Question

The teacher asked the class to change 𝑦 = 𝑥^2 − 8𝑥 + 11 to vertex form. A classmate does not understand why completing the square is necessary to do this. How would you explain this to your classmate? Change the function to vertex form in your explanation

Answer 1

Completing the square is necessary in order to convert the equation 𝑦 = 𝑥^2 − 8𝑥 + 11 to vertex form because the vertex form of a quadratic equation is in the form 𝑦 = 𝑎(𝑥 − ℎ)^2 + 𝑘, where (ℎ, 𝑘) represents the coordinates of the vertex.

To start, let's rewrite the given equation 𝑦 = 𝑥^2 − 8𝑥 + 11.

Now, let's focus on completing the square to transform this equation into vertex form. We want to rewrite the quadratic expression 𝑥^2 − 8𝑥 in the form 𝑎(𝑥 − ℎ)^2, where ℎ represents the x-coordinate of the vertex.

To do this, let's consider the coefficient of 𝑥, which is -8 in our equation. We divide this coefficient by 2, giving us -4. Now, we square this value, obtaining (-4)^2 = 16.

To maintain the equivalence of our equation, we need to add and subtract 16 inside the parentheses, like this: 𝑦 = 𝑥^2 − 8𝑥 + 16 − 16 + 11.

Now, let's rearrange the terms: 𝑦 = (𝑥^2 − 8𝑥 + 16) − 16 + 11.

By grouping the first three terms, we have: 𝑦 = (𝑥 − 4)^2 − 5.

Voila! We have successfully completed the square and transformed the equation 𝑦 = 𝑥^2 − 8𝑥 + 11 into vertex form, which is 𝑦 = (𝑥 − 4)^2 − 5.

In this form, we can easily identify the vertex of the parabola as (4, -5). The vertex form gives us important information about the vertex and the direction of the parabola's opening.