Straight Line Graphs

Spec reference: Algebra - Graphs
Key idea: Plot and interpret straight line graphs; find gradients and equations of lines.

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$y=mx+c$

The equation of a straight line is $y=mx+c$ where:

• $m$ is the gradient (steepness)
• $c$ is the y-intercept (where the line crosses the y-axis)

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Gradient

$$m=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}$$

• Positive gradient: line goes up left to right
• Negative gradient: line goes down left to right

Example

Find the gradient between $(1,3)$ and $(4,9)$.

$$m=\frac{9-3}{4-1}=\frac{6}{3}=\boxed{{\displaystyle 2}}$$

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Drawing a straight line

Method 1: Use a table of values.

Example

Draw $y=2x-1$

$x$	0	1	2	3
$y$	-1	1	3	5

Plot the points and join them.

Method 2: Use gradient and y-intercept directly.

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Finding the equation of a line

Example

A line has gradient 3 and passes through $(2,7)$.

$$y=mx+c$$$$7=3(2)+c$$$$c=7-6=1$$$$\boxed{{\displaystyle y=3x+1}}$$

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Parallel and perpendicular lines

• Parallel lines have the same gradient
• Perpendicular lines have gradients that multiply to $-1$: $m_1\times m_2=-1$

Example

A line has gradient 2. A perpendicular line has gradient $-{\displaystyle \frac{1}{2}}$.

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Special lines

Line	Equation
Horizontal line through $(0,k)$	$y=k$
Vertical line through $(k,0)$	$x=k$

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Test Yourself

Question 1 of 5

Gradient of $y = 3x - 7$?