Chapter 10.4, Problem 5AYU

To graph $y={\left(x-5\right)}^3-4$, shift the graph of $y=x^3$ to the (left/right) _______ unit(s) and then (up/down) _______ unit(s). (pp. 106-114)

To determine

To find: Number of units the graph of $y=x³$ shifts to obtain the graph of $y={\left(x-5\right)}^{³}-4$.

Answer to Problem 5AYU

Solution:

Right 5 units and down 4 units.

Explanation of Solution

Given:

$y={\left(x-5\right)}^{³}-4$ and $y=x^3$.

Formula used:

1. If the argument $x$ of a function $f$ is replaced by $x-h$, $h>0$, the graph of the new function $y=\left(x-h\right)$ is the graph of $f$ shifted horizontally right $h$ units.

2. If a positive real number $k$ is subtracted to the output of a function $y=f\left(x\right)$, the graph of the new function $y=f\left(x\right)-k$ is the graph of $f$ shifted vertically down $k$ units.

Calculation:

We plot the graph of $y=x^3$ and $y={\left(x-5\right)}^{³}-4$.

We plot the graph using the points in the table.

The graph above represented by blue line is the graph of $y={\left(x-5\right)}^3-4$ and the graph represented by red dotted line is the graph of $y=x^3$.

From the graph we see that, $y={\left(x-5\right)}^{³}-4$ is obtained by shifting the graph $y=x^3$ horizontally 5 unit to the right and shifting vertically down by 4 units again.