Question

graph the function ( y = x ^ { 2 } - 2 x - 8 ) by identifying the domain and any symmetries, finding the derivatives ( y ^ { prime } ) and ( y ^ { prime prime } ), 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. 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.

Explanation:

Step1: Analyze the function type

The function \( y = x^{2}-2x - 8\) is a quadratic function. The general form of a quadratic function is \(y=ax^{2}+bx + c\), where \(a = 1\), \(b=-2\), \(c = - 8\). Since \(a=1>0\), the parabola opens upwards.

Step2: Determine the behavior at infinity

As \(x
ightarrow\pm\infty\), \(y=x^{2}-2x - 8\approx x^{2}\). When \(x
ightarrow+\infty\), \(y
ightarrow+\infty\); when \(x
ightarrow-\infty\), \(y
ightarrow+\infty\).

Answer:

B. There is no absolute maximum.