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Chapter 2: Problem 111

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=\frac{1}{2} \sqrt[3]{x+2}$$

Short Answer

Expert verified
The transformed function \(h(x)=\frac{1}{2} \sqrt[3]{x+2}\) represents the graph of the cube root function, \(f(x) = \sqrt[3]{x}\), but shifted 2 units to the left and vertically compressed by a factor of 0.5. The effect is that the graph appears flattened compared to the parent function.

Step by step solution

01

Graph the cube root function, \(f(x) = \sqrt[3]{x}\)

Start by plotting the simple cube root function \(f(x) = \sqrt[3]{x}\). This function is known as the parent function for the family of cubic root functions. The points on the graph include \((-8,-2)\), \((-1,-1)\), \((0,0)\), \((1,1)\), \((8,2)\), since those points follow the pattern \(f(x) = \sqrt[3]{x}\). The graph starts from the 3rd quadrant, passes through the origin and ends in the 1st quadrant.
02

Apply a horizontal shift

The term \(x+2\) in the function indicates a shift to the left by 2 units. The points on the graph would now be \((-10,-2)\), \((-3,-1)\), \((-2,0)\), \((-1,1)\), \((6, 2)\). This shift of 2 units leftward is applied to all points on the original graph.
03

Apply a vertical compression

The coefficient \(\frac{1}{2}\) in front of the cube root applies a vertical compression by a factor of \(\frac{1}{2}\). Every y-coordinate of the basic graph will be multiplied by \(\frac{1}{2}\). The final points on the graph will be \((-10,-1)\), \((-3,-0.5)\), \((-2,0)\), \((-1,0.5)\), \((6, 1)\). The resulting graph remains increasing but will be shorter than the original cubic function.

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