4.4 Sketching Cubic Graphs

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1. What is a Cubic Polynomial?

A cubic polynomial is an expression where the highest power of x is 3. For example:

$y = x^3 - 6x^2 + 11x - 6$

In this topic, we always work with cubics written as a product of three linear factors. A linear factor is simply a bracket of the form $(x - a)$, $(x + a)$, or $(ax + b)$ — basically a bracket with x to the power of 1.

So a cubic in factored form looks like:

$y = (x - a)(x - b)(x - c)$

or with a constant multiplier:

$y = k(x - a)(x - b)(x - c)$

where a, b, c are numbers and k is a constant (a fixed number that stretches or flips the graph).

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3. Finding the X-Intercepts (Roots)

The x-intercepts are the points where the graph meets the x-axis. At these points, $y = 0$.

So to find them, you set each factor equal to zero and solve:

$y = (x - a)(x - b)(x - c) = 0$

This means: $x = a$, $x = b$, or $x = c$

Each of these is called a root (or zero) of the polynomial — the x-value where the curve hits the x-axis.

Example:

$y = (x - 1)(x + 2)(x - 3)$

Set each factor to zero:

• $x - 1 = 0 \Rightarrow x = 1$
• $x + 2 = 0 \Rightarrow x = -2$
• $x - 3 = 0 \Rightarrow x = 3$

So the graph crosses the x-axis at $x = -2$, $x = 1$, and $x = 3$.

Label these on your sketch as the points $(-2, 0)$, $(1, 0)$, and $(3, 0)$.

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4. Repeated Roots — When Two Factors Are the Same

Sometimes two of the factors are identical. For example:

$y = (x - 1)^2(x - 3)$

This means $x = 1$ is a repeated root (it appears twice). When a root is repeated:

• The graph touches the x-axis at that point and bounces back — it does not cross through
• The graph crosses the x-axis at the non-repeated root as usual

So for $y = (x-1)^2(x-3)$:

• At $x = 1$: the curve touches the x-axis (like the bottom of a U-shape at that point)
• At $x = 3$: the curve crosses the x-axis