Graphs & functions

Graph questions reward recognition. If you know the shape of each standard function and what gradient and area *mean*, many questions become a few seconds of reading rather than algebra.

A straight line is $y=mx+c$, where $m$ is the gradient and $c$ the **$y$-intercept**. Between two points, $m=\frac{y_2-y_1}{x_2-x_1}$.

Parallel lines have equal gradients. Perpendicular gradients multiply to $-1$, i.e. $m_2=-\frac{1}{m_1}$.

Know the families: linear (straight), quadratic (parabola), cubic (S-shaped), reciprocal $y=\frac{1}{x}$ (two branches, axes as asymptotes), and exponential $y=k^x$ (through $(0,1)$, growing fast).

For a quadratic, the roots are where $y=0$; the curve is symmetric about the vertical line through its turning point.

On a real-context graph, the gradient is a rate of change and the area under the line often has physical meaning. You may be asked to *estimate* these for a curve.

On a distance–time graph the gradient is the speed. On a speed–time graph the gradient is the acceleration and the area underneath is the distance travelled.

• Using $-\frac{1}{m}$ for parallel lines instead of perpendicular ones.
• Reading a distance–time graph as if a flat section means "stopped going backwards" (flat means stationary).
• Forgetting that a quadratic with two roots is symmetric, so the turning point is at their midpoint.