To rewrite the given quadratic function [tex]\( f(x) = -2x^2 + 20x - 43 \)[/tex] in the form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], we need to complete the square. Let's proceed step by step:
1. Start with the original quadratic function:
[tex]\[
f(x) = -2x^2 + 20x - 43
\][/tex]
2. Factor out the coefficient of [tex]\( x^2 \)[/tex] (which is -2) from the first two terms:
[tex]\[
f(x) = -2(x^2 - 10x) - 43
\][/tex]
3. To complete the square, take half of the coefficient of [tex]\( x \)[/tex] (inside the parentheses), square it, and add/subtract it inside the parentheses:
- The coefficient of [tex]\( x \)[/tex] is -10. Half of -10 is -5, and squaring -5 gives 25.
[tex]\[
f(x) = -2(x^2 - 10x + 25 - 25) - 43
\][/tex]
4. Rewrite the expression inside the parentheses to form a perfect square trinomial:
[tex]\[
f(x) = -2((x - 5)^2 - 25) - 43
\][/tex]
5. Distribute the -2 and simplify:
[tex]\[
f(x) = -2(x - 5)^2 + 50 - 43
\][/tex]
[tex]\[
f(x) = -2(x - 5)^2 + 7
\][/tex]
Now, the quadratic function is in the form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where:
- [tex]\( a = -2 \)[/tex]
- [tex]\( h = 5 \)[/tex]
- [tex]\( k = 7 \)[/tex]
So, the equation in the specified form is:
[tex]\[
f(x) = -2(x - 5)^2 + 7
\][/tex]
The vertex of the quadratic function, given in the form [tex]\( a(x-h)^2 + k \)[/tex], is:
[tex]\[
(h, k) = (5, 7)
\][/tex]
Thus, the answers are:
Writing in the form specified:
[tex]\[
f(x) = -2(x - 5)^2 + 7
\][/tex]
Vertex:
[tex]\[
(5, 7)
\][/tex]
