Use transformations of the absolute value function, \( f(x)=|x| \), to graph the function \( g(x)=-2|x+3|-5 \)

. To graph the function \( g(x)=-2|x+3|-5 \), we can follow these steps: 1. **Identify the parent function**: The parent function is \( f(x)=|x| \), which is a V-shaped graph with the vertex at the origin (0,0). 2. **Horizontal shift**: The expression inside the absolute value, \( |x+3| \), indicates a horizontal shift. Since we have \( x+3 \), the graph shifts to the left by 3 units. 3. **Vertical stretch**: The coefficient in front of the absolute value, \( -2 \), indicates a vertical stretch. The absolute value function \( |x| \) is stretched by a factor of 2. 4. **Reflection over the x-axis**: The negative sign in front of the absolute value function, \( -2|x+3| \), indicates a reflection over the x-axis. 5. **Vertical shift**: The constant term, \( -5 \), indicates a vertical shift. The graph shifts down by 5 units. Now, let's put these transformations together to graph \( g(x) \): - Start with the parent function \( f(x)=|x| \). - Shift the graph 3 units to the left to get \( f(x)=|x+3| \). - Stretch the graph vertically by a factor of 2 to get \( f(x)=2|x+3| \). - Reflect the graph over the x-axis to get \( f(x)=-2|x+3| \). - Shift the graph down by 5 units to get \( g(x)=-2|x+3|-5 \). The final graph of \( g(x) \) will have the following characteristics: - The vertex will be at the point (-3, -5). - The graph will be a V-shape, but it will be narrower than the parent function due to the vertical stretch. - The graph will be reflected over the x-axis. - The graph will be shifted down by 5 units. Here's a visual representation of the transformations: ``` Original: |x| Shift left: |x+3| Stretch: 2|x+3| Reflect: -2|x+3| Shift down: -2|x+3|-5 ``` To graph \( g(x) \), you can plot the vertex (-3, -5) and then use the transformations to sketch the rest of the graph.