To find the vertex of the quadratic function [tex]\( f(x) = -x^2 + 4 \)[/tex], follow these steps:
1. Identify the standard form of a quadratic function:
A quadratic function is generally given by [tex]\( f(x) = ax^2 + bx + c \)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants.
2. Write down the coefficients:
For the function [tex]\( f(x) = -x^2 + 4 \)[/tex], we identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 0 \)[/tex]
- [tex]\( c = 4 \)[/tex]
3. Calculate the x-coordinate of the vertex:
The x-coordinate of the vertex of a quadratic function can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
Substituting the values of the coefficients into the formula:
[tex]\[
x = -\frac{0}{2 \cdot -1} = 0
\][/tex]
4. Calculate the y-coordinate by substituting x back into the original function:
Substitute [tex]\( x = 0 \)[/tex] back into the function [tex]\( f(x) = -x^2 + 4 \)[/tex]:
[tex]\[
f(0) = -0^2 + 4 = 4
\][/tex]
5. Determine the vertex coordinates:
Hence, the vertex of the function [tex]\( f(x) = -x^2 + 4 \)[/tex] is the point [tex]\( (0, 4) \)[/tex].
So, the correct answer is [tex]\( (0, 4) \)[/tex].
