Mrs. Garcia has assigned her student the function
Alendro says that because it is a square root function, it can only have positive values and therefore his graph is only in the first quadrant.
Dako says that because of the two negative sign, all y values will be positive and therefore his graph is in the first and second quadrants.
Marisha says they are both wrong. Because it is a negative square root function, her graph is in the third and fourth quadrants.
Which one of them is correct?
Guidance
A square root function has the form
x  y 

16  4 
9  3 
4  2 
1  1 
0  0 
1  und 
Notice that this shape is half of a parabola, lying on its side. For
Example A
Graph
Solution: To graph this function, draw a table.
x  y 

2  5 
3  6 
6  7 
11  8 
After plotting the points, we see that the shape is exactly the same as the parent graph. It is just shifted up 5 and to the right 2. Therefore, we can conclude that
The domain is all real numbers such that
Example B
Graph
Solution: From the previous example, we already know that there is going to be a horizontal shift to the left one unit. The 3 in front of the radical changes the width of the function. Let’s make a table.
x  y 


0 
0  3 
3  6 
8  9 
15  12 
Notice that this graph grows much faster than the parent graph. Extracting
Example C
Graph
Solution: Extracting
x  y 

2  3 
3  2 
6  1 
11  0 
18  1 
The negative sign in front of the radical, we now see, results in a reflection over
Using the graphing calculator: If you wanted to graph this function using the TI83 or 84, press
Intro Problem Revisit If you graph the function
Guided Practice
1. Evaluate
Graph the following square root functions. Describe the relationship to the parent graph and find the domain and range. Use a graphing calculator for #3.
2.
3.
4.
Answers
1. Plug in
2. Here, the negative is under the radical. This graph is a reflection of the parent graph over the
The domain is all real numbers less than or equal to zero. The range is all real numbers greater than or equal to zero.
3. The starting point of this function is
The domain is all real numbers greater than or equal to 3. The range is all real numbers greater than or equal to zero.
4. Using the graphing calculator, the function should be typed in as:
Explore More
Evaluate the function,

f(3) 
f(6) 
f(13)  What is the domain of this function?
Graph the following square root functions and find the domain and range. Use your calculator to check your answers.

f(x)=x+2−−−−√ 
y=x−5−−−−√−2 
y=−2x+1−−−−√ 
f(x)=1+x−3−−−−√ 
f(x)=12x+8−−−−√ 
f(x)=3x+6−−−−√ 
y=21−x−−−−√ 
y=x+3−−−−√−5 
f(x)=4x+9−−−−√−8 
y=−32x−3−−−−√+6 
y=−35−x−−−−√+7 
f(x)=23−x−−−−√−9