Math Problem Statement

How to translate vertex form to standard

Solution

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

Vertex Form of a Quadratic Equation:

y=a(xh)2+ky = a(x - h)^2 + k Where:

  • aa is the same coefficient as in the standard form.
  • (h,k)(h, k) is the vertex of the parabola.

Standard Form of a Quadratic Equation:

y=ax2+bx+cy = ax^2 + bx + c Where:

  • aa, bb, and cc are constants.

Steps to Convert:

  1. Expand the Square: Use the identity (xh)2=x22hx+h2(x - h)^2 = x^2 - 2hx + h^2. y=a(x22hx+h2)+ky = a(x^2 - 2hx + h^2) + k

  2. Distribute the aa: y=ax22ahx+ah2+ky = a \cdot x^2 - 2ah \cdot x + a \cdot h^2 + k

  3. Simplify: y=ax22ahx+(ah2+k)y = ax^2 - 2ah \cdot x + (ah^2 + k)

Now, the equation is in standard form: y=ax2+bx+cy = ax^2 + bx + c Where:

  • b=2ahb = -2ah
  • c=ah2+kc = ah^2 + k

Example:

Convert y=2(x3)2+4y = 2(x - 3)^2 + 4 to standard form.

  1. Expand: (x3)2=x26x+9(x - 3)^2 = x^2 - 6x + 9 y=2(x26x+9)+4y = 2(x^2 - 6x + 9) + 4

  2. Distribute 22: y=2x212x+18+4y = 2x^2 - 12x + 18 + 4

  3. Simplify: y=2x212x+22y = 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 aa 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 aa remains the same when converting between vertex form and standard form—it dictates the direction and width of the parabola.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Vertex Form
Standard Form

Formulas

Vertex form: y = a(x - h)^2 + k
Standard form: y = ax^2 + bx + c
Expansion formula: (x - h)^2 = x^2 - 2hx + h^2

Theorems

Quadratic Expansion
Properties of Parabolas

Suitable Grade Level

Grades 9-12