Given the function \( y = x^4 - 16x^2 \), let's analyze it thoroughly. 

### 1. **Factor the expression:**

We can simplify the expression by factoring it:
\[
y = x^4 - 16x^2 = x^2(x^2 - 16)
\]
Notice that \(x^2 - 16\) is a **difference of squares**:
\[
x^2 - 16 = (x - 4)(x + 4)
\]
Thus, the fully factored form is:
\[
y = x^2 (x - 4)(x + 4)
\]

---

### 2. **Find the roots:**

To find the roots, set \(y = 0\):
\[
x^2 (x - 4)(x + 4) = 0
\]
This gives:
- \(x^2 = 0 \quad \Rightarrow \quad x = 0\)
- \(x - 4 = 0 \quad \Rightarrow \quad x = 4\)
- \(x + 4 = 0 \quad \Rightarrow \quad x = -4\)

Thus, the **roots** are:
\[
x = -4, \quad x = 0, \quad x = 4
\]

---

### 3. **Determine the behavior around the roots (sign analysis):**

For large positive or negative \(x\), the \(x^4\)-term dominates, so the function grows rapidly in both directions. Now, let's analyze the sign of \(y\) around the critical points \(x = -4, 0, 4\).

- **Interval** \( (-\infty, -4) \): All terms are positive, so \(y > 0\).
- **Interval** \( (-4, 0) \): The factor \((x + 4)\) is positive, but \((x - 4)\) is negative, so \(y < 0\).
- **Interval** \( (0, 4) \): The factor \(x^2\) is positive, but \((x - 4)\) is negative, so \(y < 0\).
- **Interval** \( (4, \infty) \): All terms are positive, so \(y > 0\).

---

### 4. **Critical points and turning points:**

To find **critical points**, take the derivative of \(y = x^4 - 16x^2\).

\[
\frac{dy}{dx} = 4x^3 - 32x
\]
Factor the derivative:
\[
\frac{dy}{dx} = 4x(x^2 - 8) = 4x(x - 2\sqrt{2})(x + 2\sqrt{2})
\]
Setting the derivative to zero:
\[
4x(x - 2\sqrt{2})(x + 2\sqrt{2}) = 0
\]
The critical points are:
\[
x = 0, \quad x = 2\sqrt{2}, \quad x = -2\sqrt{2}
\]

---

### 5. **Second derivative for concavity:**

Now, take the second derivative of \(y\):
\[
\frac{d^2y}{dx^2} = 12x^2 - 32
\]
Set this equal to 0 to find inflection points:
\[
12x^2 - 32 = 0 \quad \Rightarrow \quad 12x^2 = 32 \quad \Rightarrow \quad x^2 = \frac{8}{3}
\]
Thus, the inflection points occur at:
\[
x = \pm \frac{2\sqrt{6}}{3}
\]

---

### 6. **Graph behavior summary:**

- **Roots**: \(x = -4, 0, 4\)
- **Critical points**: \(x = 0, \pm 2\sqrt{2}\)
- **Inflection points**: \(x = \pm \frac{2\sqrt{6}}{3}\)
- The function tends towards \(+\infty\) as \(x \to \pm \infty\).

---

Would you like a detailed plot or further explanation on any step?

---

**Related questions:**

1. How does the second derivative test determine concavity?
2. How do critical points relate to maxima or minima?
3. What is the significance of inflection points in graphing?
4. How does sign analysis help determine intervals of positivity and negativity?
5. Can you describe the symmetry of this function?

**Tip:** When graphing, always look at end behavior to determine how the function behaves as \(x \to \infty\) or \(x \to -\infty\).

Learn how to solve the polynomial function y = x^4 - 16x^2 by factoring, finding roots, and analyzing critical points and concavity. This problem covers advanced algebraic concepts, including the difference of squares and the second derivative test. Suitable for students in Grades 11-12.

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