GCSE Maths · Higher · Chapter 17: Quadratic equations · Lesson 1

Quadratic graphs and key points

Grades 4–7About 50 minutesCalculator allowed39 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/gcse-quadratic-graphs-and-key-points/

NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out $3 x^{2} + 1$ when $x = - 3$ .
[1 mark]
Answer
A2
From chapter 12
Two triangles are similar. A side of 8 cm matches 16 cm. Another side is 7 cm. Find its match.
[1 mark]
Answer
A3
From chapter 15
The equation $x^{3} + 4 x = 52$ has one solution. Between which two whole numbers is it?
[1 mark]
Answer
A4
From chapter 8
Factorise fully $12 x^{2} + 8 x$ .
[1 mark]
Answer

Grades 4–6

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find where the curve $y = x^{2} - 2 x - 35$ crosses the $x$ -axis, and find the coordinates of its turning point.

Hint Factorise and set

y = 0

. The turning point is halfway between the two roots.

B1
Find where the curve $y = x^{2} - 12 x + 20$ crosses the $x$ -axis, and find the coordinates of its turning point.
[4 marks]
Answer $x = \dots\dots$ and $\dots\dots$ $(\dots\dots, \dots\dots)$

Example 2

Find the turning point of $y = 21 - 4 x - x^{2}$ . Is it a maximum or a minimum?

Hint Write it as

- (x^{2} + 4 x - 21)

and factorise to find the roots.

B2
Find the turning point of $y = 32 + 4 x - x^{2}$ . Is it a maximum or a minimum? Give a reason.
[3 marks]
Answer $(\dots\dots, \dots\dots)$ $\dots\dots\dots\dots$

Grades 5–6

C1
Write down the coordinates of the point where $y = 3 x^{2} + 5 x - 8$ crosses the $y$ -axis.
[1 mark]
Answer $(\dots\dots, \dots\dots)$
C2
For $y = x^{2} - 5 x - 7$ , work out the value of $y$ when $x = - 3$ .
[2 marks]
Answer

C3
A curve crosses the $x$ -axis at $- 11$ and $3$ . Write down its line of symmetry.
[1 mark]
Answer
C4
Find where $y = x^{2} - 15 x + 54$ crosses the $x$ -axis.
[2 marks]
Answer
C5
Complete the table of values for $y = x^{2} - x - 3$ .
[2 marks]
Answer $x = - 1$ : $\dots$ $x = 1$ : $\dots$ $x = 3$ : $\dots$
C6
A curve $y = a (x + 2) (x - 6)$ crosses the $y$ -axis at $(0, 36)$ . Work out the value of $a$ .
[2 marks]
Answer

Grades 6–7

D1
Ravi says, "The turning point of $y = x^{2} - 10 x + 16$ is $(0, 16)$ , because that is where the curve crosses the $y$ -axis."
Explain why Ravi is wrong, and find the turning point.
[3 marks]
Answer $(\dots\dots, \dots\dots)$

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find where a quadratic curve crosses the axes.
I can find a turning point and say if it is a maximum or a minimum.
I can use a drawn curve to solve a different equation.

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.