AQA Level 2 Further Maths · Level 2 Certificate · Chapter 3: Algebra III · Lesson 5

Graphs of quadratic functions

About 85 minutesNo calculator62 marks

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The first two questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Write $x^{2} - 8 x + 13$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer $\dots\dots$
A2
Needed today
Solve $x^{2} + 2 x - 15 = 0$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A3
Last lesson
Work out the equation of the line through $(1, 6)$ and $(5, - 2)$ . Give it in the form $y = m x + c$ .
[2 marks]
Answer $y = \dots\dots$
A4
From chapter 1
Expand and simplify $(2 + 3)^{2}$ .
[2 marks]
Answer $\dots\dots$

Example, then your turn

Level 2

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

Example 1

For the graph of $y = x^{2} - 6 x + 15$ , work out the vertex, the equation of the line of symmetry and where the graph meets the $y$ -axis. Sketch the graph.

Hint Complete the square.

(x - p)^{2}

is never negative, so the least value is when the bracket is

0

B1
For the graph of $y = x^{2} + 12 x + 43$ , work out the vertex, the equation of the line of symmetry and where the graph meets the $y$ -axis.
[3 marks]
Answervertex $(\dots, \dots) x = \dots (\dots, \dots)$

Example 2

For the graph of $y = 5 + 6 x - x^{2}$ , write $y$ in the form $k - (x - p)^{2}$ . Hence write down the vertex and the line of symmetry, and sketch the graph.

Hint Take out

- 1

- (x^{2} - 6 x) + 5

. Complete the square inside the bracket.

B2
For the graph of $y = 8 x - x^{2} - 1$ , write $y$ in the form $k - (x - p)^{2}$ . Hence write down the vertex and say whether it is a maximum or a minimum point.
[3 marks]
Answer $y = \dots\dots (\dots, \dots)$ , $\dots\dots$

Example 3

Sketch the graph of $y = x^{2} - 2 x - 8$ . Mark the coordinates of the points where it meets the axes, and of its vertex.

Hint Factorise to find where

y = 0

. The vertex is halfway between the two roots.

B3
Sketch the graph of $y = x^{2} + 2 x - 24$ . Mark the coordinates of the points where it meets the axes, and of its vertex.
[4 marks]

Example 4

A quadratic curve has its vertex at $(2, - 5)$ and passes through $(0, 7)$ . Work out its equation in the form $y = a x^{2} + b x + c$ .

Hint A curve with vertex

(p, q)

has equation

y = a (x - p)^{2} + q

. Use the other point to find

a

B4
The curve in the diagram has its vertex at $(- 1, 4)$ and passes through $(1, - 4)$ . Work out its equation in the form $y = a x^{2} + b x + c$ .
[3 marks]
Answer $y = \dots\dots$

Example 5

A farmer has $40$ m of fencing to make three sides of a rectangular pen. A wall is the fourth side. The two sides at right angles to the wall are $x$ m long. Show that the area is $A = 40 x - 2 x^{2}$ , and work out the greatest possible area.

Hint The side parallel to the wall is

40 - 2 x

. Complete the square on

40 x - 2 x^{2}

B5
A shop's daily profit is $£ P$ when it charges $£ x$ for an item, where $P = 18 x - x^{2} - 45$ . Write $P$ in the form $k - (x - p)^{2}$ . Hence work out the greatest daily profit and the price that gives it. Work out the two prices at which the profit is $£0$ .
[4 marks]
Answer $£ \dots\dots$ at $£ \dots\dots$ ; $£ \dots\dots$ and $£ \dots\dots$

Practice

Level 2

C1
$f (x) = x^{2} + 2 x - 4$ with domain $- 4 \leq x \leq 1$ . Work out the range of $f (x)$ .
[3 marks]
Answer $\dots\dots \leq f (x) \leq \dots\dots$
C2
Write $x^{2} - 2 x - 6$ in the form $(x + a)^{2} + b$ . Hence work out the exact values of $x$ where the graph of $y = x^{2} - 2 x - 6$ meets the $x$ -axis.
[3 marks]
Answer $\dots\dots x = \dots\dots$

