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Answer :
To find the vertex of the given quadratic function [tex]\( f(x) = 5x^2 - 40x + 94 \)[/tex], we can use the vertex formula. A quadratic function in the standard form [tex]\( ax^2 + bx + c \)[/tex] has its vertex at the point [tex]\((-b/(2a), f(-b/(2a)))\)[/tex].
1. Identify the coefficients from the quadratic equation:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = -40 \)[/tex]
- [tex]\( c = 94 \)[/tex]
2. Calculate the x-coordinate of the vertex using the formula [tex]\( -b/(2a) \)[/tex]:
[tex]\[
x_{\text{vertex}} = \frac{-b}{2a} = \frac{-(-40)}{2 \times 5} = \frac{40}{10} = 4
\][/tex]
3. To find the y-coordinate of the vertex, substitute [tex]\( x_{\text{vertex}} = 4 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(4) = 5(4)^2 - 40(4) + 94
\][/tex]
[tex]\[
= 5 \times 16 - 160 + 94
\][/tex]
[tex]\[
= 80 - 160 + 94
\][/tex]
[tex]\[
= 14
\][/tex]
Therefore, the vertex of the graph of the function [tex]\( f(x) = 5x^2 - 40x + 94 \)[/tex] is [tex]\((4, 14)\)[/tex]. So, the correct answer is [tex]\((4, 14)\)[/tex].
1. Identify the coefficients from the quadratic equation:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = -40 \)[/tex]
- [tex]\( c = 94 \)[/tex]
2. Calculate the x-coordinate of the vertex using the formula [tex]\( -b/(2a) \)[/tex]:
[tex]\[
x_{\text{vertex}} = \frac{-b}{2a} = \frac{-(-40)}{2 \times 5} = \frac{40}{10} = 4
\][/tex]
3. To find the y-coordinate of the vertex, substitute [tex]\( x_{\text{vertex}} = 4 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(4) = 5(4)^2 - 40(4) + 94
\][/tex]
[tex]\[
= 5 \times 16 - 160 + 94
\][/tex]
[tex]\[
= 80 - 160 + 94
\][/tex]
[tex]\[
= 14
\][/tex]
Therefore, the vertex of the graph of the function [tex]\( f(x) = 5x^2 - 40x + 94 \)[/tex] is [tex]\((4, 14)\)[/tex]. So, the correct answer is [tex]\((4, 14)\)[/tex].
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