Transformation: Vertical shrink by a factor of 13 and vertical translation up by 6 units. Graph:
Practice makes perfect
We want to describe the transformations of the parent function f(x)=x^(1/3) represented by g(x)= 13x^(1/3)+6. Let's start by rewriting f(x) and g(x) into the general format of cube root function.
f(x) = x^(1/3) ⇔ f(x) = sqrt(x) [0.5em]
g(x) = 1/3x^(1/3) +6 ⇔ g(x) = 1/3 sqrt(x)+6
Now, let's look at the possible transformations. Then we can more clearly identify the ones being applied.
Transformations of f(x)
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k>0
y=f(x)- k
Vertical Stretch or Shrink
Vertical stretch, a>1
y= af(x)
Vertical shrink, 0< a< 1
y= af(x)
Now, using the table, let's highlight the transformations.
g(x)= 1/3 sqrt(x)+ 6
We can describe the transformations as a vertical shrink by a factor of 13 and a vertical translation up by 6 units. Let's begin by graphing the parent function.
Next, we will multiply the y- coordinates by a= 13. This shrinks the parent graph by a factor of 13.
Next, we will translate the graph 6 units up. To do so we will add 6 to every y-coordinate.
Finally, let's show the graphs of f(x) and g(x) on the same coordinate plane.
