Lesson 1

Lesson 1: Radical Functions and Transformations

Focus

Police officers use mathematics to analyze car accident scenes. Investigators are able to determine what likely happened without even seeing the actual accident. To determine the vehicle’s speed before the collision, investigators may use the formula where S is the speed of the vehicle, L is the length of the skid marks, and f is the coefficient of friction (determined by the road surface). By looking at the formula, can you describe the relationship between speed and the length of skid marks? Would a graph of this function help you to more easily see the relationship?

In Lesson 1 you will look at graphing functions that include radical signs.

Lesson Outcomes

At the end of this lesson, you will be able to

• sketch the graph of radical functions
• state the domain and range of radical functions
• apply transformations to radical functions

Lesson Questions

You will investigate the following questions:

• How can radical functions be graphed?
• How are graphs of radical functions used to analyze real-life situations?

Assessment

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

• completion of the Lesson 1 Assignment (Download the Lesson 1 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

Self-Check activities are for your own use. You can compare your answers to suggested answers to see if you are on track. If you have difficulty with concepts or calculations, contact your teacher.

Remember that the questions and activities you will encounter provide you with the practice and feedback you need to successfully complete this course. You should complete all questions and place your responses in your course folder. Your teacher may wish to view your work to check on your progress and to see if you need help.

Time

Each lesson in Mathematics 30-1 Learn EveryWare is designed to be completed in approximately two hours. You may find that you require more or less time to complete individual lessons. It is important that you progress at your own pace, based on your individual learning requirements.

This time estimation does not include time required to complete Going Beyond activities or the Module Project.

Materials and Equipment

• graph paper

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 restrictions on variables in radical expressions
• determine the value of square roots
• determine the inverse of a function

Are You Ready?

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

1. Identify any restrictions on the variable in the following expressions.
   1. Answer
   2. Answer
   3. Answer
2. Determine the value of the following:
   1. Answer
   2. x2 = 9 Answer
3. Determine the equation of the inverse function of the function f(x) = 8x2, x ≥ 0. Answer

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

Refresher

Go to Radical at the Mathematics Glossary website to review the definition of radical.

The video titled “Domain of a Function” provides a review of restrictions on the variable in radical expressions.

For a review of square roots of value, view the video titled “Understanding Square Roots.”

The video “Function Inverses Example 2” provides a review of the inverse of a quadratic function.

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

Discover

Air resistance causes objects falling to Earth to fall at different rates. If air resistance is removed, all objects drop at the same rate. The video “The Hammer and Feather” shows two objects falling in a vacuum (i.e., there is no air resistance).

Image Credit: Apollo 15 Crew, NASA

Try This 1

For objects falling near the surface of Earth, the function d = 5t2 approximately models the time, t, in seconds, for an object to fall a distance, d, in metres, if the resistance caused by air can be ignored.

1.  
   1. Identify any restrictions on the domain of the function d = 5t2. Why are these restrictions necessary?
   2. What is the range of the function?
   3. Create and complete a table similar to the following for the function d = 5t2.

       

      t	d

   4. Create a graph showing the distance fallen as a function of time.
2. 
   
   1. Determine the equation of the inverse function of d = 5t2.
   2. Create and complete a table of values similar to the following, and then graph the inverse function.1

       

      t	d

Save your responses in your course folder.

Share 1

With a partner or in a group, compare and describe the original function and the inverse function in terms of domain, range, and shape of graph.

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

1 Adapted from Pre-Calculus 12. Whitby, ON: McGraw-Hill Ryerson, 2011. Reproduced with permission.

Explore

The inverse function from Try This 1 was this is a radical function.

In Try This 2 you graphed the simplest radical function possible: Did you notice the distinct shape that you graphed? You may remember this shape from Try This 1 in Discover. You have now seen two radical functions with this distinct shape. This shape is sometimes describes as one-half of a parabola opening to the right. The domain of the function is {x|x ≥ 0, y ∈ R} because you cannot take the square root of a negative number. The range of the function is {y|y ≥ 0, y ∈ R}.

Radical functions can be much more complex than the function you graphed in Try This 2. The general form of a radical function can be written as Do you see the similarities in this form to the general form for functions used in Module 1?

In Try This 3 you will investigate how the graph of radical functions changes in response to changes in the parameters a, h, and k of the general form of the function.

Try This 3

Open “Functions Involving Square Roots.”

Select the “Show domain” and “Show range” boxes in the gizmo.

This gizmo begins with a graph of Check that your graph, the domain, and the range from Try This 2 match this graph.

1. Use the slider to change the value of a, and record your answers in a chart like the following.
   
   

   a-value	Function	Domain	Range	Transformation of Graph and State the Stretch Factor	Sketch or Image
   1		{x|x ≥ 0, x ∈ R}	{y|y ≥ 0, y ∈ R}	none	Screenshot reprinted with permission of ExploreLearning.
   Positive value: ________
   Negative value: ________

2. Set the a-value back to 1. Use the slider and change the value of h, and record your answers in a chart like the following.
   
