https://file.aiquickdraw.com/mathbot/file/1726557299835whyrp7st.jpg

The task involves converting quadratic functions from their general form \( y = ax^2 + bx + c \) into vertex form \( y = a(x - h)^2 + k \), where \( h \) and \( k \) represent the coordinates of the vertex, and \( a \) is the same as the leading coefficient in both forms.

Here are the steps to convert each equation to vertex form by completing the square:

### 1. \( y = x^2 - 6x - 6 \)
- Start by focusing on \( x^2 - 6x \).
- Take half of \(-6\), square it, and add it inside the equation: \( \left( \frac{-6}{2} \right)^2 = 9 \).
- Add and subtract 9: \( y = (x^2 - 6x + 9) - 9 - 6 \).
- This simplifies to: \( y = (x - 3)^2 - 15 \).

Thus, the vertex form is:  
\[ y = (x - 3)^2 - 15 \]
So, \( a = 1 \), \( h = 3 \), and \( k = -15 \).

### 2. \( y = x^2 + 4x - 10 \)
- Focus on \( x^2 + 4x \).
- Take half of 4, square it, and add it: \( \left( \frac{4}{2} \right)^2 = 4 \).
- Add and subtract 4: \( y = (x^2 + 4x + 4) - 4 - 10 \).
- Simplifies to: \( y = (x + 2)^2 - 14 \).

Thus, the vertex form is:  
\[ y = (x + 2)^2 - 14 \]  
So, \( a = 1 \), \( h = -2 \), and \( k = -14 \).

### 3. \( y = 3x^2 - 18x + 2 \)
- Factor out 3 from the first two terms: \( y = 3(x^2 - 6x) + 2 \).
- Complete the square inside the parentheses: \( \left( \frac{-6}{2} \right)^2 = 9 \).
- Add and subtract \( 9 \times 3 \) inside: \( y = 3(x^2 - 6x + 9) - 27 + 2 \).
- Simplifies to: \( y = 3(x - 3)^2 - 25 \).

Thus, the vertex form is:  
\[ y = 3(x - 3)^2 - 25 \]  
So, \( a = 3 \), \( h = 3 \), and \( k = -25 \).

### 4. \( y = x^2 - 16x + 14 \)
- Focus on \( x^2 - 16x \).
- Half of \(-16\) is \(-8\), and \( \left( \frac{-16}{2} \right)^2 = 64 \).
- Add and subtract 64: \( y = (x^2 - 16x + 64) - 64 + 14 \).
- Simplifies to: \( y = (x - 8)^2 - 50 \).

Thus, the vertex form is:  
\[ y = (x - 8)^2 - 50 \]  
So, \( a = 1 \), \( h = 8 \), and \( k = -50 \).

### 5. \( y = 2x^2 + 8x - 13 \)
- Factor out 2 from the first two terms: \( y = 2(x^2 + 4x) - 13 \).
- Complete the square inside: \( \left( \frac{4}{2} \right)^2 = 4 \).
- Add and subtract \( 4 \times 2 \): \( y = 2(x^2 + 4x + 4) - 8 - 13 \).
- Simplifies to: \( y = 2(x + 2)^2 - 21 \).

Thus, the vertex form is:  
\[ y = 2(x + 2)^2 - 21 \]  
So, \( a = 2 \), \( h = -2 \), and \( k = -21 \).

---

Would you like more details on any step? Here are some related questions to explore:

1. How is completing the square useful for solving quadratic equations?
2. Can vertex form help in graphing quadratic functions more easily?
3. What is the geometric significance of the constants \( h \) and \( k \)?
4. How does factoring differ from completing the square in quadratic transformations?
5. What are some real-life applications of quadratic functions?

**Tip:** Always double-check your constants when completing the square to avoid mistakes.

Transform the given quadratic functions in general form into vertex form. Find the constants a, h, and k.

Explore the transformation of quadratic equations from general form (y = ax^2 + bx + c) to vertex form (y = a(x - h)^2 + k) using the method of completing the square. This guide provides detailed steps for finding the vertex constants a, h, and k, suitable for high school algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation