Topic guide

A cubic graph is produced by a function where the highest power of $x$ is $3$, for example $y = x^3 - 3x + 2$.

The shape of a cubic graph

The simplest cubic graph is $y = x^3$. As $x$ gets larger, $y$ gets larger very quickly. As $x$ gets more negative, $y$ gets more negative very quickly. This produces a smooth curve that generally runs from the bottom-left to the top-right.

More complex cubics may have zero or two turning points, but they always have a point of inflection (where the curve changes from bending one way to bending the other). A cubic with a negative leading coefficient, such as $y = -x^3$, will run from the top-left to the bottom-right.

Worked example: Plotting a cubic graph

Question: Complete the table of values for $y = x^3 - 3x + 2$ and plot the graph for values of $x$ from $-3$ to $3$.

Step 1: Calculate the missing values
Substitute each $x$-value into the equation carefully.

• When $x = -3$: $y = (-3)^3 - 3(-3) + 2 = -27 + 9 + 2 = -16$
• When $x = -2$: $y = (-2)^3 - 3(-2) + 2 = -8 + 6 + 2 = 0$
• When $x = 0$: $y = (0)^3 - 3(0) + 2 = 0 - 0 + 2 = 2$
• When $x = 2$: $y = (2)^3 - 3(2) + 2 = 8 - 6 + 2 = 4$

Step 2: Plot the coordinates
Plot the pairs $(-3, -16)$, $(-2, 0)$, $(-1, 4)$, $(0, 2)$, $(1, 0)$, $(2, 4)$, and $(3, 20)$ on the coordinate grid.

Step 3: Draw a smooth curve
Join the plotted points with a single, smooth continuous curve. Do not use a ruler to join them with straight lines.

Common mistakes

• Sign errors with negative numbers: Remember that $(-2)^3$ is $-8$, not $8$. Odd powers preserve the negative sign. Also, subtracting a negative number is equivalent to adding a positive number.
• Drawing straight lines: A cubic graph is a curve. Do not join the plotted points with straight line segments using a ruler.
• Sharp turning points: Any turning points on a cubic curve should be rounded and smooth, not sharp corners.
• Misplotted points: If one point breaks the smooth pattern of the curve, double-check your substitution for that $x$-value.