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:
[tex]y=\sqrt[3]{x}[/tex]
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:
[tex]y\text{ = }\sqrt[3]{x+a}\text{ + b}[/tex]
Where,
[tex]\begin{gathered} a\colon\text{ Horizontal shift} \\ b\colon\text{ Vertical shift} \end{gathered}[/tex]
To further describe the direction of each shift exployed by the constant ( a and b ) is given:
[tex]\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}[/tex]
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,
[tex]\begin{gathered} a\text{ = -1} \\ b\text{ = 0} \end{gathered}[/tex]
The parent function undergoes horizontal translation to the right. The modified relationship is expressed as follows:
[tex]y\text{ = }\sqrt[3]{x\text{ - 1}}[/tex]
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:
[tex]f\text{ ( x ) = S.F}\cdot\text{ }\sqrt[3]{x-1}[/tex]
Where,
[tex]\begin{gathered} S\mathrm{}F\colon\text{ Scale Factor} \\ SF\text{ > 0 }\ldots\text{ Stretch} \\ SF\text{ < 0 }\ldots\text{ Compresses} \end{gathered}[/tex]
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:
[tex]f\text{ ( x ) = 2}\cdot\sqrt[3]{x-1}[/tex]
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:
[tex](\text{ }-7\text{ , -4 ) , ( 0 , -2 ) , ( 1 , 0 ) , ( 2 , 2 ) , ( }9\text{ , 4 )}[/tex]
We can also use the pair of coordinates and plot them on a graph as follows:
You can connect the point plots with a free hand sketch and construct the required function f ( x ).
