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: x≥0 Range: f(x) ≥ 0
Practice makes perfect
The given function is a square root function.
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
To graph the given square root function, let's put it in general form first.
f(x)=sqrt(3x) ⇔ f(x)=sqrt(3)sqrt(x- 0)+ 0
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. Let's begin with the parent function.
Now, we will multiply the y-coordinates by a=sqrt(3). This stretches the parent graph by a factor of sqrt(3).
Since h and k are zero, neither horizantal nor vertical translation is needed. Thus, 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.
3x ≥ 0 ⇔ 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.
