https://file.aiquickdraw.com/mathbot/file/1726688475488mv2rsxx3.jpg

Let's go through the two questions shown in the image.

### Question 10:

The given equation is \( y = -a(x + j)^2 - k \).

This is a standard form of a **parabola** equation, \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex and the sign of \(a\) determines whether the parabola opens upward or downward.

- Since the coefficient of \((x + j)^2\) is **negative** (i.e., \(-a\)), the parabola opens **downward**.
- The vertex form shows that the vertex of the parabola is \((-j, -k)\).

Thus, the correct answer is **C) The graph opens downward and the vertex is (-j, -k)**.

---

### Question 11:

You are asked to simplify the expression \( \left( \frac{x}{2} - y \right)^2 \).

Expand the square:
\[
\left( \frac{x}{2} - y \right)^2 = \left( \frac{x}{2} \right)^2 - 2 \cdot \frac{x}{2} \cdot y + y^2
\]
\[
= \frac{x^2}{4} - xy + y^2
\]

This matches option **C**, so the correct answer is **C) \frac{x^2}{4} - xy + y^2**.

---

Would you like further details on these problems, or do you have any questions?

Here are 5 related questions to extend your understanding:
1. How do you derive the vertex form of a quadratic equation from standard form?
2. What is the significance of the coefficient \(a\) in determining the direction of a parabola?
3. How would the equation change if the parabola opened upward instead?
4. Can you generalize the steps to expand any binomial squared?
5. How does completing the square help in converting a quadratic equation into vertex form?

**Tip:** When expanding binomials, always keep in mind the distributive property and use it systematically to avoid mistakes.

The vertex of the parabola in the xy-plane above is (0, -c). Which of the following is true about the parabola with the equation y = -a(x + j)^2 - k? Which of the following is equivalent to (x/2 - y)^2?

Solve problems involving the vertex of a parabola and expanding binomials. Learn how to identify the vertex of the equation y = -a(x + j)^2 - k and simplify the expression (x/2 - y)^2. Suitable for high school 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