Question

What is the Number of terms, Degree, LC, X-intercepts & End Behavior


f(x) = -x⁴ + 1


f(x) = x⁴ - x²

Answer 1

Let's analyze each function:

### 1. For the function \( f(x) = -x^4 + 1 \):

- **Number of terms**: 2 (the terms are \(-x^4\) and \(1\))
- **Degree**: 4 (the highest power of \(x\) is 4)
- **Leading Coefficient (LC)**: -1 (the coefficient of the term with the highest degree, which is \(-x^4\))
- **X-intercepts**: To find the x-intercepts, set \(f(x) = 0\):
\[
-x^4 + 1 = 0 \quad \Rightarrow \quad x^4 = 1 \quad \Rightarrow \quad x = \pm 1
\]
Therefore, the x-intercepts are \(x = 1\) and \(x = -1\).
- **End Behavior**:
- As \(x \to \infty\), \(f(x) \to -\infty\)
- As \(x \to -\infty\), \(f(x) \to -\infty\)

This indicates that the ends of the graph go downward.

### 2. For the function \( f(x) = x^4 - x^2 \):

- **Number of terms**: 2 (the terms are \(x^4\) and \(-x^2\))
- **Degree**: 4 (the highest power of \(x\) is 4)
- **Leading Coefficient (LC)**: 1 (the coefficient of the term with the highest degree, which is \(x^4\))
- **X-intercepts**: To find the x-intercepts, set \(f(x) = 0\):
\[
x^4 - x^2 = 0 \quad \Rightarrow \quad x^2(x^2 - 1) = 0 \quad \Rightarrow \quad x^2 = 0 \quad \text{or} \quad x^2 = 1 \quad \Rightarrow \quad x = 0, x = 1, x = -1
\]
Therefore, the x-intercepts are \(x = 0\), \(x = 1\), and \(x = -1\).
- **End Behavior**:
- As \(x \to \infty\), \(f(x) \to \infty\)
- As \(x \to -\infty\), \(f(x) \to \infty\)

This indicates that the ends of the graph go upward.

### Summary:
For \( f(x) = -x^4 + 1 \):
- Number of terms: 2
- Degree: 4
- Leading Coefficient: -1
- X-intercepts: \(x = 1\), \(x = -1\)
- End Behavior: Both ends go down.

For \( f(x) = x^4 - x^2 \):
- Number of terms: 2
- Degree: 4
- Leading Coefficient: 1
- X-intercepts: \(x = 0\), \(x = 1\), \(x = -1\)
- End Behavior: Both ends go up.