Cubic graphs

What a cubic looks like

A cubic function has $x^3$ as its highest power, e.g. $y = x^3 - 4x$ .
Its graph is a smooth curve with a distinctive "S" bend: it can rise, dip, then rise again.
A cubic can cross the $x$ -axis once, twice or three times, so it has up to three roots.
The ends point in opposite directions: for a positive $x^3$ coefficient the curve comes up from the bottom-left and heads up to the top-right.

A cubic curve y equals x cubed minus four x, crossing the x-axis at minus two, zero and two, with a local maximum then a local minimum.

Reading roots from a factored cubic

If a cubic is written as a product of factors, each factor gives a root — just like a quadratic:

$y = a(x - p)(x - q)(x - r) \;\Rightarrow\; \text{roots } x = p,\; q,\; r$

Set each factor to zero to find where the curve meets the $x$ -axis.
A repeated factor (e.g. $(x - 2)^2$ ) means the curve touches the axis there instead of crossing it.

The $y$ -intercept of a cubic

As always, set $x = 0$ and read off $y$ .
From factored form, that is $a \times (-p)(-q)(-r)$ — just multiply the constants.

Worked ExampleSketching a factored cubic

Sketch $y = (x + 2)(x - 1)(x - 3)$ , showing the intercepts.

Step 1 — Roots (set $y = 0$ )

Each factor zero gives $x = -2,\; 1,\; 3$ — three $x$ -intercepts.

Step 2 — $y$ -intercept (set $x = 0$ )

$y = (2)(-1)(-3) = 6 \;\Rightarrow\; (0, 6)$

Step 3 — Shape

The $x^3$ coefficient is positive, so the curve rises from the bottom-left, weaves through $(-2, 0)$ , $(1, 0)$ and $(3, 0)$ , crosses the $y$ -axis at $6$ , and heads up to the top-right.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Write down the roots of $y = x(x - 4)(x + 1)$ .

Merit

Find the roots and the $y$ -intercept of $y = (x - 2)(x + 1)(x + 3)$ .

Excellence

A cubic has roots at $x = -1$ , $x = 2$ (touching the axis) and passes through $(0, -8)$ . Find its equation in the form $y = a(x - p)(x - q)^2$ .

Cubic graphs

Covers AS91257

What a cubic looks like

A cubic function has $x^3$ as its highest power, e.g. $y = x^3 - 4x$ .
Its graph is a smooth curve with a distinctive "S" bend: it can rise, dip, then rise again.
A cubic can cross the $x$ -axis once, twice or three times, so it has up to three roots.
The ends point in opposite directions: for a positive $x^3$ coefficient the curve comes up from the bottom-left and heads up to the top-right.

A cubic curve y equals x cubed minus four x, crossing the x-axis at minus two, zero and two, with a local maximum then a local minimum.

Reading roots from a factored cubic

If a cubic is written as a product of factors, each factor gives a root — just like a quadratic:

$y = a(x - p)(x - q)(x - r) \;\Rightarrow\; \text{roots } x = p,\; q,\; r$

Set each factor to zero to find where the curve meets the $x$ -axis.
A repeated factor (e.g. $(x - 2)^2$ ) means the curve touches the axis there instead of crossing it.

The $y$ -intercept of a cubic

As always, set $x = 0$ and read off $y$ .
From factored form, that is $a \times (-p)(-q)(-r)$ — just multiply the constants.

Worked ExampleSketching a factored cubic

Sketch $y = (x + 2)(x - 1)(x - 3)$ , showing the intercepts.

Step 1 — Roots (set $y = 0$ )

Each factor zero gives $x = -2,\; 1,\; 3$ — three $x$ -intercepts.

Step 2 — $y$ -intercept (set $x = 0$ )

$y = (2)(-1)(-3) = 6 \;\Rightarrow\; (0, 6)$

Step 3 — Shape

The $x^3$ coefficient is positive, so the curve rises from the bottom-left, weaves through $(-2, 0)$ , $(1, 0)$ and $(3, 0)$ , crosses the $y$ -axis at $6$ , and heads up to the top-right.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Write down the roots of $y = x(x - 4)(x + 1)$ .

Merit

Find the roots and the $y$ -intercept of $y = (x - 2)(x + 1)(x + 3)$ .

Excellence

A cubic has roots at $x = -1$ , $x = 2$ (touching the axis) and passes through $(0, -8)$ . Find its equation in the form $y = a(x - p)(x - q)^2$ .

Cubic graphs

What a cubic looks like

Reading roots from a factored cubic

The yyy-intercept of a cubic

Step 1 — Roots (set y=0y = 0y=0)

Step 2 — yyy-intercept (set x=0x = 0x=0)

Step 3 — Shape

Test yourself

Practice by grade

Cubic graphs

What a cubic looks like

Reading roots from a factored cubic

The yyy-intercept of a cubic

Step 1 — Roots (set y=0y = 0y=0)

Step 2 — yyy-intercept (set x=0x = 0x=0)

Step 3 — Shape

Test yourself

Practice by grade

The $y$ -intercept of a cubic

Step 1 — Roots (set $y = 0$ )

Step 2 — $y$ -intercept (set $x = 0$ )

The $y$ -intercept of a cubic

Step 1 — Roots (set $y = 0$ )

Step 2 — $y$ -intercept (set $x = 0$ )