C3
Draw the graph of $y = x^{2} - 2 x - 2$ for $- 1 \leq x \leq 3$ on the grid. Then write down the range of $y$ for this domain.
[3 marks]
Answer $\dots\dots \leq y \leq \dots\dots$
C4
The equations of curves (i) and (ii) are two of $y = 7 x - x^{2}$ , $y = x^{2} - 7 x$ , $y = x^{2} - 7 x + 10$ and $y = 10 + 7 x - x^{2}$ . Choose the equation of each curve.
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Exam-style questions

AQA exam style

D1
Zak says, "The least value of $x^{2} + 6 x + 16$ is $16$ , because when $x = 0$ the value is $16$ ."
Zak is wrong.
(a)
Write $x^{2} + 6 x + 16$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer $\dots\dots$
(b)
Hence write down the least value, and the value of $x$ that gives it. Explain Zak's mistake.
[2 marks]
Answerleast value $\dots\dots$ when $x = \dots\dots$
Total for D1: 4 marks

D2
Here is a sketch of $y = f (x)$ , where $f (x)$ is a quadratic function. The graph meets the $x$ -axis at $A (- 2, 0)$ and at $B$ , and has a maximum point at $(1.5, 12.25)$ .
Not drawn accurately
(a)
Work out the coordinates of $B$ .
[1 mark]
Answer $(\dots, \dots)$
(b)
The equation $f (x) = k$ has exactly one solution. Write down the value of $k$ .
[1 mark]
Answer $k = \dots\dots$
(c)
The graph meets the $y$ -axis at $(0, 10)$ . Work out $f (x)$ in the form $a x^{2} + b x + c$ .
[3 marks]
Answer $f (x) = \dots\dots$
Total for D2: 5 marks
D3
The least value of $x^{2} + 8 x + c$ is $- 9$ .
(a)
Work out the value of $c$ .
[3 marks]
Answer $c = \dots\dots$
(b)
Hence work out where the graph of $y = x^{2} + 8 x + c$ meets the $x$ -axis.
[2 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$
(c)
Write down the values of $k$ for which $x^{2} + 8 x + c = k$ has two solutions.
[1 mark]
Answer $\dots\dots$
Total for D3: 6 marks
D4
A stone is thrown upwards from the top of a cliff. Its height above the sea, $h$ metres, $t$ seconds after it is thrown is $h = 35 + 30 t - 5 t^{2}$ .
(a)
Show that $h = 80 - 5 (t - 3)^{2}$ .
[2 marks]
(b)
Write down the greatest height of the stone and the time when it reaches it.
[2 marks]
Answer $\dots\dots$ m at $t = \dots\dots$
(c)
Work out the time when the stone reaches the sea.
[2 marks]
Answer $t = \dots\dots$
Total for D4: 6 marks

Extension

Stretch

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

E1
The curve $y = x^{2} + b x + 8$ meets the $y$ -axis at $(0, 8)$ . Its vertex lies on the line $y = 2 x$ . Work out the two possible values of $b$ , and the vertex of each curve.
[5 marks]
Hint 1 · What to try
Complete the square to write the vertex in terms of $b$ .
Hint 2 · The first line
$y = (x + \frac{b}{2})^{2} + 8 - \frac{b ^{2}}{4}$ , so the vertex is $(- \frac{b}{2}, 8 - \frac{b ^{2}}{4})$ . Put it into $y = 2 x$ .
Hint 3 · The full method
$8 - \frac{b ^{2}}{4} = - b$ , so $b^{2} - 4 b - 32 = 0$ and $(b - 8) (b + 4) = 0$ . $b = 8$ gives the vertex $(- 4, - 8)$ ; $b = - 4$ gives the vertex $(2, 4)$ .
Answer $b = \dots\dots$ , $(\dots, \dots)$ ; $b = \dots\dots$ , $(\dots, \dots)$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the vertex and line of symmetry of a parabola by completing the square.
I can sketch a quadratic graph, marking its intercepts and vertex.
I can find the equation of a parabola from its vertex, or its roots, and one other point.
I can use a quadratic model to find a greatest or least value.

Answers and mark scheme

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