State the x-intercepts.

b

State the y-intercept.

c

Find the equation of the axis of symmetry.

d

State the coordinates of the vertex.

e

State whether the following statements are true about the vertex:

i

The x-value of the vertex is the average of the x-values of the two x-intercepts.

ii

The vertex is the minimum value of the graph.

iii

The vertex is the maximum value of the graph.

iv

The vertex lies on the axis of symmetry.

v

The y-value of the vertex is the same as the y-value of the y-intercept.

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20

Consider the table of values generated from a quadratic function:

x 4 5 6 7 8 9 10
y 23 13 7 5 7 13 23
a

What are the coordinates of the vertex?

b

Is the vertex a maximum or minimum point?

c

What is the minimum value of the function?

21

Find the coordinates of the vertex of y = 3 x^{2} - 6 x - 9.

22

A parabola has an x-intercept at \left(-1, 0\right) and a vertex at \left(1, - 6 \right). Find the coordinates of the other x-intercept.

23

Consider the parabola defined by equation y = 2 \left(x - \dfrac{3}{4}\right)^{2} - \dfrac{1}{4}.

a

Write down the coordinates of the vertex.

b

Find the vertical axis of symmetry for this parabola.

c

Write down the coordinates of the y-intercept.

24

Consider the function y = x^{2} - 4 x + 6.

a

Find the coordinates of the vertex.

b

Is the vertex the maximum or minimum of this parabola?

c

How many real roots does this parabola have?

Graphs of quadratics
25

Consider the quadratic function h \left( x \right) = x^{2} - 2.

a

Sketch the graph of the parabola given by h \left( x \right).

b

Sketch the axis of symmetry of the parabola on the same axes.

c

What is the vertex of this parabola?

26

Consider the parabola described by the function y = \dfrac{x^{2}}{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

What are the coordinates of the vertex of the parabola?

d

Sketch the graph of y = \dfrac{x^{2}}{2} - 2.

27

A parabola of the form y = a x^{2} goes through the point \left(2, - 4 \right).

a

What is the value of a?

b

What are the coordinates of the vertex?

c

Sketch the graph of the parabola.

28

Consider the quadratic function f \left( x \right) = \left(x - 3\right)^{2}.

a

What are the coordinates of the vertex of this parabola?

b

What is the equation of the axis of symmetry of this parabola?

c

Graph the parabola corresponding to f \left( x \right).

d

Sketch the axis of symmetry of the parabola on the same axes.

29

For each of the following parabolas:

i

Find the y-intercept.

ii

Find the x-intercepts.

iii

State the equation of the axis of symmetry.

iv

Find the coordinates of the turning point.

v

Sketch the graph of the parabola.

a

y = x \left(x + 6\right)

b

y = x \left(6 - x\right)

c

y = \left(2 - x\right) \left(4 - x\right)

30

Graph the following quadratic functions:

a
y=(x+2)(x-3)
b
y=(x+1)(x-3)
31

Consider the quadratic function y = 16 - x^{2}.

a

Find the x-intercepts.

b

Find the y-intercept.

c

Sketch the graph of the function.

d

State the vertex of the parabola.

32

The equation of a parabola is of the form y = \left(x - a\right) \left(x - b\right).

a

The parabola has x-intercepts x = \sqrt{3} and x = - \sqrt{3}. Write down its equation in expanded form.

b

What is the y-value of the point on the parabola where x = 1?

c

Sketch the graph of the curve.

33

Consider the parabola y = x^{2} + x - 12.

a

Factorise the equation of the parabola.

b

Find the x-intercepts of the curve.

c

Find the y-intercept of the curve.

d

What is the equation of the axis of symmetry?

e

Find the coordinates of the vertex.

f

Is the parabola concave up or down?

g

Sketch the curve of y = x^{2} + x - 12.

