Linear Graphs — y = mx + c
The Equation of a Straight Line
Every straight-line graph can be written as:
$$y = mx + c$$
- $m$ is the gradient (slope) — how steep the line is.
- $c$ is the y-intercept — where the line crosses the y-axis.
Example: For $y = 3x - 2$, the gradient is $3$ and the y-intercept is $(0, -2)$.
Calculating the Gradient
$$m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$$
Example: Find the gradient of the line through $(1, 4)$ and $(5, 12)$.
$$m = \frac{12 - 4}{5 - 1} = \frac{8}{4} = 2$$
Sign of gradient:
- Positive gradient → line goes up from left to right.
- Negative gradient → line goes down from left to right.
- Zero gradient → horizontal line.
- Undefined gradient → vertical line.
Finding the Equation of a Line
Given gradient $m$ and a point $(x_1, y_1)$:
Use $y - y_1 = m(x - x_1)$ and rearrange to $y = mx + c$.
Example: Gradient $= 3$, passes through $(2, 7)$.
$$y - 7 = 3(x - 2) \implies y = 3x - 6 + 7 = 3x + 1$$
Parallel Lines
Parallel lines have the same gradient.
- $y = 2x + 5$ and $y = 2x - 3$ are parallel (both have $m = 2$).
- A line parallel to $y = 4x - 1$ through $(0, 3)$ is $y = 4x + 3$.
Perpendicular Lines
Perpendicular lines have gradients whose product is $-1$:
$$m_1 \times m_2 = -1 \quad \Longleftrightarrow \quad m_2 = -\frac{1}{m_1}$$
This is called the negative reciprocal.
Example: A line has gradient $\frac{2}{3}$. A perpendicular line has gradient $-\frac{3}{2}$.
Example: Find the equation of the line perpendicular to $y = 5x - 4$ passing through $(5, 3)$.
Gradient of given line: $5$. Perpendicular gradient: $-\frac{1}{5}$.
$$y - 3 = -\frac{1}{5}(x - 5) \implies y = -\frac{1}{5}x + 1 + 3 = -\frac{1}{5}x + 4$$
Drawing a Straight-Line Graph
- Choose 3 values of $x$, calculate $y$.
- Plot the three points.
- Draw a straight line through them.
WJEC Exam Tips
- Always label axes and the equation of the line.
- To find where two lines intersect, solve simultaneously.
- "Show the gradient" means draw a right-angled triangle on your line and calculate rise ÷ run.
- Watch for lines written as $ax + by = c$ — rearrange to $y = mx + c$ first.
AI-generated · claude-opus-4-7 · v3-wjec-maths