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Answer :
Let's find the vertex and the minimum value of the quadratic function [tex]\( f(x) = x^2 - 16x + 71 \)[/tex].
1. Identify the coefficients: In the quadratic function [tex]\( ax^2 + bx + c \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -16 \)[/tex]
- [tex]\( c = 71 \)[/tex]
2. Find the x-coordinate of the vertex: The formula for the x-coordinate of the vertex of a parabola is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[
x = -\frac{-16}{2 \times 1} = \frac{16}{2} = 8
\][/tex]
3. Find the y-coordinate of the vertex: Substitute [tex]\( x = 8 \)[/tex] back into the function to find the y-coordinate.
- Calculate [tex]\( f(8) \)[/tex]:
[tex]\[
f(8) = (8)^2 - 16 \cdot 8 + 71 = 64 - 128 + 71 = 7
\][/tex]
4. Determine the vertex: The vertex of the function is [tex]\( (8, 7) \)[/tex].
5. Find the minimum value: Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = 1 \)[/tex]) is positive, the parabola opens upwards. Therefore, the minimum value of the function [tex]\( f(x) \)[/tex] occurs at the vertex.
- The minimum value of [tex]\( f(x) \)[/tex] is the y-coordinate of the vertex, which is 7.
Thus, the vertex of the function is [tex]\( (8, 7) \)[/tex] and the minimum value of the function is [tex]\( 7 \)[/tex].
1. Identify the coefficients: In the quadratic function [tex]\( ax^2 + bx + c \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -16 \)[/tex]
- [tex]\( c = 71 \)[/tex]
2. Find the x-coordinate of the vertex: The formula for the x-coordinate of the vertex of a parabola is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[
x = -\frac{-16}{2 \times 1} = \frac{16}{2} = 8
\][/tex]
3. Find the y-coordinate of the vertex: Substitute [tex]\( x = 8 \)[/tex] back into the function to find the y-coordinate.
- Calculate [tex]\( f(8) \)[/tex]:
[tex]\[
f(8) = (8)^2 - 16 \cdot 8 + 71 = 64 - 128 + 71 = 7
\][/tex]
4. Determine the vertex: The vertex of the function is [tex]\( (8, 7) \)[/tex].
5. Find the minimum value: Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = 1 \)[/tex]) is positive, the parabola opens upwards. Therefore, the minimum value of the function [tex]\( f(x) \)[/tex] occurs at the vertex.
- The minimum value of [tex]\( f(x) \)[/tex] is the y-coordinate of the vertex, which is 7.
Thus, the vertex of the function is [tex]\( (8, 7) \)[/tex] and the minimum value of the function is [tex]\( 7 \)[/tex].
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