https://file.aiquickdraw.com/mathbot/file/1736948392017n1djho74.jpg

Let's go through each part of the worksheet:

### 1. End Behavior of \( f(x) = 4x^4 \)
The function \( f(x) = 4x^4 \) is a polynomial of degree 4, which is an even degree with a positive leading coefficient. For such functions:
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).

So, both ends of the graph rise towards infinity.

### 2. Analyzing the Polynomial \( f(x) = -2x^4 + 3x^3 - 4x^2 + 9x + 2x - 5 \)
- **Degree**: The highest power of \( x \) is 4, so the degree is \( 4 \).
- **Leading Coefficient**: The coefficient of the term with the highest power is \( -2 \).

Now, to find the value of the function when \( x = 2 \):
\[ f(2) = -2(2)^4 + 3(2)^3 - 4(2)^2 + 9(2) + 2(2) - 5 \]
\[ f(2) = -2(16) + 3(8) - 4(4) + 9(2) + 2(2) - 5 \]
\[ f(2) = -32 + 24 - 16 + 18 + 4 - 5 \]
\[ f(2) = -32 + 24 = -8 \]
\[ -8 - 16 = -24 \]
\[ -24 + 18 = -6 \]
\[ -6 + 4 = -2 \]
\[ -2 - 5 = -7 \]
So, \( f(2) = -7 \).

### 3. Domain and Range of \( f(x) = 3x^9 \)
- **Domain**: Since \( x^9 \) is a polynomial, the domain is \( (-\infty, \infty) \).
- **Range**: Because \( x^9 \) is an odd power and \( 3x^9 \) is positive, the function takes all values from \( 0 \) to \( \infty \) when \( x \) is positive, and from \( -\infty \) to \( 0 \) when \( x \) is negative. Hence, the range is \( (-\infty, \infty) \).

Would you like further clarification on any of these parts?

Here are 5 related questions you might find interesting:
1. What is the end behavior of \( f(x) = -5x^6 \)?
2. How do you find the zeros of a polynomial function?
3. What is the range of \( f(x) = x^{10} - 1 \)?
4. Can a polynomial of even degree have an infinite range?
5. How do you determine the sign of the leading coefficient's effect on the graph's shape?

**Tip**: For polynomials, the leading term usually dictates the end behavior of the function.

Analyze the end behavior of f(x) = 4x^4, determine the degree and leading coefficient of f(x) = -2x^4 + 3x^3 - 4x^2 + 9x + 2x - 5, and identify the domain and range of f(x) = 3x^9.

Explore the end behavior of f(x) = 4x^4, find the degree and leading coefficient of f(x) = -2x^4 + 3x^3 - 4x^2 + 9x + 2x - 5, and identify the domain and range of f(x) = 3x^9. This solution includes detailed steps and explanations suitable for high school algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation