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

Graphs of functions and linear functions

About 75 minutesNo calculator62 marks

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Date

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

A1
Needed today
Rearrange $3 x - 2 y = 8$ to make $y$ the subject.
[2 marks]
Answer $y = \dots\dots$
A2
Needed today
The point $(k, 19)$ lies on the line $y = 4 - 3 x$ . Work out $k$ .
[2 marks]
Answer $k = \dots\dots$
A3
Last lesson
$f (x) = 3 x + 1$ and $g (x) = x^{2} - 4$ . Work out $g f (- 2)$ .
[2 marks]
Answer $g f (- 2) = \dots\dots$
A4
From chapter 2
Make $x$ the subject of $3 (x - a) = b x + 2$ .
[2 marks]
Answer $x = \dots\dots$

Example, then your turn

Level 2

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

Example 1

Work out the gradient of the line through $A (- 3, 5)$ and $B (3, - 4)$ . Does the line go uphill or downhill from left to right?

Hint Gradient

= \frac{change in y}{change in x}

. Take both changes from

A

B

B1
Work out the gradient of the line through $P (- 2, 9)$ and $Q (4, - 6)$ , and the gradient of the line through $R (- 5, 1)$ and $S (3, 7)$ . Which of the two lines goes downhill from left to right?
[3 marks]
Answer $P Q$ : $\dots\dots R S$ : $\dots\dots$ downhill: $\dots\dots$

Example 2

Sketch the lines $y = 4 - 2 x$ and $y = - 1$ on the same axes. Mark the coordinates of each point where a line meets an axis.

Hint For

y = m x + c

, put

x = 0

and then

y = 0

. The line

y = - 1

is parallel to the

x

-axis.

B2
Sketch the lines $y = 3 x - 6$ and $x = - 2$ on the same axes. Mark the coordinates of each point where a line meets an axis.
[3 marks]

Example 3

The line $L$ has equation $5 x - 3 y + 15 = 0$ . Work out where $L$ meets each axis, and the gradient of $L$ .

Hint Put

x = 0

, then

y = 0

. For the gradient, make

y

the subject.

B3
The line $M$ has equation $2 x + 5 y - 20 = 0$ . Work out where $M$ meets each axis, and the gradient of $M$ .
[3 marks]
Answer $x$ -axis $(\dots, \dots) y$ -axis $(\dots, \dots)$ gradient $\dots\dots$

Example 4

A line has equation $\frac{x}{4} - \frac{y}{5} - 1 = 0$ . Write it in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers, and write down where it meets each axis.

Hint Multiply every term by

20

. The form

\frac{x}{a} + \frac{y}{b} = 1

shows the crossings:

(a, 0)

and

(0, b)

B4
Write $6 x - 5 y = 30$ in the form $\frac{x}{a} + \frac{y}{b} = 1$ . Hence write down where the line meets each axis.
[3 marks]
Answer $a = \dots\dots b = \dots\dots (\dots, \dots)$ and $(\dots, \dots)$

Example 5

A plumber charges $£ C$ for a job that lasts $h$ hours, where $C = 45 + 32 h$ . No job lasts more than $8$ hours. Work out the charge for a $3$ -hour job and the cost per hour of that job. State the fixed charge and the charge per hour, and sketch $C$ against $h$ .

Hint The constant term is the charge for

0

hours. The coefficient of

h

is the gradient: the extra charge for each hour.

B5
A phone plan costs $£ C$ in a month when $g$ gigabytes of data are used, where $C = 12 + 1.5 g$ . Work out the cost for $10$ gigabytes and for $40$ gigabytes, and the cost per gigabyte in each case. State the fixed monthly charge.
[4 marks]
Answer $£ \dots\dots$ , $£ \dots\dots$ ; per GB $£ \dots\dots$ , $£ \dots\dots$ ; fixed $£ \dots\dots$

Practice

Level 2

C1
For each line, say whether its gradient is positive, negative, zero or infinite. (i) $y = 5 - x$ (ii) $x = - 4$ (iii) $y = 2$ (iv) $3 y = x$
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$ (iii) $\dots\dots$ (iv) $\dots\dots$
C2
$f (x) = \frac{1}{2} x + 1$ with domain $- 2 \leq x \leq 4$ . Draw the graph of $y = f (x)$ on the grid. Then write down the range of $f (x)$ .
[3 marks]
Answer $\dots\dots \leq f (x) \leq \dots\dots$

