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Let's work through the given function \( f(x) = x^4 - 16x^2 \) and address the steps requested:

### a) Factoring to find the x-intercepts:
The function can be factored as:
\[
f(x) = x^2(x^2 - 16) = x^2(x - 4)(x + 4)
\]
Setting \( f(x) = 0 \) to find the x-intercepts:
\[
x^2(x - 4)(x + 4) = 0
\]
This gives the solutions:
\[
x = 0, \, x = 4, \, x = -4
\]
These are the x-intercepts.

### b) Behavior of the graph:
- The graph **crosses** the x-axis at \( x = 4 \) and \( x = -4 \). This happens because at these points the factors \( (x - 4) \) and \( (x + 4) \) have a power of 1 (odd power), meaning the sign of the function changes.
- The graph **touches** the x-axis and turns around at \( x = 0 \). This occurs because the factor \( x^2 \) has an even power, so the function does not change sign here, resulting in a touch and turn around at this point.

### c) Finding the y-intercept:
The y-intercept is found by calculating \( f(0) \):
\[
f(0) = 0^4 - 16(0^2) = 0
\]
Thus, the y-intercept is \( f(0) = 0 \).

Would you like more details on any of these points?

Here are 5 related questions for further exploration:
1. How do we classify the behavior of the graph at points where the function crosses versus touches the x-axis?
2. What is the significance of the multiplicity of roots in polynomial functions?
3. Can you explain how to determine the end behavior of \( f(x) = x^4 - 16x^2 \)?
4. How would the graph change if we modified the function to \( f(x) = x^4 - 16x^2 + 1 \)?
5. What are some other methods for solving polynomial equations beyond factoring?

**Tip**: Always look at the degree and leading coefficient of a polynomial to predict its end behavior.

Use the given function f(x) = x^4 - 16x^2 to complete parts (a) through (e). Find the x-intercepts, determine where the graph crosses and touches the x-axis, and compute the y-intercept by evaluating f(0).

Learn how to solve the polynomial function f(x) = x^4 - 16x^2 by finding the x-intercepts, analyzing where the graph crosses or touches the x-axis, and computing the y-intercept. This step-by-step explanation includes factoring and determining behavior at the intercepts. Suitable for high school students in Grades 10-12.

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