To find the points where the function \( y = (x+4)(x+2)(x-1)(x-3) \) intersects the x-axis, we need to determine the values of \( x \) where \( y = 0 \). This occurs when any of the factors in the equation equals zero. 

### Step 1: Set \( y = 0 \):
\[
(x+4)(x+2)(x-1)(x-3) = 0
\]

### Step 2: Solve for \( x \) by setting each factor equal to zero:
- \( x+4 = 0 \) gives \( x = -4 \)
- \( x+2 = 0 \) gives \( x = -2 \)
- \( x-1 = 0 \) gives \( x = 1 \)
- \( x-3 = 0 \) gives \( x = 3 \)

Thus, the function intersects the x-axis at \( x = -4 \), \( x = -2 \), \( x = 1 \), and \( x = 3 \).

### Step 3: Find corresponding y-values for some additional points.
You can calculate the values of \( y \) at different points of interest (for example, points between or outside these roots).

- At \( x = 0 \):
  \[
  y = (0+4)(0+2)(0-1)(0-3) = 4 \times 2 \times (-1) \times (-3) = 24
  \]
  So, \( y(0) = 24 \).

- At \( x = -3 \):
  \[
  y = (-3+4)(-3+2)(-3-1)(-3-3) = 1 \times (-1) \times (-4) \times (-6) = -24
  \]
  So, \( y(-3) = -24 \).

The full set of intercepts for the function are \( (-4, 0) \), \( (-2, 0) \), \( (1, 0) \), and \( (3, 0) \), along with other points like \( (0, 24) \) and \( (-3, -24) \).

Would you like further details on graphing or behavior of the function?

### Follow-up questions:
1. What are the turning points of this polynomial function?
2. What is the behavior of the function as \( x \to \infty \) or \( x \to -\infty \)?
3. How would the graph look near each root?
4. Can you calculate the derivative of this function to find critical points?
5. How does this function compare to a standard cubic or quartic function?

### Tip:
To understand polynomial behavior, study the leading term for end behavior and use the derivative to find local extrema.

Explore how to find the roots of the polynomial y = (x+4)(x+2)(x-1)(x-3), where the function intersects the x-axis, and additional points like (0, 24) and (-3, -24). Learn key algebraic concepts such as polynomial roots, the fundamental theorem of algebra, and how to calculate y-values for various x. Suitable for Grades 10-12.

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