the image shows a geometric representation of the function f(x)=x² - 2x - 6 written in standard form. what…

the image shows a geometric representation of the function f(x)=x² - 2x - 6 written in standard form. what is this function written in vertex form? f(x)=(x - 1)² - 7 f(x)=(x + 1)² - 7 f(x)=(x - 1)² - 5 f(x)=(x + 1)² - 5

the image shows a geometric representation of the function f(x)=x² - 2x - 6 written in standard form. what is this function written in vertex form? f(x)=(x - 1)² - 7 f(x)=(x + 1)² - 7 f(x)=(x - 1)² - 5 f(x)=(x + 1)² - 5

Answer

Explanation:

Step1: Recall vertex - form formula

The vertex - form of a quadratic function is $f(x)=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. For a quadratic function $y = ax^{2}+bx + c$, we can complete the square. Given $f(x)=x^{2}-2x - 6$, where $a = 1$, $b=-2$, $c=-6$.

Step2: Complete the square for the $x$ - terms

[ \begin{align*} f(x)&=x^{2}-2x - 6\ &=(x^{2}-2x)-6\ &=(x^{2}-2x + 1-1)-6\ &=(x - 1)^{2}-1-6 \end{align*} ]

Step3: Simplify the expression

[ \begin{align*} f(x)&=(x - 1)^{2}-7 \end{align*} ]

Answer:

$f(x)=(x - 1)^{2}-7$