What differences do you see between the given function and the parent function? Apply those transformations to the graph of the parent function, y=sqrt(x).
Graph:
Domain: All real numbers Range: All real numbers
Practice makes perfect
The given function is a cube root function.
f(x)= 4sqrt(x- 2)+ 1
The graph of it will be a transformed version of the parent function y=sqrt(x). Cube root functions typically follow the same general format.
f(x)= asqrt(x- h)+ k
Graphing the Function
To graph the given function, let's show the possible transformations of f(x)=sqrt(x).
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
Horizontal Translations
Translation right h units, h>0
y=f(x- h)
Translation left h units, h>0
y=f(x+ h)
Vertical Stretch or Shrink
Vertical stretch, a>1
y= af(x)
Vertical shrink, 0< a<1
y= af(x)
Reflections
In the x-axis
y= - f(x)
In the y-axis
y= f(- x)
Using the table, we can graph the function as a series of transformations. Let's begin with the parent function.
Next, we will multiply the y-coordinates by a= 4. This stretches the graph of the parent function by a factor of 4.
Now, we will translate the graph 1 unit up by adding 1 to each of the y-coordinates.
As our last transformation, we will translate the graph 2 units to the right To do this, we will add 2 to each x-coordinate.
Finally, we have the graph of the given function.
Finding the Domain and Range
Looking at the graph of the function, notice that it continues to infinity in both the positive and negative directions. This means that the domain and range of the function are all real numbers.
Domain:& All real numbers
Range:& All real numbers
