The given equation is \( y = \sqrt{x^2 + 2x} \).

To better understand the properties of this function, let's analyze a few aspects of it:

### 1. **Domain**:
For \( y = \sqrt{x^2 + 2x} \) to be defined, the expression inside the square root must be non-negative:
\[
x^2 + 2x \geq 0
\]
Factoring the quadratic expression:
\[
x(x + 2) \geq 0
\]
The solution to this inequality will give the domain of the function.

- \( x(x + 2) = 0 \) at \( x = 0 \) and \( x = -2 \).
- The sign chart analysis for the intervals determined by these points shows that the expression is non-negative when \( x \leq -2 \) or \( x \geq 0 \).

Thus, the domain is \( (-\infty, -2] \cup [0, \infty) \).

### 2. **Range**:
Since the square root function always gives non-negative results, \( y \geq 0 \). Therefore, the range is \( [0, \infty) \).

### 3. **Behavior at key points**:
- At \( x = 0 \): 
\[
y = \sqrt{0^2 + 2(0)} = 0
\]
- At \( x = -2 \):
\[
y = \sqrt{(-2)^2 + 2(-2)} = \sqrt{4 - 4} = 0
\]

### 4. **Asymptotic behavior**:
- As \( x \to \infty \), \( x^2 + 2x \) becomes large, so \( y \approx x \) for large positive \( x \).
- As \( x \to -\infty \), the function is not defined, since the domain restricts negative values of \( x \) to \( x \leq -2 \).

Do you want more details or have any questions?

Here are five related questions:
1. How would the function change if we altered the coefficient of \(x^2\)?
2. Can we find the critical points of this function?
3. What happens to the graph for values outside the domain?
4. What is the slope of the tangent at \(x = 1\)?
5. How would you graph this function?

**Tip**: When solving inequalities involving quadratic expressions, sketching a number line helps visualize where the expression is positive or negative.

Explore the square root function y = sqrt(x^2 + 2x) in detail, including its domain, range, and asymptotic behavior. This solution provides a step-by-step analysis of quadratic inequalities and the behavior of the function at key points. Ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation