Question

1. (#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)= - 1, lim_{x→5^{-}}g(x)=∞, and lim_{x→ - 1^{+}}g(x)= - ∞

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 \(\lim_{x
ightarrow5^{-}}g(x)=\infty\) indicates a vertical asymptote at \(x = 5\) (the function goes to positive infinity as \(x\) approaches \(5\) from the left). The limit \(\lim_{x
ightarrow - 1^{+}}g(x)=-\infty\) 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).