https://file.aiquickdraw.com/mathbot/file/1727758698579ju5cbd3m.jpg

Let's analyze the given function step by step.

### Function: 
\( f(x) = x^4 - 16x^2 \)

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### (c) Find the y-intercept by computing \( f(0) \):

To find the y-intercept, substitute \( x = 0 \) into the function:

\[
f(0) = 0^4 - 16(0^2) = 0
\]

Thus, the y-intercept is \( 0 \), which is correct as shown in the image.

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### (d) Determine the symmetry of the graph:

To check for symmetry:

- **Even function**: A function is even if \( f(-x) = f(x) \) for all \( x \).
- **Odd function**: A function is odd if \( f(-x) = -f(x) \).

Let’s check the symmetry of \( f(x) \):
\[
f(-x) = (-x)^4 - 16(-x)^2 = x^4 - 16x^2 = f(x)
\]

Since \( f(-x) = f(x) \), the function is even and has **y-axis symmetry**, which is correctly selected.

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### (e) Determine the graph of the function:

To understand the shape of the graph, let’s analyze the function \( f(x) = x^4 - 16x^2 \):

- The function is a quartic (degree 4) polynomial.
- It has the form \( f(x) = x^4 - 16x^2 = x^2(x^2 - 16) = x^2(x - 4)(x + 4) \).
- The function has **zeros at \( x = -4, 0, 4 \)**.

Now, considering the behavior of the graph:

- For large positive or negative \( x \), \( x^4 \) dominates, so the graph rises in both directions.
- The local minima and maxima suggest that there will be turning points between these zeros.

Based on this, the correct graph should have:

- **y-axis symmetry (even function)**
- Zeros at \( x = -4, 0, 4 \)
- Behavior where the graph opens upward as \( x \to \pm \infty \).

The correct graph appears to be **B**, which matches these conditions with local maxima and minima at the appropriate places.

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Do you need further clarification or details on any of these steps?

### Here are 5 related questions to explore:
1. How do you determine the end behavior of higher-degree polynomials?
2. What are the turning points of a quartic function, and how can you find them?
3. How does factoring help in finding the roots of polynomial functions?
4. What is the difference between even and odd functions in terms of symmetry?
5. How can you determine local maxima and minima for polynomial functions?

**Tip:** When solving for symmetry, checking \( f(-x) \) is a quick way to determine if the function is even, odd, or neither.

Use the given function f(x) = x^4 - 16x^2 to complete parts (a) through (e) below, focusing on finding the y-intercept, determining symmetry, and selecting the appropriate graph for the function.

Learn how to analyze the polynomial function f(x) = x^4 - 16x^2 by finding its y-intercept, determining symmetry, and identifying its graph. This example explores key concepts such as even function symmetry, graph behavior, and polynomial root analysis, suitable for students in Grades 11-12.

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