13.1 Cubic, reciprocal and exponential graphs — Learn

Prerequisites: You can plot coordinates, recognise quadratic graphs and substitute values into expressions.

Objectives: You will sketch cubic, reciprocal and exponential graphs and interpret their intercepts, asymptotes and growth.

Cubic graphs

A cubic rule has highest power three. The graph of $y=x^3$ falls to the left and rises to the right. To find an $x$-intercept, set $y=0$ and solve.

Example 1. Find the intercepts of $y=x^3-4x$.

Set $y=0$:

Factor out $x$:

Factor the difference of squares:

So the intercepts are $(-2,0),(0,0),(2,0)$. The $y$-intercept is found by setting $x=0$.

Reciprocal graphs

A reciprocal rule has the form $y=\frac{k}{x}$. The curve has two branches and approaches both axes. An asymptote is a line that a curve approaches without reaching.

Example 2. Find values for $y=\frac1x$ at $x=-2,-1,1,2$.

Substitute each input:

The rule has no value when $x=0$, so its branches do not join across that line.

Exponential graphs

An exponential rule has the variable in the power, such as $y=2^x$. For each unit increase in $x$, the output is multiplied by the same factor.

Example 3. Find values of $y=2^x$ for $x=-1,0,1,2$.

Substitute each input:

The graph passes through $(0,1)$ and increases by a factor of two each step right.

Watch out. The reciprocal rule is undefined at $x=0$. Leave a gap there; joining the branches would include points that do not satisfy the rule.

Progress check

The chapter signal crosses its middle value while rising, then reaches a peak. Which graph family could model this repeating pattern? Give one reason.

You should now be able to

• sketch cubic, reciprocal and exponential graphs using key points;
• find intercepts by setting a coordinate to zero;
• identify reciprocal asymptotes and exponential growth;
• interpret how a graph changes as its input changes.