To translate a quadratic equation from **vertex form** to **standard form**, follow these steps:

### Vertex Form of a Quadratic Equation:
\[
y = a(x - h)^2 + k
\]
Where:
- \(a\) is the same coefficient as in the standard form.
- \((h, k)\) is the vertex of the parabola.

### Standard Form of a Quadratic Equation:
\[
y = ax^2 + bx + c
\]
Where:
- \(a\), \(b\), and \(c\) are constants.

### Steps to Convert:

1. **Expand the Square**: Use the identity \((x - h)^2 = x^2 - 2hx + h^2\).
   \[
   y = a(x^2 - 2hx + h^2) + k
   \]

2. **Distribute the \(a\)**:
   \[
   y = a \cdot x^2 - 2ah \cdot x + a \cdot h^2 + k
   \]

3. **Simplify**:
   \[
   y = ax^2 - 2ah \cdot x + (ah^2 + k)
   \]
   
Now, the equation is in standard form:
\[
y = ax^2 + bx + c
\]
Where:
- \(b = -2ah\)
- \(c = ah^2 + k\)

### Example:
Convert \(y = 2(x - 3)^2 + 4\) to standard form.

1. Expand: \((x - 3)^2 = x^2 - 6x + 9\)
   \[
   y = 2(x^2 - 6x + 9) + 4
   \]
   
2. Distribute \(2\):
   \[
   y = 2x^2 - 12x + 18 + 4
   \]

3. Simplify:
   \[
   y = 2x^2 - 12x + 22
   \]

This is the quadratic in standard form.

---

Would you like more examples or details on specific steps?

### 5 Follow-up Questions:
1. How do you identify the vertex from the vertex form?
2. What is the purpose of the coefficient \(a\) in the vertex form?
3. How can you find the axis of symmetry from the standard form?
4. What happens when the vertex is at the origin?
5. How does completing the square relate to converting from standard to vertex form?

### Tip:
The value of \(a\) remains the same when converting between vertex form and standard form—it dictates the direction and width of the parabola.

Learn how to translate a quadratic equation from vertex form (y = a(x - h)^2 + k) to standard form (y = ax^2 + bx + c) with detailed steps and examples. Understand key algebraic concepts such as expanding squares, distributing constants, and simplifying. Suitable 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