2. (#9 exam 1 review) sketch the graph of the function g for which: g(0)=g(2)=g(4)=0, g(1)=g(3)=0, g(2)=…

Explanation:

Step1: Analyze zero - points

The function has zeros at (x = 0), (x=2) and (x = 4), so the graph crosses or touches the (x) - axis at these points.

Step2: Analyze critical points

Since (g^{\prime}(1)=g^{\prime}(3)=0), the function has horizontal tangents at (x = 1) and (x = 3). And (g^{\prime}(2)=-1) means the slope of the tangent line at (x = 2) is (- 1).

Step3: Analyze asymptotes

The limit (\lim_{x\rightarrow5^{-}}g(x)=\infty) indicates a vertical asymptote at (x = 5) (the function goes to positive infinity as (x) approaches (5) from the left). The limit (\lim_{x\rightarrow - 1^{+}}g(x)=-\infty) indicates a vertical asymptote at (x=-1) (the function goes to negative infinity as (x) approaches (-1) from the right).

Step4: Sketch the graph

Based on the above - mentioned properties, start by marking the zero - points, critical points and asymptotes on the coordinate plane. Then, draw a smooth curve that satisfies all the given conditions.

Answer:

A hand - sketched graph with zeros at (x = 0), (x = 2), (x = 4), horizontal tangents at (x = 1) and (x = 3), a tangent with slope (-1) at (x = 2), a vertical asymptote at (x=-1) (where the function goes to (-\infty) as (x) approaches (-1) from the right) and a vertical asymptote at (x = 5) (where the function goes to (\infty) as (x) approaches (5) from the left).