Linear, Cubic and Reciprocal Graphs

Recognising a graph from its equation

A graph shows all the coordinate pairs $(x,y)$ that make an equation true. To sketch one, show its important shape and position: it need not be plotted to the precision of a full table-of-values graph. A few carefully chosen points can help you place the shape. The highest power of $x$ is often a clue to the family of graph.

Straight-line graphs

A linear equation such as $y=2x$ gives a straight line. Each increase of 1 in $x$ increases $y$ by 2, so points such as $(0,0)$ and $(1,2)$ fix the line. More generally, $y=mx+c$ is straight: $m$ tells you its gradient, or steepness, and $c$ tells you where it crosses the $y$-axis. If $m=0$, the line is horizontal: $y=2$ stays at height 2 for every $x$.

The equation $x=1$ also gives a straight line, but it is vertical: every point on it has $x$-coordinate 1. It is not in the form $y=mx+c$, because one value of $x$ has many possible -values.

Quadratic graphs

A quadratic such as $y=x^2$ makes a smooth U-shaped curve called a parabola. Squaring $-2$ and $2$ gives the same value, 4, so $(-2,4)$ and lie at the same height. The passes through and rises on both sides. A negative coefficient of , as in , turns the U upside down. Other can be shifted, but keep this smooth curved shape. Locating their intercepts and turning points precisely is studied separately.

The basic cubic graph

For $y=x^3$, cubing a negative number gives a negative result and cubing a positive number gives a positive result. The graph passes through $(-1,-1)$, $$ and . It curves from the bottom left to the top right, flattening as it passes through the origin rather than forming a corner there. This is the basic cubic shape to recognise here; not every cubic equation has exactly the same appearance.

The reciprocal graph

In $y=\frac{1}{x}$, each $y$-value is the reciprocal of its $x$-value: for example, $x=$ gives , while gives . Positive gives positive , and negative gives negative . The therefore has two separate curved branches, one in the top-right quadrant and one in the bottom-left.

You cannot divide by zero, so the graph has no point with $x=0$. Also, $\frac{1}{x}$ is never zero, so it has no point on the $x$-axis. Each branch gets closer to the axes without crossing them: the axes are asymptotes. Sketch the branches separately, never joined through the origin.

Recognise the shapes

• $y=mx+c$: straight line; $y=c$ is horizontal. $x=c$ is vertical and is not of the form .