2.1 Linear Functions And Graphs

Learning Objectives

Use coordinates and tables of values to represent linear relationships.
Plot straight-line graphs accurately from equations and practical data.
Interpret gradient as a constant rate of change and identify intercepts.
Use linear graphs to estimate unknown values and solve contextual problems.
Distinguish a linear function from a non-linear relationship.

Key Terms

Function: A rule that assigns exactly one output to each permitted input.
Coordinate: An ordered pair $(x,y)$ that locates a point on a Cartesian grid.
Linear function: A function whose graph is a straight line.
Gradient: The constant rate of change of $y$ with respect to $x$ .
Intercept: A point where a graph crosses an axis.
Independent variable: The input variable, normally placed on the horizontal axis.
Dependent variable: The output variable, normally placed on the vertical axis.

8th Edition Chapter Map

Coordinates, variables and function rules
Tables of values and plotting straight lines
Gradient and intercepts
Practical linear graphs and interpolation
Recognising linear and non-linear patterns

From A Rule To A Graph

A function connects an input to an output. In the rule $y=3x-2$ , a chosen value of $x$ is multiplied by 3 and then 2 is subtracted to produce $y$ . Each input gives one output, so ordered pairs such as $(0,-2)$ , $(1,1)$ and $(2,4)$ lie on the graph.

A linear function has a constant rate of change. Equal increases in $x$ produce equal increases or decreases in $y$ . Its graph is a straight line. The general form is $y=mx+c$ , where $m$ is the gradient and $c$ is the y-intercept.

y=mx+c

The gradient is $m$ and the graph crosses the y-axis at $(0,c)$ .

Making A Table Of Values

Choose suitable $x$ -values, substitute each one into the equation and calculate the matching $y$ -value. Negative inputs must be placed in brackets when powers or signs are involved. At least two correct points determine a straight line, but a third point is a useful accuracy check.

Worked Example: Complete A Table

Question: Complete a table for $y=2x+3$ when $x=-2,-1,0,1,2$ .

$x=-2\Rightarrow y=2(-2)+3=-1$ .
$x=-1\Rightarrow y=1$ .
$x=0\Rightarrow y=3$ .
$x=1\Rightarrow y=5$ and $x=2\Rightarrow y=7$ .

Answer: The coordinate pairs are $(-2,-1),(-1,1),(0,3),(1,5),(2,7)$ .

Plotting Accurately

Label both axes, show units where appropriate and use a uniform scale. Plot points as small crosses rather than large dots. Join points belonging to a linear function with a ruled straight line. Do not join unrelated discrete data unless the context allows intermediate values.

When a graph represents a continuous quantity such as distance, time or temperature, interpolation between plotted points is usually meaningful. Extrapolation beyond the observed range is less reliable because the relationship may change.

Gradient As Rate Of Change

The gradient tells how much $y$ changes for each increase of 1 in $x$ . For two points $(x_1,y_1)$ and $(x_2,y_2)$ , calculate the vertical change and divide by the horizontal change. The order used in the numerator and denominator must be consistent.

m=\frac{y_2-y_1}{x_2-x_1}

Worked Example: Gradient From Two Points

Question: Find the gradient of the line through $(-3,8)$ and $(5,-4)$ .

Change in $y=-4-8=-12$ .
Change in $x=5-(-3)=8$ .
$m=-12/8=-3/2$ .

Answer: The gradient is $-\frac32$ .

Understanding The Sign Of A Gradient

Type of line	Gradient	Meaning
Rises from left to right	Positive	The output increases as the input increases.
Falls from left to right	Negative	The output decreases as the input increases.
Horizontal	Zero	The output is constant.
Vertical	Undefined	The horizontal change is zero, so division by zero would be required.

Intercepts

The y-intercept is found by setting $x=0$ . For $y=4x-7$ , the y-intercept is $-7$ . The x-intercept is found by setting $y=0$ . In the same example, $0=4x-7$ , so $x=7/4$ .

Practical Linear Graphs

A conversion graph may connect two scales such as kilometres and miles, or one currency and another. A straight line through the origin represents direct proportionality. A fixed starting charge produces a non-zero intercept. For example, a taxi fare of 200 plus 75 per kilometre can be modelled by $C=75d+200$ . The gradient is the price per kilometre and the intercept is the fixed charge.

Worked Example: Interpret A Cost Function

Question: A printing service charges 450 fixed setup cost plus 18 per booklet. The cost is $C=18n+450$ . Find the cost of 35 booklets and explain the two constants.

$C=18(35)+450=630+450=1080$ .
The gradient 18 is the additional cost per booklet.
The intercept 450 is the fixed setup cost when no booklet has yet been printed.

Answer: The cost is 1080.

Recognising A Linear Pattern

In a table, a relationship is linear if equal changes in $x$ give equal first differences in $y$ . A curved graph or changing first difference indicates a non-linear function. A rule such as $y=x^2+1$ is not linear because the power of $x$ is 2.

Examination Guidance

Use a ruler for a straight-line graph and a smooth curve for a non-linear graph.
Write units beside numerical answers obtained from practical graphs.
Read graph values using guide lines and give only the accuracy justified by the scale.
When finding gradient from a graph, choose two well-separated points on the line, not necessarily the original plotted points.

Common Mistakes

Interchanging $x$ and $y$ when plotting a coordinate.
Using unequal intervals without clearly marking the scale.
Calculating gradient as horizontal change divided by vertical change.
Assuming every straight line shows direct proportion; it must also pass through the origin.

Knowledge Check And Practice

1. For $y=5-2x$ , find $y$ when $x=-3$ .

Answer:

y=5-2(-3)=11

2. State the gradient and y-intercept of $y=7x+4$ .

Answer: Gradient 7 and y-intercept 4.

3. Find the x-intercept of $y=3x-12$ .

Answer: Set

y=0

3x=12

, so

x=4

4. A line passes through $(2,5)$ and $(8,20)$ . Find its gradient.

Answer:

(20-5)/(8-2)=15/6=5/2

5. Explain why $y=4x+3$ is not a direct proportion.

Answer: Its graph does not pass through the origin because the y-intercept is 3.

Extended Worked Practice

Equation Of A Line Through Two Points

Question: Find the equation of the line through $(2,5)$ and $(8,17)$ .

Calculate the gradient: $m=\frac{17-5}{8-2}=\frac{12}{6}=2$ .
Use $y=mx+c$ and substitute $(2,5)$ : $5=2(2)+c$ .
So $c=1$ .

Answer: $y=2x+1$ . Check with the second point: $2(8)+1=17$ .

Interpreting A Linear Model

Question: A tank contains 900 litres and is drained at 35 litres per minute. Write a formula for the volume $V$ after $t$ minutes and find when 375 litres remain.

The initial value is 900 and the rate of change is $-35$ , so $V=900-35t$ .
Set $V=375$ : $375=900-35t$ .
$35t=525$ , hence $t=15$ .

Answer: $V=900-35t$ , and 375 litres remain after 15 minutes.

Testing Collinearity

Question: Determine whether $A(1,4)$ , $B(4,10)$ and $C(7,16)$ lie on one straight line.

$m_{AB}=\frac{10-4}{4-1}=2$ .
$m_{BC}=\frac{16-10}{7-4}=2$ .
The gradients are equal and the segments share point $B$ .

Answer: The three points are collinear.

Maths Syllabus D Complete Subject Notes (O Level | 4024) 2.1 Linear Functions And Graphs

Maths Syllabus D Complete Subject Notes (O Level | 4024)