 iGCSE (2021 Edition)

Worksheet
Key features of parabolas
1

Consider the general quadratic equation y = a x^{2} + b x + c, a \neq 0.

a

If a \lt 0, in what direction will the parabola open?

b

If a \gt 0, in what direction will the parabola open?

2

Does the parabola represented by the equation y = x^{2} - 8 x + 9 open upward or downward?

3

Does the graph of y = x^{2} + 6 have any x-intercepts? Explain your answer.

4

State whether the following parabolas have 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
5

Consider the given graph:

a

What are the x-intercepts?

b

What is the y-intercept?

c

What is the maximum value?

d

What is the range of the quadratic function?

6

Consider the given graph:

a

Is the curve concave up or concave down?

b

State the y-intercept of the graph.

c

What is the minimum value?

d

At which value of x does the minimum value occur?

e

Determine the interval of x for which the graph is decreasing.

7

Consider the graph of the parabola:

a

State the coordinates of the x-intercept.

b

State the coordinates of the vertex.

c

State whether the following statements are true about the vertex:

i

The vertex is the minimum value of the graph.

ii

The vertex occurs at the x-intercept.

iii

The vertex lies on the axis of symmetry.

iv

The vertex is the maximum value of the graph.

8

Suppose that a particular parabola is concave down, and its vertex is located in quadrant 2.

a

How many x-intercepts will the parabola have?

b

How many y-intercepts will the parabola have?

9

Suppose that a particular parabola has two x-intercepts, and its vertex is located in quadrant 4. Will such a parabola be concave up or concave down?

10

Consider the quadratic function defined in the table on the right:

a

What are the coordinates of the vertex?

b

What is the minimum value of the function?

11

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

12

State whether the following can be found, without any calculation, from the equation of the form y = \left(x - h\right)^{2} + k but not from the equation of the form y = x^{2} + b x + c:

a

x-intercepts

b

y-intercept

c

vertex

13

Quadratic function A is represented graphically as shown. Quadratic function B, which is concave down, shares the same x-intercepts as quadratic function A, but has a y-intercept closer to the origin. Which of the functions has a greater maximum value?

14

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

15

Consider the equation y = 25 - \left(x + 2\right)^{2}. What is the maximum value of y?

16

Consider the function y = \left(14 - x\right) \left(x - 6\right).

a

State the zeros of the function.

b

Find the axis of symmetry.

c

Is the graph of the function concave up or concave down?

d

Determine the maximum y-value of the function.

e

State the range of the function.

17

Consider the parabola of the form y = a x^{2} + b x + c, where a \neq 0.

Complete the following statement:

The x-coordinate of the vertex of the parabola occurs at x = ⬚. The y-coordinate of the vertex is found by substituting x = ⬚ into the parabola's equation and evaluating the function at this value of x.

18

Find the x-coordinate of the vertex of the parabola represented by P \left( x \right) = p x^{2} - \dfrac{1}{2} p x - q.

19

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.

20

True or false:

a

The quadratic formula can be used to find the y-intercept.

b

If the parabola has only one x-intercept , then the x-intercept is also the vertex.

21

Consider the parabola whose equation is y = 3 x^{2} + 3 x - 7. Find the x-intercepts of the parabola in exact form.

22

Consider the parabola described by the function y = - 2 x^{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 = - 2 x^{2} + 2.

23

Consider the two graphs. One of them has equation f(x) = x^{2} + 5.

What is the equation of the other graph?

24

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

a

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

b

Plot the axis of symmetry of the parabola on the same graph.

c

What is the vertex of the parabola?

d

What is the range of the parabola?

25

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

26

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

a

Find the x-intercepts.

b

Find the y-intercept.

c

Determine the coordinates of the vertex.

d

Sketch the graph.

27

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 of this parabola?

c

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

d

Sketch the graph of the parabola.

e

Plot the axis of symmetry of the parabola on the same graph.

28

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
29

Consider the parabola y = \left(2 - x\right) \left(x + 4\right).

a

State the y-intercept.

b

State the x-intercepts.

c

Complete the table of values:

d

Determine the coordinates of the vertex of the parabola.

e

Sketch the graph of the parabola.

