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
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GPT 3.5
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.
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.
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