https://file.aiquickdraw.com/mathbot/file/1729523661775bepy9vzd.jpg

Let's analyze the quadratic functions given and determine their vertices. The general form of a vertex in a quadratic equation of the form \( y = a(x - h)^2 + k \) is \( (h, k) \), where \( h \) and \( k \) are the coordinates of the vertex.

Here are the given functions:

1. \( y = 2(x - 1)^2 + 4 \)
   - The vertex form is clear: \( h = 1 \), \( k = 4 \).
   - So the vertex is \( (1, 4) \), which corresponds to **F**.

2. \( y = (x + 3)^2 + 2 \)
   - This can be written as \( y = (x - (-3))^2 + 2 \), so \( h = -3 \), \( k = 2 \).
   - The vertex is \( (-3, 2) \), which corresponds to **A**.

3. \( y = (x - 3)^2 - 2 \)
   - The vertex form gives \( h = 3 \), \( k = -2 \).
   - The vertex is \( (3, -2) \), which corresponds to **E**.

4. \( y = 2x^2 + 4x - 2 \)
   - First, we need to convert this to vertex form. Let's complete the square:
     \[
     y = 2(x^2 + 2x) - 2
     \]
     \[
     y = 2((x + 1)^2 - 1) - 2
     \]
     \[
     y = 2(x + 1)^2 - 2 - 2 = 2(x + 1)^2 - 4
     \]
     The vertex form is \( y = 2(x + 1)^2 - 4 \), so \( h = -1 \), \( k = -4 \).
     - The vertex is \( (-1, -4) \), which corresponds to **D**.

5. \( y = (x - 3)^2 + 2 \)
   - The vertex form gives \( h = 3 \), \( k = 2 \).
   - The vertex is \( (3, 2) \), which corresponds to **C**.

### Final Answers:
- \( y = 2(x - 1)^2 + 4 \): **F**
- \( y = (x + 3)^2 + 2 \): **A**
- \( y = (x - 3)^2 - 2 \): **E**
- \( y = 2x^2 + 4x - 2 \): **D**
- \( y = (x - 3)^2 + 2 \): **C**

Let me know if you need further details or have any questions! 

Here are some follow-up questions:
1. What does the "a" value in the vertex form equation \( y = a(x - h)^2 + k \) represent?
2. How can you determine if the parabola opens upwards or downwards?
3. How would you find the vertex if the quadratic equation is in standard form \( ax^2 + bx + c \)?
4. What role does the vertex play in identifying the axis of symmetry of a parabola?
5. How does completing the square help in converting from standard form to vertex form?

**Tip:** When converting a quadratic equation to vertex form, completing the square is a reliable technique to easily identify the vertex.

Find the vertex of each quadratic function. Choose your answer from the choices below. Use CAPITAL letters.

This problem involves identifying the vertices of quadratic functions from equations in vertex form and standard form. It covers algebra concepts such as completing the square and vertex properties of parabolas. 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