Linear Functions and Their Graphs

A linear function is a function whose graph is a straight line. Many linear functions are written in the form y = mx + c. In this form, m tells us the gradient, or steepness, of the line, and c tells us where the line crosses the y-axis. This point is called the y-intercept.

What you will learn

• Recognise a linear function
• Identify the general form y = mx + c
• Explain the meaning of m and c
• Identify c as the y-intercept
• Recognise horizontal, vertical, and sloping straight lines
• Make a table of values for a linear function
• Draw the graph of a linear function from a table of values

Key Vocabulary

Term	Meaning
Linear function	A function whose graph is a straight line.
Gradient	The steepness of a line. In y = mx + c, the gradient is m.
y-intercept	The point where the graph crosses the y-axis.
c	In y = mx + c, c is the y-intercept.
Horizontal line	A line written in the form y = c. The value of y is constant.
Vertical line	A line written in the form x = k. The value of x is constant.

Key Ideas

The Form y = mx + c

The common form of a linear function is:

In this equation:

• m is the gradient.
• c is the y-intercept.

The y-intercept is the point where the line crosses the y-axis. At this point, the value of x is always 0. So the y-intercept is usually written as the point (0, c).

Graph Showing c as the y-intercept

In the graph above, the equation is y = x + 2. This matches the form y = mx + c, where:

• m = 1, so the gradient is 1.
• c = 2, so the y-intercept is 2.
• The graph crosses the y-axis at (0, 2).

Finding c from an Equation

To find c, look for the constant term in the equation y = mx + c.

Equation	Gradient m	y-intercept c	Point where graph crosses y-axis
y = 2x + 3	2	3	(0, 3)
y = -x + 4	-1	4	(0, 4)
y = 5x - 6	5	-6	(0, -6)
y = x	1	0	(0, 0)

Types of Straight Lines

Type of Line	Form	Example	Description
Sloping line	y = mx + c	y = 2x - 6	The line rises or falls as x changes.
Horizontal line	y = c	y = 4	The value of y is always the same.
Vertical line	x = k	x = 3	The value of x is always the same.

Diagrams of Different Straight Lines

How to Draw a Linear Function

To draw the graph of a linear function, use a table of values.

Worked Examples

Example 1: Identify m and c in y = 3x + 5.

1. Compare y = 3x + 5 with y = mx + c.
2. The coefficient of x is 3, so m = 3.
3. The constant term is 5, so c = 5.
4. The graph crosses the y-axis at (0, 5).

Answer: m = 3, c = 5, and the y-intercept is (0, 5).

Example 2: Make a table of values for y = 2x - 6 when x = 0, 1, 3.

x	Substitution	y	Ordered Pair
0	y = 2(0) - 6	-6	(0, -6)
1	y = 2(1) - 6	-4	(1, -4)
3	y = 2(3) - 6	0	(3, 0)

Answer: The points are (0, -6), (1, -4), and (3, 0).

Example 3: State the y-intercept of y = 2x - 6.

1. The equation is in the form y = mx + c.
2. In y = 2x - 6, the value of c is -6.
3. So the graph crosses the y-axis at (0, -6).

Answer: The y-intercept is (0, -6).

Example 4: What kind of line is y = 4?

1. There is no x term.
2. This means y is always 4.
3. The graph is a horizontal line crossing the y-axis at (0, 4).

Answer: y = 4 is a horizontal line.

Example 5: What kind of line is x = 3?

1. The equation says that x is always 3.
2. The graph is a vertical line through x = 3.
3. This line is not a function of x because one input has many outputs.

Answer: x = 3 is a vertical line.

Recognising Linear and Non-Linear Functions

Equation	Linear?	Reason
y = 3x + 1	Yes	It is in the form y = mx + c.
y = -2x + 5	Yes	It has only the first power of x.
y = x² + 1	No	It contains x², so the graph is not a straight line.
y = 4	Yes	It is a horizontal straight line.
x = 3	Straight line, but not a function of x	It is vertical and fails the vertical line test.

Common Mistakes

Multiple Choice Questions

Practice Questions

1. Write the general form of a linear function.
2. In y = mx + c, what does m represent?
3. In y = mx + c, what does c represent?
4. State the gradient and y-intercept of y = 4x - 7.
5. State the y-intercept of y = 2x + 3.
6. Make a table of values for y = x + 2 when x = 0, 1, 2.
7. What type of line is x = 3?
8. What type of line is y = -2?
9. State whether y = x² + 3 is linear or non-linear.
10. Explain how to draw the graph of a linear function from a table of values.

Answers

1. y = mx + c
2. m represents the gradient.
3. c represents the y-intercept.
4. Gradient = 4, y-intercept = -7. The graph crosses the y-axis at (0, -7).
5. The y-intercept is 3. The graph crosses the y-axis at (0, 3).

6. x	0	1	2
   y = x + 2	2	3	4

   Ordered pairs: (0, 2), (1, 3), (2, 4).
7. x = 3 is a vertical line.
8. y = -2 is a horizontal line.
9. y = x² + 3 is non-linear because it contains x².
10. Choose values of x, calculate the matching y-values, plot the ordered pairs, and join the points with a straight line.

Lesson Summary