Problem 117 Begin by graphing the cube root ... [FREE SOLUTION]

Chapter 2: Problem 117

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

Short Answer

Expert verified

The graph of the function $g(x)=\sqrt[3]{-x-2}$ can be obtained by graphing the function $f(x)= \sqrt[3]{x}$, then reflecting this graph about the y-axis, and shifting it 2 units to the right.

Step by step solution

Graph the parent function

Start by graphing the parent function $f(x)=\sqrt[3]{x}$. The cube root function appears as a curve that increases as $x$, goes from left to right passing through the point (0,0), negative values of $x$ give negative output while the positive ones give positive values.

Identify the transformations

Observe that the given function $g(x)=\sqrt[3]{-x-2}$ can be derived from the parent function $f(x)=\sqrt[3]{x}$ through the transformations: reflection about the y-axis - because of the negative sign in front of $x$, and horizontal shift - 2 units to the right.

Apply the transformations to the graph

Finally, the graph of $g(x)$ can be obtained by applying the transformations to the graph of $f(x)$. Start by reflecting the graph of $f(x)$ about the y-axis. Then shift this reflected graph 2 units to the right. Draw the curve. This new curve is the graph of $g(x)=\sqrt[3]{-x-2}$.

Verification

In order to confirm that you have the correct graph, you may want to choose specific points on $f(x)$ and the corresponding points on $g(x)$, to see if the transformations have been applied correctly. Choosing the point (1,1) on $f(x)$ for example, through reflection it becomes (-1, 1) on $g(x)$ and after a shift 2 units to the right it becomes (1,1). Likewise for many more points on $f(x)$.

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91影视

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

Short Answer

Step by step solution

Graph the parent function

Identify the transformations

Apply the transformations to the graph

Verification

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