Describe how the given function can be obtained from one of the basic graphs. Then graph the function. \( g(x)=-(x+1)^{3} \)

1. **Starting Graph:** Begin with the basic cubic function \[ f(x)=x^3. \] 2. **Horizontal Translation:** Replace \( x \) with \( x+1 \) to shift the graph to the left by 1 unit. The new function becomes \[ h(x)=(x+1)^3. \] 3. **Reflection:** Multiply the function by \(-1\) to reflect it across the \( x \)-axis. This gives the function \[ g(x) = -(x+1)^3. \] 4. **Graphing the Function:** - **Shift:** The original graph of \( f(x)=x^3 \) is shifted left by 1 unit. - **Reflection:** The graph is reflected across the \( x \)-axis. - **Key Points:** - When \( x = -1 \), \[ g(-1) = -((-1)+1)^3 = -0^3 = 0. \] - When \( x = 0 \), \[ g(0) = -(0+1)^3 = -1. \] - When \( x = -2 \), \[ g(-2) = -((-2)+1)^3 = -(-1)^3 = 1. \] - **End Behavior:** - As \( x \to \infty \), \( g(x) \to -\infty \). - As \( x \to -\infty \), \( g(x) \to \infty \). 5. **Summary of Transformations:** - Start with \( f(x)=x^3 \). - Shift left by 1: \( f(x) \to (x+1)^3 \). - Reflect across the \( x \)-axis: \( (x+1)^3 \to -(x+1)^3 \). 6. **Graph Sketch:** To sketch the graph of \( g(x)=-(x+1)^3 \): - Plot the key point \((-1, 0)\). - Note that the point originally at \((0,0)\) for \( f(x) = x^3 \) now moves to \((0,-1)\) after the transformation. - Similarly, the point \((-1, -1)\) for \( f(x)=x^3 \) becomes \((-2,1)\) after transformation. - Draw a smooth curve through these points ensuring that: - The graph passes through the point \((-1,0)\). - The left side of the curve approaches \(\infty\) as \( x \to -\infty \). - The right side descends to \(-\infty\) as \( x \to \infty \). The resulting graph is a reflected and left-shifted version of the standard cubic graph.