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Sagot :
The function [tex]f(x)=\left(x-5\right)\left(x+1\right)[/tex] has a vertex at (2, -9).
Quadratic function
The quadratic function can be represented by a quadratic equation in the Standard form: [tex]ax^2+bx+c[/tex] where: a, b and c are your respective coefficients. In the quadratic function the coefficient "a" must be different than zero ([tex]a\neq 0[/tex]) and the degree of the function must be equal to 2.
For solving this question, you should find the vertex of quadratic equation (h,k). Where, [tex]h=-\frac{b}{2a}[/tex] . On the other hand, the value of the k-coordinate will be found when you replace the value of h in the equation of your question.
1. [tex]f(x)=-(x-3)^2[/tex]
Write this equation in Standard form:
[tex]f(x)=-(x^2-6x+9)\\ \\ f(x)=-x^2+6x-9[/tex]
Find h, using the formula [tex]h=-\frac{b}{2a}[/tex]
[tex]h=-\frac{b}{2a} =-\frac{6}{2*(-1)} =-\frac{6}{-2}=-(-3)=3[/tex]
Find k, replacing x for the value of h
[tex]-x^2+6x-9\\ \\ -(3)^2+6*3-9=-9+18-9=-18+18=0[/tex]
The vertex of quadratic equation is (3,0). Incorrect Option.
2. [tex]f(x)=(x+8)^2[/tex]
Write this equation in Standard form:
[tex]f(x)=x^2+16x+64[/tex]
Find h, using the formula [tex]h=-\frac{b}{2a}[/tex]
[tex]h=-\frac{b}{2a} =-\frac{16}{2*1} =-\frac{16}{2}=-8[/tex]
Find k, replacing x for the value of h
[tex]x^2+16x+64\\ \\ (-8)^2+16*(-8)+64=64-128+64=128-128=0[/tex]
The vertex of quadratic equation is (-8,0). Incorrect Option.
3. [tex]f(x)=\left(x-5\right)\left(x+1\right)[/tex]
Write this equation in Standard form:
[tex]f(x)=(x^2+x-5x-5)\\ \\ f(x)=(x^2-4x-5)[/tex]
Find h, using the formula [tex]h=-\frac{b}{2a}[/tex]
[tex]h=-\frac{b}{2a} =-\frac{-4}{2*1} =-\frac{-4}{2}=-(-2)=2[/tex]
Find k, replacing x for the value of h
[tex]x^2-4x-5\\ \\ 2^2-4*2-5=4-8-5=-4-5=-9[/tex]
The vertex of quadratic equation is (2,-9). Correct Option.
4. [tex]f(x)=-\left(x-1\right)\left(x-5\right)[/tex]
Write this equation in Standard form:
[tex]f(x)=-\left(x-1\right)\left(x-5\right)\\ \\ f(x)=-(x^2-5x-x+5)\\ \\ f(x)=-(x^2-6x+5)\\ \\ f(x)=-x^2+6x-5[/tex]
Find h, using the formula [tex]h=-\frac{b}{2a}[/tex]
[tex]h=-\frac{b}{2a} =-\frac{6}{2*(-1)} =-\frac{6}{-2}=-(-3)=3[/tex]
Find k, replacing x for the value of h
[tex]-x^2+6x-5\\ \\ -(3)^2+6*3-5=-9+18-5=18-14=4[/tex]
The vertex of quadratic equation is (3,4). Incorrect Option.
Learn more about the vertex of the quadratic function here:
https://brainly.com/question/26203210
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