C3
The line $2 x + 7 y = k$ passes through the point $(- 3, 2)$ . Work out the value of $k$ . Then work out where the line meets the $x$ -axis.
[3 marks]
Answer $k = \dots\dots (\dots, \dots)$
C4
Show that the points $A (- 2, - 1)$ , $B (1, 5)$ and $C (4, 11)$ lie on one straight line.
[2 marks]
C5
Write the equation $y = \frac{2}{3} x - 5$ in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.
[2 marks]
Answer $\dots\dots = 0$

Exam-style questions

AQA exam style

D1
Which line has gradient $- \frac{3}{2}$ ? Circle your answer. $2 x + 3 y = 63 x + 2 y = 63 x - 2 y = 62 x - 3 y = 6$
[1 mark]

D2
The points $A (p, 3 p)$ and $B (4, 2)$ lie on a straight line with gradient $4$ .
(a)
Work out the value of $p$ .
[3 marks]
Answer $p = \dots\dots$
(b)
The line meets the $y$ -axis at $C$ . Work out the coordinates of $C$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
Total for D2: 5 marks
D3
The line $L$ has equation $2 x - 5 y + 15 = 0$ . Sam says, "The gradient of $L$ is $2$ , and $L$ crosses the $y$ -axis at $(0, 15)$ ."
Sam is wrong.
(a)
Work out the gradient of $L$ .
[2 marks]
Answer $\dots\dots$
(b)
Write down where $L$ crosses the $y$ -axis, and explain Sam's mistake.
[2 marks]
Answer $(\dots, \dots)$
Total for D3: 4 marks
D4
The line $3 x + 5 y = 30$ meets the $x$ -axis at $P$ and the $y$ -axis at $Q$ . $O$ is the origin.
Not drawn accurately
(a)
Work out the coordinates of $P$ and of $Q$ .
[2 marks]
Answer $P (\dots, \dots) Q (\dots, \dots)$
(b)
Work out the area of triangle $O P Q$ .
[1 mark]
Answer $\dots\dots$ square units
(c)
A straight line through $O$ meets $P Q$ at $M$ and cuts triangle $O P Q$ into two triangles of equal area. Work out the gradient of $O M$ .
[3 marks]
Answer $\dots\dots$
Total for D4: 6 marks
D5
Water drains from a tank. The volume, $V$ litres, left in the tank $t$ minutes after the tap is opened is $V = 480 - 15 t$ . The model holds until the tank is empty.
(a)
Write down the volume when the tap is opened, and the rate at which the volume falls.
[2 marks]
Answer $\dots\dots$ litres; $\dots\dots$ litres per minute
(b)
Work out the value of $t$ when the volume is $180$ litres.
[2 marks]
Answer $t = \dots\dots$
(c)
Sketch the graph of $V$ against $t$ for the values of $t$ where the model holds. Mark where your graph meets the axes.
[2 marks]
Total for D5: 6 marks

Extension

Stretch

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

E1
$A$ is the point $(1, 1)$ and $B$ is the point $(3, 7)$ . $C$ is the point $(k, k^{2} - 2)$ and is not the same point as $B$ . $A$ , $B$ and $C$ lie on one straight line. Work out the coordinates of $C$ .
[4 marks]
Hint 1 · What to try
Work out the gradient of $A B$ . The gradient of $A C$ must be the same.
Hint 2 · The first line
Gradient $A B = 3$ , so $\frac{k ^{2} - 2 - 1}{k - 1} = 3$ .
Hint 3 · The full method
$k^{2} - 3 = 3 k - 3$ , so $k^{2} - 3 k = 0$ and $k = 0$ or $k = 3$ . $k = 3$ gives $(3, 7)$ , which is $B$ , so $k = 0$ and $C$ is $(0, - 2)$ .
Answer $(\dots\dots, \dots\dots)$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the gradient of a line from two points and say what its sign means.
I can sketch a straight line from any form of its equation, marking where it meets the axes.
I can use the form $\frac{x}{a} + \frac{y}{b} = 1$ and rewrite equations with integer coefficients.
I can use a linear model, explaining its gradient and fixed term and the domain where it holds.

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