To determine the coordinates of the turning point for the curve given by the equation \( y = 4x^2 - 56x \), we will complete the square for the quadratic expression. Here are the steps in detail:
1. Rewrite the given equation:
The original equation is:
[tex]\[
y = 4x^2 - 56x
\][/tex]
2. Factor out the coefficient of \( x^2 \) from the quadratic expression:
We have the term \( 4x^2 \), so we factor out 4 from both terms involving \( x \):
[tex]\[
y = 4(x^2 - 14x)
\][/tex]
3. Complete the square inside the parentheses:
To complete the square, we take half of the coefficient of \( x \) (which is -14), square it, and add-subtract it inside the parentheses:
[tex]\[
x^2 - 14x \rightarrow (x^2 - 14x + 49 - 49)
\][/tex]
Notice that \( \left(\frac{-14}{2}\right)^2 = 49 \).
4. Rewrite the expression inside the parentheses with the added and subtracted square term:
[tex]\[
y = 4[(x^2 - 14x + 49) - 49]
\][/tex]
5. Combine terms to form a perfect square trinomial:
The expression \( x^2 - 14x + 49 \) can be written as \( (x - 7)^2 \), so:
[tex]\[
y = 4[(x - 7)^2 - 49]
\][/tex]
6. Simplify the expression by distributing the constant outside the parentheses:
[tex]\[
y = 4(x - 7)^2 - 4 \cdot 49
\][/tex]
[tex]\[
y = 4(x - 7)^2 - 196
\][/tex]
7. Identify the turning point from the completed square form:
The expression \( y = 4(x - 7)^2 - 196 \) reveals that the turning point occurs when \( (x - 7)^2 \) is minimized. Since \( (x - 7)^2 \) is non-negative and reaches its minimum value of 0 when \( x = 7 \):
[tex]\[
x - 7 = 0 \implies x = 7
\][/tex]
8. Find the corresponding \( y \)-coordinate by substituting \( x = 7 \) back into the completed square form:
[tex]\[
y = 4(7 - 7)^2 - 196
\][/tex]
[tex]\[
y = 4(0)^2 - 196
\][/tex]
[tex]\[
y = -196
\][/tex]
The coordinates of the turning point are therefore:
[tex]\[
(7, -196)
\][/tex]
This completes the step-by-step process of completing the square and finding the turning point for the curve given by [tex]\( y = 4x^2 - 56x \)[/tex]. The turning point is at [tex]\( (7, -196) \)[/tex].
