1 Answer

Shira · Answer 1 · 2023-12-28T04:51:27+0000

We will investigate the techniques used to graph a plot on a cartesian grid.

To plot any function on a cartesian grid we employ a systematic method to construct the function from its parent. A parent function is the base function which describes the preliminary relationship between the two variables.

We see that the function f ( x ) involves a radical i.e " cube root ". So we can say that the parent function follows the relationship of a cube root function as follows:

$y=\sqrt[3]{x}$

The above is the parent relation of the given function f ( x ). We can graph the above parent function as the first step to future transformation:

The above graph expresses a relationship of the parent function.

Now we will attempt to transform the parent function by the help of translation and dilation.

The general rule of translation is expressed as follows:

$y\text{ = }\sqrt[3]{x+a}\text{ + b}$

Where,

$\begin{gathered} a\colon\text{ Horizontal shift} \\ b\colon\text{ Vertical shift} \end{gathered}$

To further describe the direction of each shift exployed by the constant ( a and b ) is given:

$\begin{gathered} a\text{ }>\text{ 0 }\ldots\text{ Left shift} \\ a\text{ < 0 }\ldots\text{ Right shift} \\ \\ b\text{ > 0 }\ldots\text{ Up shift} \\ b\text{ < 0 }\ldots\text{ Down shift} \end{gathered}$

The magnitude of each parameter ( a and b ) will denote the number of unit step that entire function will shift towards.

To model the given function f ( x ) from the parent function we see that there is a horizontal shift of one unit towards the right. The amount of vertical shift is is zero!

Therefore,

$\begin{gathered} a\text{ = -1} \\ b\text{ = 0} \end{gathered}$

The parent function undergoes horizontal translation to the right. The modified relationship is expressed as follows:

$y\text{ = }\sqrt[3]{x\text{ - 1}}$

The plot of the modified function is given as follows:

We see that every single point of the parent function is shifted to the right by 1 unit!

We will undergo an another step of transformation i.e dilation. Dilation is specified by a scale factor that either stretches or compresses the initial function.

The general guideline for a scale factor is expressed as follows:

$f\text{ ( x ) = S.F}\cdot\text{ }\sqrt[3]{x-1}$

Where,

$\begin{gathered} S\mathrm{}F\colon\text{ Scale Factor} \\ SF\text{ > 0 }\ldots\text{ Stretch} \\ SF\text{ < 0 }\ldots\text{ Compresses} \end{gathered}$

We see that the given function has a scale factor of 2. This means we expect the initial function to be expanded or stretched vertically! The function is:

$f\text{ ( x ) = 2}\cdot\sqrt[3]{x-1}$

The plot of the given function would stretch the modified function vertically by 2 units. This means we expect each and every coordinate of the function to be pulled out! The plot is given as:

From the above plot of the given function f ( x ) we can extract 5 coordinate pairs that completely describes the entire function as follows: