# MathMaster Blog

aх^3 + bх^2 + cx + d = 0 is a generalized form of a cubic equation. The graph of a cubic function can have a maximum of 3 roots.

The basic cubic function $which is also known as the parent cubic function$ is f$x$ = х^3

• The domain and range of a cubic function is the set of all real numbers.
• To find the x-intercept$s$ of a cubic function, we just substitute y = 0 $or f(x$ = 0) and solve for x-values.
• To find the y-intercept of a cubic function, we just substitute x = 0 and solve for the y-value.
• To find the critical points of a cubic function f$x$ = aх^3 + bх^2 + cx +d = 0, we set the first derivative to zero and solve via quadratic equation formula.

For instance:

The end behavior of the cubic function is as follows:

• If the leading coefficient is positive $a > 0$:

The graph is from bottom to top in this case.

• If the leading coefficient is negative $a < 0$:

The graph is from top to bottom.

## Example:

Graph f$x$ = х^3 - 4х^2 + x - 4 .

### Solution:

Step1:

Find the x-intercept$s$. To do this, substitute f$x$ = 0. Then, solve:

х^3 - 4х^2 + x - 4 = 0

х^2$x - 4$ + 1$x - 4$ = 0

$x - 4$$х^2 + 1$ = 0

x- 4 = 0; х^2 + 1 = 0

x = 4; х^2 = -1

x = 4; x = +- i

As complex numbers can’t be x-intercepts, f$x$ has only one x-intercept - $4, 0$.

Step2:

Find the y-intercept. To do this, substitute x = 0.

Then, f $x$ = -$0$^3 + 4$0$^2 - 4 = -4

Therefore, the y-intercept of the function is $0, -4$.

Step3:

Find the critical point$s$ by setting f'$x$ = 0.

3x^2 - 8x + 1 = 0

By quadratic formula, $$-8 \pm \sqrt{64 - 12}$ \div 6 \approx 0.131 and 2.535$

Step4:

Substitute each of the critical points' y-coordinates into the above function to find the matching y-coordinate$s$.

f$0.131$ $\approx$ -3.935

f$2.535$ $\approx$ -10.879

So, the critical points are $0.131, -3.935$ and $2.535, -10.879$.

Step5:

Find the end behavior of the function.

Since the leading coefficient of the function is 1, its end behavior is

f$x$ -> +∞ as x -> +∞

f$x$ -> -∞ as x -> -∞

Step6:

Plot the points from steps 1, 2, 4.

Step7:

Connect the points by the curve.