Chapter 10.2, Problem 14E

To determine

To compare the graph.

Answer to Problem 14E

  $k=4$

Explanation of Solution

Given information: $f\left(x\right)=\sqrt[3]{x}$ and $g\left(x\right)=\sqrt[3]{x}+k$

Formula Used: Graph Comparison

Calculation:

• To translate a function vertically is to shift the function up or down. If a positive number is added, the function shifts up the y-axis by the amount added.  If a positive number is subtracted, the function shifts down the y-axis by the amount subtracted.  In general, a vertical translation is given by the equation:

  $y=f\left(x\right)+b$

Where  $f\left(x\right)$ is some given function and  $b$  is the constant that we are adding to cause a translation.

• To translate a function horizontally is the shift the function left or right. While vertical shifts are caused by adding or subtracting a value outside of the function parameters, horizontal shifts are caused by adding or subtracting a value inside the function parameters.  The general equation for a horizontal shift is given by:

  $y=f\left(x-a\right)$

Where  $f\left(x\right)$  would be the original function, and  $a$  is the constant being added or subtracted to cause a horizontal shift.  When  $a$  is positive, the function is shifted to the right.  When  $a$  is negative, the function is shifted to the left.

• A reflection swaps all of the x or y values across the x or y-axis, respectively. It can be visualized by imagining that a mirror lies across that axis.
A vertical reflection is given by the equation  $y=-f\left(x\right)$ and results in the curve being “reflected” across the x-axis.

A horizontal reflection is given by the equation  $y=f\left(-x\right)$ and results in the curve being “reflected” across the y-axis.

As in the graph there is a vertical shift of 4 units up, so $k=4$