Linear and Quadratic Functions and Their Graphs

Linear functions

A linear function has the form

Its graph is always a straight line:

• $a$ is the slope — the change in $f(x)$ for a one-unit increase in $x$. A positive slope means the line rises left to right; a negative slope means it falls.
• $b$ is the intercept — the value of $f(x)$ when $x = 0$ (where the line crosses the vertical axis).

To sketch a linear function, only two points are needed — e.g. the intercept $(0, b)$ and one more point obtained by substituting any convenient value of $x$.

Quadratic functions

A quadratic function has the form

Its graph is always a parabola — U-shaped if $a > 0$ (opens upward, has a minimum point) and an inverted U if $a < 0$ (opens downward, has a maximum point).

Vertex (turning point). The parabola's vertex — its minimum (if $a>0$) or maximum (if $a<0$) point — occurs at

In business problems (maximum revenue, maximum profit) it is usually enough to find $x = -\dfrac{b}{2a}$ and then substitute this value back into $f(x)$ directly to get the maximum or minimum value.

Roots (zeros). The values of $x$ for which $f(x) = 0$ are found either by factorising $ax^2+bx+c$, or, when it does not factorise neatly, by the quadratic formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ …