Lesson 2

Lesson 2: Square Root of a Function

 

Focus

 

In previous mathematics courses you have determined the square root of a number. Is it possible to determine the square root of a function? If so, what do you think the graph of the function would look like?

 

Accelerated motion occurs when the speed of an object changes. In Lesson 1 you looked at the formula d = 5t2 that models an object when it is dropped near Earth. You can use this formula to determine the time an object takes to fall if you know the distance the object falls by rearranging the formula to

 

In this form the formula is a linear function. When graphed, the formula will be a straight line. To algebraically solve this equation for t, you would determine the square root of both sides of the equation. How would the graph of the function's square root be different from the graph of the original function?

In Module 1 you looked at ways to change graphs of functions through transformations and by finding the inverse function. In Lesson 2 you will look at another method to change the graph of a given function by taking the square root of the function.

 

Lesson Outcomes

 

At the end of this lesson you will be able to

• sketch the graph of given the graph of y = f(x), and explain the strategies used
• compare the domain and range of the functions  and y = f(x) and explain why the domain and range might be different

Lesson Question

 

In this lesson you will investigate the following question:

• How does determining the square root of a given function change the graph?

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 2 Assignment (Download the Lesson 2 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms and Formula Sheet
• work under Project Connection

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

 

Before beginning this lesson you should be able to identify x- and y-intercepts and maximum or minimum values of quadratic functions.

Are You Ready?

 

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. Determine the x-intercept(s) and y-intercept for these quadratic functions. Also determine the maximum or minimum values.
   1. y = x2 − 9 Answer
   2. y = −x2 + 16 Answer

If you answered the Are You Ready? questions without difficulty, move to Discover.

 

If you found the Are You Ready? questions difficult, complete Refresher.

Refresher

 

Do a review of graphing and solving quadratic functions in “Quadratic Functions 3.”

---

Do a review of quadratic functions in the form y = a(x − h)2 + k in “Quadratics in Vertex Form—Activity A.”

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

 

Try This 1

 

How does the graph of a function compare to the graph of the square root of the function? Compare the graphs for the functions y = x + 4 and

1. Fill in the table of values. Create a graph using the values.
   
   

   x	y = x + 4
   −6
   −4
   −3
   0
   5
   12

2. Fill in the table of values. Create a graph using the values on the same grid that you graphed y = x + 4.
   
   

   x
   −6
   −4
   −3
   0
   5
   12

3. Compare the two functions.
   1. State the domain and range of the two functions. Explain why you think they are different.
   2. When y = 0 for the original function, y = 0 for the square root of the function. Why?
   3. When y = 1 for the original function, y = 1 for the square root of the function. Why?
   4. When y = 4 for the original function, y = 2 for the square root of the function. Why?

Save your responses in your course folder.

 

Share 1

 

With a partner or group, discuss the following question based on your graphs created for Try This 1:

 

Identify a pattern between the coordinates (x, y) of the first function y = x + 4 and the coordinates of the second function

 

If required, save a record of your discussion in your course folder.

Explore

 

In Try This 1 you compared the graphs of y = x + 4 and You may have noticed that the mapping  shows the correspondence between points on the first graph and those points on the second graph.

 

What other patterns do you see when you compare the original function y = x + 4 and the square root of the original function

 

In Try This 2 you will compare a given function to the square root of the given function.

 

Try This 2

 

Use Visualizing Square Root to compare a function f(x) and the square root of that function

 

 

Comparison 1 has been done for you as an example.

 

Comparison 1: Compare the functions f(x) = −x + 1 and

 

Step 1: Select the linear function in Visualizing Square Root.

 

Step 2: Use the sliders to set a to −1 and b to 1.

 

Step 3: Select the "Show Square Root" box and compare the two graphs. Record your observations in a table.

 

Solution

 

	f(x) = –x + 1
Domain	{x|x ∈ R}	{x|x ≤ 1, x ∈ R}
Range	{y|y ∈ R}	{y|y ≥ 0, y ∈ R}
Invariant Point(s)	(1, 0) and (0, 1)
Sketch/Image of Both Functions

 

Comparison 2: Compare the functions f(x) = 0.5x − 1 and

Step 1: Change a to 0.5 and b to −1.

Step 2: Record your observations in a table similar to the following one.

