Standard Form

Today, we will introduce cubic functions, which are polynomial functions of degree 3. Can anyone tell me what that means?

Does it mean the highest power of x is 3?

Exactly! The general form is 𝑓(𝑥) = 𝑎𝑥³ + 𝑏𝑥² + 𝑐𝑥 + 𝑑, where 𝑎 is not zero. This ensures it’s indeed a cubic function.

What happens if a equals zero?

If 𝑎 = 0, then it's not a cubic function anymore; it won't have the degree of 3.

Remember this: Cubic functions always have that 'S-like' shape or inverted shape on the graph due to their degree. Let's hold onto that concept for our next discussion.

Let's dive deeper into the key features of cubic functions. What do you think determines the shape of the graph?

Is it the coefficients a, b, c, and d?

You're on the right track! The sign of 𝑎 affects the end behavior. If it's positive, the graph rises on both ends, and if it's negative, it falls. This S-shape is crucial!

What about the turning points?

Good question! A cubic function can have up to two turning points, which correspond to local maximum and minimum values. Identifying these can help us sketch the graph more accurately.

Now, let's summarize: Cubic functions have an S-shaped graph, can cross the x-axis up to three times, and depend on the sign of 𝑎 for their end behavior.

We have discussed the standard form of cubic functions. Who can tell me the standard form again?

It's 𝑓(𝑥) = 𝑎𝑥³ + 𝑏𝑥² + 𝑐𝑥 + 𝑑.

Exactly! Now, there's also the factored form, which helps us find the roots easier. What is the factored form?

It's 𝑓(𝑥) = 𝑎(𝑥 − 𝑟₁)(𝑥 − 𝑟₂)(𝑥 − 𝑟₃), where 𝑟₁, 𝑟₂, 𝑟₃ are the roots.

Perfect! Remember, roots can be real or complex. The factored form is especially useful for graphing because it gives us the x-intercepts directly. Keep this in mind as we move on to finding those roots!

Now that we know the forms, let's focus on how to find the roots of cubic equations. Who can remind us of the Rational Root Theorem?

It suggests possible rational roots based on the factors of the constant and leading coefficient!

Exactly! Using this theorem, we can guess possible rational roots. And what do we do next?

We can use synthetic division or long division to reduce it to a quadratic form.

Right! Once reduced to quadratic, we can apply the quadratic formula or factor it. Remember, finding roots is crucial since it tells us where our graph intersects the x-axis.

To graph a cubic function, we start with its end behavior and intercepts. What does the y-intercept tell us?

It's where the graph crosses the y-axis, which is found from 𝑓(0).

Correct! And how do we find the x-intercepts?

By solving 𝑓(𝑥) = 0 for x!

Exactly. After plotting the intercepts, we make sure to sketch the characteristic S-shape of a cubic function. Practice this by sketching 𝑓(𝑥) = 𝑥³ - 3𝑥² - 4𝑥 + 12 together.