Lesson 115: Graphing Cubic Functions

New Concept

A cubic function is a polynomial function in which the greatest power of any variable is 3.

What’s next

Next, you'll see how these functions behave by graphing the parent function, $y=x^3$, and using its graph to solve equations.

Property

A cubic function is a polynomial function of degree 3. Its parent function is $y = x^3$.

Explanation

Think of cubics as the cool, curvy cousins of quadratics. Instead of a simple U-shape from $x^2$, cubics are powered by $x^3$, creating a signature S-shaped graph with ends pointing in opposite directions.

Examples

• The parent function is the simplest cubic: $y = x^3$.
• A transformed cubic function could look like: $y = -2x^3 - 9$.
• A cubic function in standard form: $y = 10x^3 + 3x^2 - 5$.

Property

To solve a cubic equation, rewrite it so one side equals zero. The solutions are the x-intercepts of the related function's graph.

Explanation

Got a tough cubic equation? Turn it into a picture! Shove everything to one side to equal zero, graph that new function, and find where it crosses the x-axis. Those x-intercepts are your answers.

Examples

• To solve $x^3 - 1 = 0$, graph $y = x^3 - 1$ and find the x-intercept at $x = 1$.
• To solve $2 = -2x^3 - 7$, first rewrite it as $0 = -2x^3 - 9$, then graph $y = -2x^3 - 9$.
• Or, solve $2 = -2x^3 - 7$ by graphing $y=2$ and $y=-2x^3-7$ to find their intersection.

What if an equation's solution isn't a neat integer? Graphing gives us a powerful way to find it. This demonstrates our second key idea: solving cubic equations by graphing.

Example Problem

Solve the equation $4 = -2x^3 - 4$ by graphing.

Property

The zeros of a function are its x-intercepts or solutions.

Explanation

Zeros are just a cool math name for where a graph crosses the x-axis. They're called 'zeros' because at these exact spots, the function's y-value is zero. Think of it as the graph's touchdown point!

Examples

• If a graph crosses the x-axis at $x = -2$, then $-2$ is a zero of the function.
• The function $y = x^3 - 8$ has a zero at $x = 2$, because $2^3 - 8 = 0$.
• Finding the zeros of $y = x^3 + 1$ is the same as solving the equation $0 = x^3 + 1$.