Question

Graphing Transformations Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations.

$f\left(x\right)=\sqrt[3]{-x}$

Short answer

The required graph of the function is shown in Figure below.

Step-by-step solution

Step 1. Concept of transformation of function.

1). Vertical shifting: To shift a function in the upward direction, a constant is added to the function that is positive and in order to the shift the function downward then a negative constant is added to the function.

Consider the function for the vertical shifting.

$\begin{array}{l}y=f\left(x\right)+c\\ y=f\left(x\right)-c\end{array}$

2). Horizontal shifting: To shift the graph of the function horizontally that is either left or right the horizontal shifting is used. The function$f\left(x\right)$ is shifted n units to the right as$f\left(x-n\right)$ and the same function is shifted n unit to the left by $f\left(x+n\right)$.

3). Reflecting graph: To reflect the y coordinate of each point of the graph of a function$y=f\left(x\right)$ just change the sign of the function as $y=-f\left(x\right)$. To change the x coordinate of the graph, just shift the graph the required function will be $f\left(-x\right)$.

4). Vertical stretch and shrink: The graph of the function$y=f\left(x\right)$ is stretched if it is multiplied by a constant c that is $y=cf\left(x\right)$, and the function shrink if the function is divided by the constant c and is $y=\frac{1}{c}f\left(x\right)$.

5). Horizontal stretch and shrink: The graph of the function$y=f\left(x\right)$ is stretched for $y=f\left(cx\right)$, and the function shrink if the function $y=f\left(\frac{x}{c}\right)$.

Step 2. Graph of standard function.

The given function is $f\left(x\right)=\sqrt[3]{-x}$

Here, the standard function is $f\left(x\right)=\sqrt[3]{x}$.

The graph for the standard function is shown in Figure below.

Step 3. Shifted function.

From the concept of the reflecting of graph. The graph of the function$\sqrt[3]{x}$ is reflected along the y axis and forms the function $f\left(x\right)=\sqrt[3]{-x}$.

Hence, the graph of the function$f\left(x\right)=\sqrt[3]{-x}$ is obtained.

Step 4. Graphical Interpretation.

The graph of the function$f\left(x\right)=\sqrt[3]{-x}$ that is reflected form of the graph$\sqrt[3]{x}$ along the y axis is shown in figure below.