Given the function
P ( x )

Given the function:

$P(x) = (x - 4)(x + 2)(x - 6)$

1. P-intercept (y-intercept)

The P-intercept or y-intercept is the point where the graph of the function crosses the y-axis. This occurs when $x = 0$.

To find the P-intercept, substitute $x = 0$ into the function:

$P(0) = (0 - 4)(0 + 2)(0 - 6) = (-4)(2)(-6) = 48$

So, the coordinates of the P-intercept are: $(0, 48)$

2. x-intercepts

The x-intercepts are the points where the graph crosses the x-axis, which occurs when $P(x) = 0$.

Set the function equal to 0: $(x - 4)(x + 2)(x - 6) = 0$

The solutions occur when any of the factors equals zero: $x - 4 = 0 \quad \Rightarrow \quad x = 4$ $x + 2 = 0 \quad \Rightarrow \quad x = -2$ $x - 6 = 0 \quad \Rightarrow \quad x = 6$

Thus, the coordinates of the x-intercepts are: $(4, 0), (-2, 0), (6, 0)$

Summary:

• P-intercept (y-intercept): $(0, 48)$
• x-intercepts: $(4, 0), (-2, 0), (6, 0)$

Do you want more details or have any questions?
Here are some related questions:

1. How can we determine the symmetry of this cubic function?
2. What is the behavior of the graph near each x-intercept?
3. How does the sign of the leading coefficient affect the shape of the graph?
4. What are the local maximum and minimum points of this function?
5. Can this function be factored further or simplified?

Tip: The degree of a polynomial gives a good indication of how many x-intercepts and turning points the graph might have.