Chapter 10.2, Problem 12Q

To determine

To compare the graph.

Answer to Problem 12Q

  $h=-1$

Explanation of Solution

Given information: $f\left(x\right)=\sqrt[3]{x}$ and $p\left(x\right)=\sqrt[3]{x-h}$

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 they-axis by the amount added. If a positive number is subtracted, the function shifts down they-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 thexoryvalues across thexory-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 horizontal shift of 1 units left, so $h=-1$