 

	f(x) = 0.5x − 1
Domain
Range
Invariant Point(s)
Sketch/Image of Both Functions

 

Comparison 3: Compare the functions f(x) = x2 − 1 and

Step 1: Change the function to quadratic.

Step 2: Change a to 1, h to 0, and k to −1.

Step 3: Record your observations in a table similar to the following one.

 

	f(x) = x2 − 1
Domain
Range
Invariant Point(s)
Sketch/Image of Both Functions

 

Comparison 4: Compare the functions f(x) = 3x2 + 4 and

Step 1: Change a to 3, h to 0, and k to 4.

Step 2: Record your observations in a similar table.

 

	f(x) = 3x2 + 4
Domain
Range
Invariant Point(s)
Sketch/Image of Both Functions

1. Look at the location of the invariant points. Is there a pattern? Why are the invariant points located where they are?
2. How do the domains compare? Why are there differences?
3. How do the ranges compare? Explain the differences.
4. Refer to the sketches of each comparison. What happens to the value of as the value of f(x) changes? Use the following table to summarize the common patterns you observed for all four comparisons.
   
   

   Value of f(x) (original graph)	When the y values are less than 0 (f(x) < 0)	When the y value is 0 (f(x) = 0)	When the y values are between 0 and 1 (0 < f(x) < 1)	When the y value is 1 (f(x) = 1)	When the y values are greater than 1 (f(x) > 1)
   Value of

Save your responses in your course folder.

 

Share 2

 

With a partner or group, discuss the following questions based on your graphs created in Try This 2:

 

Based on the patterns you identified in Try This 2, create a method you could use to graph the square root of any function.

 

If required, save a record of your discussion in your course folder.

During Try This 1 and Try This 2, you began to notice patterns between the graphs of y = f(x) and Do these patterns help you graph the square root of any function?

 

Your chart from Try This 2, which summarizes the patterns between the graphs of y = f(x) and , may or may not look similar to the following chart. However, the patterns should be similar. This chart arranges the patterns by the value of the original function, y = f(x), and the effect on the graph of the square root of the function,

 

f(x)	f(x) < 0	f(x) = 0	0 < f(x) < 1	f(x) = 1	f(x) > 1
graph Note: Take the square root of the y-values of y = f(x), and the range must be positive.	graph undefined	and y = f(x) graphs intersect on x-axis. This is an invariant point.	graph is above y = f(x) graph	graph intersects y = f(x) graph. This is an invariant point.	graph is below y = f(x) graph

 

Based on the patterns you have seen throughout Lesson 2, you will see how to graph when given the graph y = f(x). Go to Graphing the Square Root of a Function.

 

 

Self-Check 1

In Try This 1 and Try This 2 you discovered some of the patterns between the domain and range of y = f(x) and

 

The domains of the square root functions are restricted in the areas of the original graph where y < 0. That is, any portion of the original graph that lies below the x-axis will not appear in the graph of the square root function. The range is restricted because the square roots of negative values are not real numbers. As a result, the range contains only values that are greater than or equal to zero.

 

You will now examine methods to graphically or algebraically compare the domains and ranges of a function and its square root.

Connect

 

Lesson 2 Assignment

Lesson 2 Summary

 

In this lesson you looked at the graph of when given the graph of y = f(x). You can use the values from the function f(x) to predict the values of the function The y-values of the points of  are the square roots of the y-values of the points on the original function y = f(x).

 

In terms of mapping: so The invariant points occur when f(x) = 0 and f(x) = 1 because the square root of 0 is 0 and the square root of 1 is 1. The domain of  are the x-values of f(x) for which f(x) is greater than or equal to zero. The range of  are the y-values in the range of f(x) for which f(x) is defined.

 

f(x)	f(x) < 0	f(x) = 0	0 < f(x) < 1	f(x) = 1	f(x) > 1
graph Note: Take the square root of the y-values of y = f(x), and the range must be positive.	graph undefined	and y = f(x) graphs intersect on x-axis	graph is above y = f(x) graph	graph intersects y = f(x) graph	graph is below y = f(x) graph

 

Some of the key lesson points are highlighted on the following graph.

 

 

In Lesson 3 you will study how the graphs of radical functions can help solve radical equations.