Square Root Graphs
Next in our lineup up basic graphs is the square root
function. Let's draw the graph of 
.
Basic Graph of
Let's start with a table of values. Notice that we are picking perfect squares for our x-values to get nice whole numbers for our function (y) values.
|
x
|
-1
|
0
|
1
|
4
|
9
|
|
f(x)
|
DNE
|
0
|
1
|
2
|
3
|
Notice that for an x-value of -1 we put 'DNE' for f(x).
This stands for Does Not Exist - as there is NO square root of -1 (or
ANY negative number for that matter.) Therefore the basic graph of
only exists for POSITIVE values of x.
Graphing the four points that we got from our table of values gives us:
These points are connected with a smooth
curve to give us the basic graph of :
Notice that there is NO ARROW on the left hand side, as this graph does NOT continue past the point (0,0).
Transformation Example
As with the other graphs you have examined, the transformations work the same way for the square root graph. The standard form of the square root function is:
where
a is the vertical stretch
h is the horizontal shift and
k is the vertical shift.
Example
Sketch the graph of 
.
Solution
Start by drawing the basic graph of :
Now perform the transformations in order:
1. In this example, our a value is 3. That means we must multiply all our y - coordinates by 3. This results in a graph of:
This is the graph of 
.
2. Our h value is negative 2. (Again, if this does not make sense, refer back to the lesson on graphing quadratic functions for an explanation of how h works.) This means that we must shift all of our points to the left 2 units. The graph after this step looks like:
This is the graph of 
.
3. Our final transformation is the k value, which in this case is 1. This means that each point must be moved up one unit. That gives us our final graph:
This is your final graph: .
Now that you have seen the basic graph of the square root function and explored some transformations, you can head back to the Polynomial and Rational Functions main page.