Chapter 8.5, Problem 96E

To determine

the quadratic function can be written in standard form and parabolic form

Answer to Problem 96E

When a quadratic function is in standard form, completing the square can be applied so that it can be transformed (or converted) to vertex form.

Explanation of Solution

Given information:

The quadratic function can be written in standard form and parabolic form

Formula used:

A quadratic function can be written in standard form $y=a{x}^2+bx+c$ or in vertex form $y=a{\left(x-h\right)}^2+k$

Calculation:

My friend is correct,

A quadratic function can be written in standard form $y=a{x}^2+bx+c$ or in vertex form $y=a{\left(x-h\right)}^2+k$

This is because:

(a) any quadratic function can be written in vertex form. When a quadratic function is written in vertex form, it can be expanded to have:

  $y=a\left(x^2-2hx+h^2\right)+k$

  $y=a{x}^2-2ahx+h^2+k$ , which is already in standard form with $b=-2ahx$ and $c=h^2+k$

(b) any quadratic function can be written in standard form. When a quadratic function is in standard form, completing the square can be applied so that it can be transformed (or converted) to vertex form.

Conclusion:

When a quadratic function is in standard form, completing the square can be applied so that it can be transformed (or converted) to vertex form.