Answer :
To find the vertex and minimum value of the quadratic function [tex]\( f(x) = x^2 - 16x + 71 \)[/tex], we can use properties of quadratic equations:
1. Identify the coefficients: For the quadratic equation [tex]\( ax^2 + bx + c \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -16 \)[/tex]
- [tex]\( c = 71 \)[/tex]
2. Find the vertex: The vertex of a parabola given by a quadratic equation is at the point [tex]\((h, k)\)[/tex].
- The x-coordinate of the vertex, [tex]\( h \)[/tex], is calculated using the formula:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
Substituting the values, we get:
[tex]\[
h = -\frac{-16}{2 \times 1} = \frac{16}{2} = 8
\][/tex]
- To find the y-coordinate, [tex]\( k \)[/tex], substitute [tex]\( x = 8 \)[/tex] back into the function [tex]\( f(x) \)[/tex]:
[tex]\[
k = f(8) = (8)^2 - 16 \times 8 + 71
\][/tex]
[tex]\[
k = 64 - 128 + 71
\][/tex]
[tex]\[
k = 7
\][/tex]
- Therefore, the vertex of the parabola is [tex]\((8, 7)\)[/tex].
3. Determine the minimum value: Since the parabola opens upwards (the coefficient of [tex]\( x^2 \)[/tex] is positive, [tex]\( a = 1 > 0 \)[/tex]), the vertex represents the minimum point. Thus, the minimum value of the function is the y-coordinate of the vertex, which is [tex]\( 7 \)[/tex].
In summary:
- The vertex of the function [tex]\( f(x) = x^2 - 16x + 71 \)[/tex] is [tex]\((8, 7)\)[/tex].
- The minimum value of the function is [tex]\( 7 \)[/tex].
1. Identify the coefficients: For the quadratic equation [tex]\( ax^2 + bx + c \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -16 \)[/tex]
- [tex]\( c = 71 \)[/tex]
2. Find the vertex: The vertex of a parabola given by a quadratic equation is at the point [tex]\((h, k)\)[/tex].
- The x-coordinate of the vertex, [tex]\( h \)[/tex], is calculated using the formula:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
Substituting the values, we get:
[tex]\[
h = -\frac{-16}{2 \times 1} = \frac{16}{2} = 8
\][/tex]
- To find the y-coordinate, [tex]\( k \)[/tex], substitute [tex]\( x = 8 \)[/tex] back into the function [tex]\( f(x) \)[/tex]:
[tex]\[
k = f(8) = (8)^2 - 16 \times 8 + 71
\][/tex]
[tex]\[
k = 64 - 128 + 71
\][/tex]
[tex]\[
k = 7
\][/tex]
- Therefore, the vertex of the parabola is [tex]\((8, 7)\)[/tex].
3. Determine the minimum value: Since the parabola opens upwards (the coefficient of [tex]\( x^2 \)[/tex] is positive, [tex]\( a = 1 > 0 \)[/tex]), the vertex represents the minimum point. Thus, the minimum value of the function is the y-coordinate of the vertex, which is [tex]\( 7 \)[/tex].
In summary:
- The vertex of the function [tex]\( f(x) = x^2 - 16x + 71 \)[/tex] is [tex]\((8, 7)\)[/tex].
- The minimum value of the function is [tex]\( 7 \)[/tex].
