https://file.aiquickdraw.com/mathbot/file/17253826206646fi0dyc6.jpg

The image you uploaded contains two polynomial functions, each asking for the degree and the end behaviors. Let's analyze both functions one by one.

### 1. **First Polynomial:**
   \[
   y = -6x + 5 + 2x - 2
   \]

   **Simplify the expression**:
   \[
   y = (-6x + 2x) + (5 - 2) = -4x + 3
   \]

   **Degree**: The degree of the polynomial is determined by the highest power of \( x \). In this case, the highest power of \( x \) is 1, so the degree is **1**.

   **End Behavior**: Since this is a linear function with a negative leading coefficient (\(-4x\)), the end behavior is:
   - As \( x \) approaches \( +\infty \), \( y \) approaches \( -\infty \).
   - As \( x \) approaches \( -\infty \), \( y \) approaches \( +\infty \).

### 2. **Second Polynomial:**
   \[
   y = 4x + \frac{1}{2} - 3x + x^2 + \frac{1}{2}
   \]

   **Simplify the expression**:
   \[
   y = x^2 + (4x - 3x) + \left(\frac{1}{2} + \frac{1}{2}\right) = x^2 + x + 1
   \]

   **Degree**: The highest power of \( x \) is 2, so the degree of the polynomial is **2**.

   **End Behavior**: Since this is a quadratic function with a positive leading coefficient (\(x^2\)):
   - As \( x \) approaches \( +\infty \), \( y \) approaches \( +\infty \).
   - As \( x \) approaches \( -\infty \), \( y \) also approaches \( +\infty \).

Would you like more details on how these conclusions were reached, or do you have any further questions?

Here are 5 related questions that can expand your understanding:
1. How do we determine the degree of a polynomial with multiple variables?
2. What would the end behavior be if the leading coefficient of a quadratic term is negative?
3. How do higher-degree polynomials (degree 3 or more) influence end behavior?
4. What is the importance of the constant term in a polynomial when analyzing its graph?
5. How does the sign of the leading coefficient affect the slope of a linear function?

**Tip:** Always simplify polynomials by combining like terms to accurately determine the degree and end behavior of the function.

Determine the degree and end behaviors for the following polynomials: y = -6x + 5 + 2x - 2 and y = 4x + 1/2 - 3x + x^2 + 1/2.

Learn how to determine the degree and end behaviors for the polynomials y = -6x + 5 + 2x - 2 and y = 4x + 1/2 - 3x + x^2 + 1/2. This tutorial covers algebraic concepts, including polynomial functions, simplifying expressions, and analyzing end behaviors. Ideal for high school students in Grades 9-12.

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