Answer :
To find the vertex and the minimum value of the quadratic function [tex]\( f(x) = x^2 - 16x + 71 \)[/tex], we can follow these steps:
1. Identify the coefficients: In the function [tex]\( f(x) = ax^2 + bx + c \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -16 \)[/tex]
- [tex]\( c = 71 \)[/tex]
2. Use the vertex formula: The formula to find the x-coordinate of the vertex of a quadratic function is:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
3. Calculate the x-coordinate of the vertex:
[tex]\[
x = -\frac{-16}{2 \times 1} = \frac{16}{2} = 8
\][/tex]
4. Find the y-coordinate of the vertex: Substitute [tex]\( x = 8 \)[/tex] back into the function to find the y-coordinate (which is also the minimum value since the parabola opens upwards):
[tex]\[
f(8) = (8)^2 - 16(8) + 71 = 64 - 128 + 71 = 7
\][/tex]
5. Conclusion: The vertex of the function is [tex]\((8, 7)\)[/tex], and the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( 7 \)[/tex].
Therefore, the answers for the question are:
- Vertex: [tex]\((8, 7)\)[/tex]
- Minimum value: [tex]\(7\)[/tex]
1. Identify the coefficients: In the function [tex]\( f(x) = ax^2 + bx + c \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -16 \)[/tex]
- [tex]\( c = 71 \)[/tex]
2. Use the vertex formula: The formula to find the x-coordinate of the vertex of a quadratic function is:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
3. Calculate the x-coordinate of the vertex:
[tex]\[
x = -\frac{-16}{2 \times 1} = \frac{16}{2} = 8
\][/tex]
4. Find the y-coordinate of the vertex: Substitute [tex]\( x = 8 \)[/tex] back into the function to find the y-coordinate (which is also the minimum value since the parabola opens upwards):
[tex]\[
f(8) = (8)^2 - 16(8) + 71 = 64 - 128 + 71 = 7
\][/tex]
5. Conclusion: The vertex of the function is [tex]\((8, 7)\)[/tex], and the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( 7 \)[/tex].
Therefore, the answers for the question are:
- Vertex: [tex]\((8, 7)\)[/tex]
- Minimum value: [tex]\(7\)[/tex]
