Problem 115 - College Algebra Essentials

Question

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=-\sqrt[3]{x+2}$$

Step-by-Step Solution

Answer

The function \(h(x) = -\sqrt[3]{x + 2}\) is a transformation of \(f(x) = \sqrt[3]{x}\) that involves a reflection over the x-axis and a horizontal shift 2 units to the left.

Step 1: Start with the Base Function

Start by graphing the base function \(f(x)=\sqrt[3]{x}\). This is a function whose value for a given x is the cube root of x. It is asymmetric with respect to the origin. The graph starts at \(-\infty\) and rises with an increasing slope, crosses through the origin, and then slows down its growth as it extends toward \(+\infty\).

Step 2: Identify the Transformation

Next, it is necessary to identify the transformations that will occur based on the function \(h(x) = -\sqrt[3]{x + 2}\). There are two transformations here: a reflection in the x-axis (indicated by the negative sign) and a horizontal move to the left by 2 units.

Step 3: Apply the Reflection

The negative sign in \(h(x)\) indicates that the transformed function, when compared to the original function, will be reflected in the x-axis. The result will be a downward opening function.

Step 4: Apply the Horizontal Shift

Now, apply the horizontal transformation. The \((x+2)\) in the cube root indicates a horizontal shift of 2 units to the left(which is opposite to the sign of 2). Adjust every point on the reflected graph by moving it two places to the left.

Step 5: Graph the Final Function

Finally, draw out the final graph. This new graph represents \(h(x) = -\sqrt[3]{x + 2}\), a reflection of \(f(x)=\sqrt[3]{x}\) in the x-axis and shifted two units to the left.