3. Let \( g(x) \) be a function with the following properties. The graph has vertical asymptotes \( x=-2, x=4 \). - The graph has horizontal asymptote \( y=-3 \). - \( g^{\prime}(1)=0 ; g^{\prime}(7)=0 \) \( g^{\prime}(x)>0 \) when \( 17 \) - \( g^{\prime}(x)<0 \) when \( x<-2,-20 \) when \( -29 \) - \( g(1)=2, g(7)=-6, g(9)=-4 \) Use the techniques taught in this course and the information above to sketch a graph of \( g(x) \). For full credit, you must show work (6)

To sketch the graph of the function \( g(x) \) based on the provided properties, we will analyze each piece of information step by step. ### Step 1: Identify Asymptotes 1. **Vertical Asymptotes**: The graph has vertical asymptotes at \( x = -2 \) and \( x = 4 \). This means that as \( x \) approaches these values, \( g(x) \) will approach \( \pm \infty \). 2. **Horizontal Asymptote**: The graph has a horizontal asymptote at \( y = -3 \). This indicates that as \( x \) approaches \( \pm \infty \), \( g(x) \) approaches -3. ### Step 2: Analyze Critical Points and Behavior 1. **Critical Points**: We have \( g'(1) = 0 \) and \( g'(7) = 0 \). These points are where the function has local extrema. 2. **Increasing/Decreasing Intervals**: - \( g'(x) > 0 \) when \( 1 < x < 4 \) and \( x > 7 \): The function is increasing in these intervals. - \( g'(x) < 0 \) when \( x < -2 \), \( -2 < x < 1 \), and \( 4 < x < 7 \): The function is decreasing in these intervals. ### Step 3: Analyze Concavity 1. **Second Derivative**: We have \( g''(0) = 9 \), indicating that the function is concave up at \( x = 0 \). 2. **Concavity Intervals**: - \( g''(x) > 0 \) when \( -2 < x < 4 \) and \( 4 < x < 9 \): The function is concave up in these intervals. - \( g''(x) < 0 \) when \( x > 9 \): The function is concave down in this interval. ### Step 4: Evaluate Function Values 1. **Function Values**: We know: - \( g(1) = 2 \) - \( g(7) = -6 \) - \( g(9) = -4 \) ### Step 5: Sketch the Graph Now, we can combine all this information to sketch the graph of \( g(x) \): 1. **Start with the vertical asymptotes** at \( x = -2 \) and \( x = 4 \). The graph will approach \( \pm \infty \) as it approaches these lines. 2. **At \( x = 1 \)**, the function has a local maximum (since \( g'(1) = 0 \) and \( g'(x) < 0 \) before and \( g'(x) > 0 \) after). The value is \( g(1) = 2 \). 3. **From \( x = 1 \) to \( x = 4 \)**, the function decreases (since \( g'(x) < 0 \)). It approaches the vertical asymptote at \( x = 4 \). 4. **At \( x = 4 \)**, the function approaches \( -\infty \). 5. **From \( x = 4 \) to \( x = 7 \)**, the function is decreasing and reaches \( g(7) = -6 \). 6. **At \( x = 7 \)**, the function has a local minimum (since \( g'(7) = 0 \)). 7. **From \( x = 7 \) to \( x = 9 \)**, the function increases (since \( g'(x) > 0 \)) and reaches \( g(9) = -4 \). 8. **For \( x > 9 \)**, the function is concave down and approaches the horizontal asymptote \( y = -3 \). ### Final Sketch - The graph starts from \( -\infty \) as \( x \) approaches \( -2 \). - It increases to a maximum at \( (1, 2) \), then decreases to \( -\infty \) at \( x = 4 \). - It continues decreasing to \( (7, -6) \), then increases to \( (9, -4) \) and approaches the horizontal asymptote \( y = -3 \) as \( x \) goes to \( +\infty \). This analysis provides a comprehensive understanding of the behavior of \( g(x) \) based on the given properties.