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≥- 4 Range: f(x) ≤ - 1
Practice makes perfect
The given function is a square root function.
f(x)=- (1/3sqrt(x+ 4)+ 1)
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
Let's put the minus sign aside before we start to graph the given function. We will first graph f(x)=1/3sqrt(x+4)+1 and then we will reflect it across the x-axis.
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=13. This shrinks the parent graph by a factor of 13.
Now, we will translate the graph 1 unit up by adding 1 to each of the y-coordinate.
Next, we will translate the graph 4 units to the left. To do this, we will subtract 4 to each x-coordinate.
The last thing to do is to reflect the graph of f(x)= 13sqrt(x+4)+1 across the x-axis because there is a minus sign in front of the given fuction. To do this, we will put (-) sign to each y-coordinate.
Points
Reflection in the x-axis
(0,1.6)
(0,-1.6)
(5,2)
(5,-2)
Now we can plot the reflected graph.
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+4 ≥ 0 ⇔ x ≥ - 4
Therefore, the possible values of x are those such that x≥ - 4.
We have a decreasing function and it always takes negative values. By substituting the minimum value of the domain into the function, we can find the maximum value of the range.
