Graphs are used to show the relationship between variables. Some of the most important straight line graphs at GCSE level are those of the form
\( x = a \), \( y = b \) and \( y = ax + b \).

 

Being able to draw, interpret, recognise and sketch these graphs is essential.

 

 

 

The Graph of x = a

The equation:

$$
x = a
$$

 

represents a vertical straight line.

• The value of \( x \) is fixed
• The value of \( y \) can be anything

 

For example:

$$
x = 3
$$

 

This is a vertical line that crosses the x axis at \( 3 \).

 

Every point on the line has an x coordinate of \( 3 \), such as:

$$
(3,\ 1),\ (3,\ -2),\ (3,\ 5)
$$

 

Key features:
• The line is parallel to the y axis
• It has no gradient

 

 

The Graph of y = b

The equation:

$$
y = b
$$

 

represents a horizontal straight line.

• The value of \( y \) is fixed
• The value of \( x \) can be anything

 

For example:

$$
y = -2
$$

 

This is a horizontal line that crosses the y axis at \( -2 \).

 

Every point on the line has a y coordinate of \( -2 \), such as:

$$
(1,\ -2),\ (-4,\ -2),\ (6,\ -2)
$$

 

Key features:
• The line is parallel to the x axis
• The gradient is zero

 

 

The Graph of y = ax + b

The equation:

$$
y = ax + b
$$

 

represents a straight line with a gradient.

• \( a \) is the gradient
• \( b \) is the y intercept

 

The y intercept is the point where the line crosses the y axis.

 

 

 

Understanding the Gradient a

The gradient \( a \) shows how steep the line is.

• If \( a \) is positive, the line slopes upwards from left to right
• If \( a \) is negative, the line slopes downwards from left to right
• A larger value of \( a \) gives a steeper line

 

For example:

$$
y = 2x + 1
$$

 

The gradient is \( 2 \), meaning the line goes up 2 units for every 1 unit moved to the right.

 

 

Understanding the Intercept b

The intercept \( b \) tells you where the line crosses the y axis.

 

For example:

$$
y = -x + 3
$$

 

The line crosses the y axis at:

$$
(0,\ 3)
$$

 

 

Sketching y = ax + b

To sketch a graph of the form \( y = ax + b \):

• Plot the y intercept \( (0,\ b) \)
• Use the gradient to find another point
• Draw a straight line through the points

 

For example, for:

$$
y = 3x - 2
$$

 

The y intercept is \( (0,\ -2) \).

Using the gradient of \( 3 \), another point is \( (1,\ 1) \).

 

 

Recognising Graphs from Equations

• \( x = a \) always gives a vertical line
• \( y = b \) always gives a horizontal line
• \( y = ax + b \) gives a sloping straight line

 

Recognising these forms quickly helps when interpreting graphs or sketching without full calculations.

 

 

Key Points to Remember

\( x = a \) gives a vertical line.
\( y = b \) gives a horizontal line.
\( y = ax + b \) gives a straight line with gradient \( a \) and intercept \( b \).
Straight line graphs extend infinitely in both directions.

 

Understanding these basic graphs is the foundation for all further work with linear graphs and coordinate geometry.