Use transformations of the standard cubic function, \( f(x)=x^{3} \), to graph the function \( h(x)=-(x-4)^{3} \). Use the graphing tool to graph the function. Click to enlarge graph

To graph the function \( h(x) = -(x-4)^{3} \) using transformations of the standard cubic function \( f(x) = x^{3} \), we can break down the transformations step by step: 1. **Start with the standard cubic function**: \[ f(x) = x^{3} \] 2. **Horizontal Shift**: The term \( (x-4) \) indicates a horizontal shift to the right by 4 units. This transforms the function to: \[ g(x) = (x-4)^{3} \] 3. **Vertical Reflection**: The negative sign in front of the function indicates a reflection across the x-axis. This transforms the function to: \[ h(x) = -(x-4)^{3} \] ### Summary of Transformations: - Start with \( f(x) = x^{3} \). - Shift right by 4 units to get \( g(x) = (x-4)^{3} \). - Reflect across the x-axis to get \( h(x) = -(x-4)^{3} \). ### Key Points to Plot: - The original cubic function \( f(x) \) has a point at \( (0, 0) \). - After shifting right by 4, the point becomes \( (4, 0) \). - Reflecting this point across the x-axis gives \( (4, 0) \) (it remains the same since it is on the x-axis). - As \( x \) approaches \( \infty \), \( h(x) \) approaches \( -\infty \), and as \( x \) approaches \( -\infty \), \( h(x) \) approaches \( \infty \). ### Additional Points: - For \( x = 3 \): \[ h(3) = -(3-4)^{3} = -(-1)^{3} = 1 \] - For \( x = 5 \): \[ h(5) = -(5-4)^{3} = -1^{3} = -1 \] - For \( x = 6 \): \[ h(6) = -(6-4)^{3} = -2^{3} = -8 \] ### Points to Plot: - \( (3, 1) \) - \( (4, 0) \) - \( (5, -1) \) - \( (6, -8) \) You can use these points and the transformations described to sketch the graph of \( h(x) = -(x-4)^{3} \). If you have access to graphing software or a graphing calculator, you can input the function directly to visualize it.