https://file.aiquickdraw.com/mathbot/file/1727295341155ejwszvf1.jpg

Let's analyze the graphs from the image step by step, based on the instructions: determine when the function is positive, negative, increasing, or decreasing, and describe its end behavior.

### Problem 5:
**Equation: \( y = -x - 2 \) (Linear function)**

- **Positive Region:** The function is positive when \( y > 0 \). This occurs for \( x < -2 \).
- **Negative Region:** The function is negative when \( y < 0 \), which happens when \( x > -2 \).
- **Increasing:** The function is always decreasing, as it has a negative slope.
- **End Behavior:**
  - As \( x \to -\infty \), \( y \to \infty \).
  - As \( x \to \infty \), \( y \to -\infty \).

### Problem 6:
**Equation: \( y = |x + 4| - 1 \) (Absolute value function)**

- **Positive Region:** The function is positive when \( y > 0 \). This occurs for \( x < -5 \) or \( x > -3 \).
- **Negative Region:** The function is negative when \( y < 0 \), which happens for \( -5 < x < -3 \).
- **Increasing:** The function increases on \( x > -4 \).
- **Decreasing:** The function decreases on \( x < -4 \).
- **End Behavior:**
  - As \( x \to -\infty \), \( y \to \infty \).
  - As \( x \to \infty \), \( y \to \infty \).

### Problem 7:
**Equation: \( y = -x^2 + 6x - 5 \) (Parabola)**

- **Positive Region:** The function is positive when \( y > 0 \). This happens approximately for \( 1 < x < 5 \).
- **Negative Region:** The function is negative when \( y < 0 \), which happens for \( x < 1 \) and \( x > 5 \).
- **Increasing:** The function increases on \( x < 3 \).
- **Decreasing:** The function decreases on \( x > 3 \).
- **End Behavior:**
  - As \( x \to -\infty \), \( y \to -\infty \).
  - As \( x \to \infty \), \( y \to -\infty \).

### Problem 8:
**Equation: \( y = -x^4 + 8x^2 + 9 \) (Quartic function)**

- **Positive Region:** The function is positive when \( y > 0 \). This occurs approximately for \( -2 < x < 2 \).
- **Negative Region:** The function is negative when \( y < 0 \), which happens for \( x < -2 \) and \( x > 2 \).
- **Increasing:** The function increases when \( -2 < x < 0 \) and \( x > 2 \).
- **Decreasing:** The function decreases when \( x < -2 \), \( 0 < x < 2 \).
- **End Behavior:**
  - As \( x \to -\infty \), \( y \to -\infty \).
  - As \( x \to \infty \), \( y \to -\infty \).

Would you like further details on any of these or have any additional questions?

Here are 5 related questions for you:
1. How do you find the vertex of a quadratic function like in problem 7?
2. What does the slope of a line tell you about the function’s behavior?
3. Can you explain why the absolute value function in problem 6 has two regions of increasing and decreasing behavior?
4. How does the degree of a polynomial, like the quartic function in problem 8, influence its end behavior?
5. How can you algebraically determine where a function is positive or negative?

**Tip:** For end behavior of polynomials, focus on the term with the highest degree!

In Exercises 5-8, approximate when the function is positive, negative, increasing, or decreasing. Then describe the end behavior of the function.

This exercise covers linear, absolute value, quadratic, and polynomial functions. Students will determine when these functions are positive, negative, increasing, or decreasing, and describe their end behaviors. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation