graph the function y = x² - 2x - 8 by identifying the domain and any symmetries, finding the derivatives y…

graph the function y = x² - 2x - 8 by identifying the domain and any symmetries, finding the derivatives y and y, finding the critical points and identifying the functions behavior at each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. then find coordinates of absolute extreme points, if any. identify the absolute maximum value and where it occurs. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. o a. the absolute maximum value occurs at x = (use a comma to separate answers as needed. type each answer only once.) o b. there is no absolute maximum. identify the absolute minimum value and where it occurs. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. o a. the absolute minimum value occurs at x = (use a comma to separate answers as needed. type each answer only once.) o b. there is no absolute minimum.

graph the function y = x² - 2x - 8 by identifying the domain and any symmetries, finding the derivatives y and y, finding the critical points and identifying the functions behavior at each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. then find coordinates of absolute extreme points, if any. identify the absolute maximum value and where it occurs. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. o a. the absolute maximum value occurs at x = (use a comma to separate answers as needed. type each answer only once.) o b. there is no absolute maximum. identify the absolute minimum value and where it occurs. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. o a. the absolute minimum value occurs at x = (use a comma to separate answers as needed. type each answer only once.) o b. there is no absolute minimum.

Answer

Explanation:

Step1: Find the domain

The function $y = x^{2}-2x - 8$ is a polynomial. The domain of a polynomial is all real - numbers, $(-\infty,\infty)$.

Step2: Check for symmetries

For a quadratic function $y = ax^{2}+bx + c$, the axis of symmetry is given by $x=-\frac{b}{2a}$. Here $a = 1$, $b=-2$, so the axis of symmetry is $x =-\frac{-2}{2\times1}=1$.

Step3: Find the first - derivative

Differentiate $y=x^{2}-2x - 8$ with respect to $x$. Using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we get $y'=2x-2$.

Step4: Find the critical points

Set $y' = 0$. Then $2x-2=0$, which gives $x = 1$.

Step5: Determine increasing and decreasing intervals

Test intervals:

  • For $x\lt1$, let $x = 0$. Then $y'(0)=2\times0 - 2=-2\lt0$, so the function is decreasing on the interval $(-\infty,1)$.
  • For $x\gt1$, let $x = 2$. Then $y'(2)=2\times2 - 2 = 2\gt0$, so the function is increasing on the interval $(1,\infty)$.

Step6: Find the second - derivative

Differentiate $y'=2x - 2$ with respect to $x$. We get $y'' = 2\gt0$.

Step7: Find the inflection points

Since $y''=2\neq0$ for all $x$, there are no inflection points. And since $y''\gt0$ for all $x$, the curve is concave up on $(-\infty,\infty)$.

Step8: Find the intercepts

  • x - intercepts: Set $y = 0$. Then $x^{2}-2x - 8=(x - 4)(x+2)=0$. So $x=-2$ and $x = 4$.
  • y - intercept: Set $x = 0$. Then $y=-8$.

Step9: Find the absolute extrema

Since the function is a parabola opening upwards ($a = 1\gt0$) and is decreasing on $(-\infty,1)$ and increasing on $(1,\infty)$, the absolute minimum occurs at $x = 1$. Substitute $x = 1$ into the function: $y=(1)^{2}-2\times1 - 8=1 - 2 - 8=-9$. There is no absolute maximum.

Answer:

For the absolute maximum: B. There is no absolute maximum. For the absolute minimum: A. The absolute minimum value $-9$ occurs at $x = 1$.