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Find the coordinates of the vertex for the parabola defined by the given quadratic function.

[tex]
f(x)=-x^2-8x+4
[/tex]

The vertex is [tex]\square[/tex]. (Type an ordered pair.)


Sagot :

To find the coordinates of the vertex for the quadratic function [tex]\( f(x) = -x^2 - 8x + 4 \)[/tex], we need to follow these steps:

1. Identify the coefficients: The quadratic function is generally in the form [tex]\( ax^2 + bx + c \)[/tex]. Here, we have:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 4 \)[/tex]

2. Calculate the x-coordinate of the vertex: The formula to find the x-coordinate of the vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]

3. Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ x = -\frac{-8}{2 \cdot (-1)} = \frac{8}{-2} = -4 \][/tex]

4. Calculate the y-coordinate of the vertex: To find the y-coordinate, substitute [tex]\( x = -4 \)[/tex] back into the original quadratic function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-4) = -(-4)^2 - 8(-4) + 4 \][/tex]
[tex]\[ f(-4) = -(16) + 32 + 4 \][/tex]
[tex]\[ f(-4) = -16 + 32 + 4 = 20 \][/tex]

5. Write the coordinates of the vertex: Combining the x-coordinate and y-coordinate, we get the vertex of the parabola.

The vertex is [tex]\((-4, 20)\)[/tex].