What differences do you see between the given function and the parent function? Apply those transformations to the graph of the parent function, f(x)=sqrt(x).
Graph:
Domain: D = { x | x ≥ 0 } Range: R = { f(x) | f(x) ≥ - 2 }
Practice makes perfect
The given function is a square root function.
f(x)=sqrt(x)- 2
The graph of it will be a transformed version of the parent function, y=sqrt(x). Square 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.
Now, we will translate the graph 2 units down by subtracting 2 to each of the y-coordinates.
Finally, we have the graph of the given function.
Finding the Domain and Range
To determine the domain of the function, recall that the radicand cannot be negative.
x ≥ 0
Therefore, the possible values of x are those such that x≥ 0. By substituting the minimum value of the domain into the function, we can find the minimum value of the range.
