Linear and Quadratic Functions

Linear functions have the form $f(x) = mx + c$, where $m$ is the gradient (slope) and $c$ is the $y$-intercept. The gradient measures steepness: between two points $(x_1, y_1)$ and $(x_2, y_2)$ it equals $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. Parallel lines share the same gradient; perpendicular lines have gradients whose product is $-1$, so $m_1 m_2 = -1$. A line can also be written in point-gradient form $y - y_1 = m(x - x_1)$, which is the fastest way to build an equation from a point and a slope.

Worked example. Find the line through $(1, 5)$ perpendicular to $y = 2x - 3$. The given gradient is $2$, so the perpendicular gradient is $-\tfrac{1}{2}$. Point-gradient form gives $y - 5 = -\tfrac{1}{2}(x - 1)$, which simplifies to $y = -\tfrac{1}{2}x + \tfrac{11}{2}$.

Forms of a quadratic

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