30

Consider the parabola y = \left(x - 3\right) \left(x - 1\right).

a

Find the y-intercept.

b

Find the x-intercepts.

c

State the equation of the axis of symmetry.

d

Find the coordinates of the turning point.

e

Sketch the graph of the parabola.

31

Consider the parabola y = x \left(x + 6\right).

a

Find the y-intercept.

b

Find the x-intercepts.

c

State the equation of the axis of symmetry.

d

Find the coordinates of the turning point.

e

Sketch the graph of the parabola.

32

Sketch the graph of the following:

a
y = (x + 2)(x - 3)
b
y = (x - 3)(x + 1)
33

Consider the function y = \left(x + 5\right) \left(x + 1\right).

a

Sketch the graph.

b

Sketch the graph of y = - \left(x + 5\right) \left(x + 1\right) on the same set of axes.

34

Consider the equation y = x^{2} - 6 x + 8.

a

Factorise the expression x^{2} - 6 x + 8.

b

Hence, or otherwise, find the x-intercepts of the quadratic function y = x^{2} - 6 x + 8

c

Find the coordinates of the turning point.

d

Sketch the graph of the function.

35

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

a

Find the x-intercepts of the curve.

b

Find the y-intercept of the curve.

c

What is the equation of the vertical axis of symmetry for the parabola?

d

Find the coordinates of the vertex of the parabola.

e

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

36

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

a

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

b

Find the y-intercept of the parabola.

c

Find the vertex of the parabola.

d

Is the parabola concave up or down?

e

Hence, sketch the graph of y = x^{2} + 4 x-1.

37

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 by completing the square.

c

Find the coordinates of the vertex of the parabola.

d

Hence, sketch the graph.

38

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

a

Determine the axis of symmetry.

b

Hence, determine the minimum value of y.

c

Sketch the graph of the function.

39

Consider the function P \left( x \right) = - 2 x^{2} - 8 x + 2.

a

Find the coordinates of the vertex.

b

Sketch the graph.

40

Consider the equation y = 6 x - x^{2}.

a

Find the x-intercepts of the quadratic function.

b

Find the coordinates of the turning point.

c

Sketch the graph.

41

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 for this curve.

c

Find the axis of symmetry.

d

Find the y-coordinate of the vertex of the parabola.

e

Sketch the graph.

42

Use your calculator or other handheld technology to graph the equations below. Then answer the following questions:

i

What is the vertex of the graph?

ii
What is the y-intercept?
a

y = 4 x^{2} - 64 x + 263

b

y = - 4 x^{2} - 48 x - 140

43

Use your calculator or other handheld technology to graph y = - 3 x^{2} - 12.

a

What is the vertex of the graph?

b

Are there any x-intercepts?

c

For what values of x is the parabola decreasing?

44

Using technology, graph the curve y = x^{2} + 6.2 x - 7.

a

Determine the axis of symmetry.

b

Determine the minimum value of y.

45

Use technology to graph the parabola y = - 2 x^{2} + 16 x - 24.

a

Find the x-intercepts of the parabola.

b

Find the y-intercept of the parabola.

c

Find the axis of symmetry of the parabola.

d

Find the y-coordinate of the vertex of the parabola.

46

Consider the function y = - 0.72 x^{2} + \sqrt{5} x + 1.21.

a

Find the x-coordinate of the vertex to two decimal places.

b

Find the y-coordinate of the vertex to two decimal places.

c

Is the graph shown a possible viewing window on your calculator that shows the vertex and the x-intercepts?

d

Use a graphing calculator or technology to find the x-intercepts to two decimal places.

47

Consider the function y = 0.91 x^{2} - 5 x - \sqrt{5}.

a

Find the x-coordinate of the vertex to two decimal places.

b

Find the y-coordinate of the vertex to two decimal places.

c

Is the graph shown a possible viewing window on your calculator that shows the vertex and the x-intercepts?

d

Use a graphing calculator or technology to find the x-intercepts to two decimal places.

Outcomes

0606C2.1

Find the maximum or minimum value of the quadratic function f : x ↦ ax^2 + bx + c by any method.

0606C2.2

Use the maximum or minimum value of f(x) to sketch the graph or determine the range for a given domain.

0606C8.2

Transform given relationships, including y = ax^n and y = Ab^x, to straight line form and hence determine unknown constants by calculating the gradient or intercept of the transformed graph.