Linear Graphs
Year 8 📊 Algebra Plot straight line graphs, find gradient and y-intercept.
📈 Gradient and y-intercept
The equation $y = mx + c$ describes a straight line, where $m$ is the gradient and $c$ is the y-intercept.
⚡ Gradient Formula
$$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$$Example: Line through $(1, 3)$ and $(4, 9)$: $m = \frac{9-3}{4-1} = \frac{6}{3} = 2$. Using $(1,3)$: $3 = 2(1)+c$ so $c=1$. Equation: $y = 2x + 1$
✏️ Drawing Straight-Line Graphs
Plot a straight line by finding at least two coordinate pairs (three for safety).
- Choose two or three values of $x$ (e.g. 0, 2, 4).
- Substitute into the equation to find each $y$ value.
- Plot the $(x, y)$ pairs and draw a ruled line through them.
Example: Draw $y = 3x - 1$: when $x=0$, $y=-1$; when $x=2$, $y=5$. Plot and draw the line.
↔️ Parallel and Perpendicular Lines
Lines with the same gradient are parallel. The gradients of perpendicular lines multiply to give $-1$.
⚡ Perpendicular Gradient
If one line has gradient $m$, a perpendicular line has gradient $-\dfrac{1}{m}$Example: A line with gradient 3 is perpendicular to a line with gradient $-\frac{1}{3}$, since $3 \times -\frac{1}{3} = -1$.
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