34

Consider the quadratic function y = - x^{2} + 4 x + 12.

a

What is the concavity of the parabola?

b

What is the y-intercept?

c

Find the x-intercepts.

d

Find the equation of the axis of symmetry.

e

Find the coordinates of the vertex.

f

Sketch the graph of the function y = - x^{2} + 4 x + 12.

35

A parabola is described by the function y = 2 x^{2} + 9 x + 9.

a

Find the x-intercepts of the parabola.

b

Find the y-intercept of this curve.

c

Find the equation of the axis of symmetry.

d

Find the y-coordinate of the vertex.

e

Sketch the curve of y = 2 x^{2} + 9 x + 9.

36

Consider the quadratic function f \left( x \right) = - 3 \left(x + 2\right)^{2} - 4.

a

What are the coordinates of the vertex of this parabola?

b

What is the equation of the axis of symmetry?

c

What is the y-coordinate of the graph of f \left( x \right) at x = -1?

d

Sketch the graph of the parabola f \left( x \right).

e

Sketch the axis of symmetry of the parabola on the same axes.

37

Consider the equation y = \left(x - 3\right)^{2} - 1.

a

Find the x-intercepts.

b

Find the y-intercept.

c

Find the coordinates of the vertex.

d

Sketch the graph of the equation.

38

Consider the parabola described by the function y = - \dfrac{1}{5} \left(x - 1\right)^{2} + 1.

a

Is the parabola concave up or down?

b

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

c

What are the coordinates of the vertex?

d

Sketch the graph of the parabola.

39

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

a

Find the zeros of the quadratic function.

b

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

c

Find the coordinates of the vertex.

d

Sketch the graph of the parabola.

40

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

a

Write the equation of the parabola.

b

Sketch the graph of the parabola.

41

On a number plane, sketch the shape of a parabola of the form y = a \left(x - h\right)^{2} + k that has the following signs for a, h and k:

a
a\gt 0, h\gt 0, k\gt 0
b
a\lt0, h\gt0, k\gt0
c
a\gt0, h\gt0, k\lt0
d
a\lt0, h\gt0, k\lt0
Equations of quadratic functions
42

Consider the two curves shown. The top curve has equation f(x) = x^{2} + 5.

State the equation of the g(x).

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43

Consider a parabola whose x-intercepts are - 10 and 4, and whose y-intercept is - 40. Find the equation of the parabola.

44

Consider the following graph:

a

What is the y-intercept of the graph?

b

Write the equation of this graph in factored form.

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A parabola has its turning point at x = - 3 and one of the x-intercepts at x = 1.

a

What is the other x-intercept?

b

If it has a y-intercept at 3, state the equation of the parabola.

c

What are the coordinates of the turning point?

46

Consider the following graph:

a

Determine the coordinates of the vertex.

b

Find the equation of the parabola.

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47

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

a

Determine the axis of symmetry of the curve.

b

Hence or otherwise find the equation of the curve.

48

Suppose a parabola has its vertex at \left( - 1 , - 8 \right) and coefficient a = \dfrac{6}{5}.

a

Write the equation of the parabola in vertex form.

b

Write the equation of the parabola in general form.

49

A family of quadratics is defined as having a vertex of \left(1, 3\right).

a

Letting a be the coefficient of x^{2}, write the equation that represents this family of quadratics.

b

Find the equation of the quadratic in this family that has a vertex at \left(1, 3\right) and passes through the point \left(9, 5\right).

c

Find the equation of the quadratic in this family that has x = 5 as one of its zeros.

50

Find the equation of the quadratic function that has a vertex at \left( - 12 , 3\right) and that passes through the point \left( - 4 , 19\right).

Application
51

Over the summer, Susana and her friends build a bike ramp to launch themselves into a lake. Susana decides that the shape of the ramp will be parabolic, and that the best parabola is given by the equation y = \dfrac{1}{4} \left(x + 2\right)^{2} + 2, where 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, where x = -6, how high is the start of the ramp?

b

At what height is the end of the ramp?

c

At what other distance x is the rider also at this height?

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MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

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