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≥1 Range: y ≥ 3
Practice makes perfect
The given function is a square root function.
f(x)= 1/2sqrt(x- 1)+ 3
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.
Next, we will multiply the y-coordinates by a= 12. This shrinks the parent graph by a factor of 12.
Now, we will translate the graph 3 units up by adding 3 to each of the y-coordinates.
For the last transformation, we will translate the graph 1 unit to the right. To do this, we will add 1 to each x-coordinate.
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-1 ≥ 0 ⇔ x ≥ 1
Therefore, the possible values of x are those such that x≥ 1. By substituting the minimum value of the domain into the function, we can find the minimum value of the range.
