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. graph the function y = x² - 2x - 8. choose the correct graph below. 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. a. the absolute maximum value occurs at x = (use a comma to separate answers as needed. type each answer only once.) b. there is no absolute maximum.

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. graph the function y = x² - 2x - 8. choose the correct graph below. 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. a. the absolute maximum value occurs at x = (use a comma to separate answers as needed. type each answer only once.) b. there is no absolute maximum.

Answer

Explanation:

Step1: Find the domain

The function $y = x^{2}-2x - 8$ is a polynomial function. The domain of a polynomial function 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 the formula $x=-\frac{b}{2a}$. 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: Analyze increasing and decreasing intervals

Test intervals around the critical point. If $x\lt1$, let $x = 0$, then $y'(0)=2\times0 - 2=-2\lt0$, so the function is decreasing on the interval $(-\infty,1)$. If $x\gt1$, 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 $y'' = 2\gt0$. Since $y''\gt0$ for all $x$, the function is concave up on $(-\infty,\infty)$ and there are no inflection points.

Step7: Find the intercepts

For the $y$ - intercept, set $x = 0$. Then $y=0^{2}-2\times0 - 8=-8$. For the $x$ - intercepts, set $y = 0$. Then $x^{2}-2x - 8=(x - 4)(x+2)=0$, so $x=-2$ or $x = 4$.

Step8: Analyze absolute extrema

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

Answer:

B. There is no absolute maximum.