   

   h-value	Function	Domain	Range	Transformation of Graph and State the Value of the Translation	Sketch or Image
   0		{x|x ≥ 0, x ∈ R}	{y|y ≥ 0, y ∈ R}	none	Screenshot reprinted with permission of ExploreLearning.
   Positive value: ________
   Negative value: ________

3. Set the h-value back to zero. Use the slider, change the value of k, and record your answers in a chart like the following.
   
   

   k-value	Function	Domain	Range	Transformation of Graph and State the Value of the Translation	Sketch or Image
   0		{x|x ≥ 0, x ∈ R}	{y|y ≥ 0, y ∈ R}	none	Screenshot reprinted with permission of ExploreLearning.
   Positive value: ________
   Negative value: ________

Open Multiple Transformations. Click on “Quadratic” to deselect it, and click on “Square root” to select this type of function.

4. Change the a, b, h, and k sliders to explore what happens to the graph when these values are changed. In a chart like the following, summarize how the parameters a, b, h, and k transform the graph given in the general form
   
   

   Parameter	Value > 0	Value < 0
   a
   b
   h
   k

Save your responses in your course folder.

In Try This 3 you may have noticed that the parameters a, b, h, and k from the general form affect the graph of the radical equation in the following ways.

Parameter	Value > 0	Value < 0
a	Vertical stretch of graph of by a factor of a.	Vertical stretch of graph of by a factor of a. Graph of reflected in x-axis.
b	Horizontal stretch of graph of by a factor of	Horizontal stretch of graph of by a factor of Graph of reflected in y-axis.
h	Graph of is translated to the right h units.	Graph of is translated to the left h units.
k	Graph of is translated up by k units.	Graph of  is translated down by k units.

Self-Check 1

1. Complete the four questions at the bottom of “Functions Involving Square Roots.” Check your answers.
   
   
    
   
   Screenshot reprinted with
   permission of ExploreLearning.

Why did the applet used in Try This 3 use the form and not the form

Look at an example of two radical functions to help understand how the a and b parameters can both be used to describe stretches of radical functions.

Try This 4

You will look at the graphs of the function and to see how these two functions are related.

1. Describe how to transform the graph of to produce the graph of the function
2. Describe how to transform the graph of to produce the graph of the function
3. Graph each of the two functions and Use a graphing calculator or graphing program; or graph them by hand.
4. Are the two functions and equivalent? Explain your reasoning.

Save your work in your course folder.

Radical functions can be written as having a vertical stretch or a horizontal stretch. In Try This 4 you looked at the radical functions and and discovered that the graphs of these two functions are the same. When you look at the functions, you can see how the second function can be rearranged to be the same as the first function.

You can compare these functions to the function and describe the function as having a vertical stretch by a factor of 2.

When you compare to you can describe the transformation as a horizontal stretch by a factor of

Even though the stretches are different, the resulting graphs are the same. Note: When analyzing radical functions, any stretch can be described as either a vertical stretch or a horizontal stretch.

Self-Check 3

1. Consider the function
   1. Describe the transformation represented by f(x) as compared to Answer
   2. Write a function equivalent to f(x) in the form Describe the transformation represented by g(x) as compared to Answer
   3. Graph both functions f(x) and g(x). How do the graphs compare? Answer

Up to this point, you have graphed radical functions using transformations. Now you will look at a graph and determine the radical function. View Determining a Radical Function from a Graph, which shows an example of how to write the radical function from a graph.

Radical functions are used in different applications. Following is an example of how radical functions are used in physics.

Try This 5

After an accident, police determine the speed at which the car had been moving by applying a simple radical formula, The formula can be used to approximate the speed, v, in miles per hour, of a car that has left a skid mark of length, L, in feet.

1. Describe the transformations required to produce the function from the function
2. Graph the function
3. State the domain and range. What do these values mean in the context of this question?
4. Use the graph to determine how far a car will skid at 65 mph and at 100 mph.

Connect

Lesson 1 Assignment

Lesson 1 Summary

In this lesson you saw how radical functions can be used in a variety of applications, including accident-scene investigations. You graphed radical functions and then answered questions using the graph and you determined the equation of a radical function from a given graph.

A radical function is a function where the variable is part of the radicand. The base radical function of has the shape of half of a parabola opening to the right. The domain is {x|x ≥ 0, x ∈ R} and the range is {y|y ≥ 0, y ∈ R}. You can graph radical functions by using transformations of the base function Using the form you can describe the transformations of the base function

Parameter	Value > 0	Value < 0
a	Vertical stretch of graph of by a factor of a.	Vertical stretch of graph of by a factor of a. Graph of reflected in x-axis.
b	Horizontal stretch of graph of by a factor of	Horizontal stretch of graph of by a factor of Graph of reflected in y-axis.
h	Graph of is translated to the right h units.	Graph of is translated to the left h units.
k	Graph of is translated up by k units.	Graph of is translated down by k units.

You can also determine the domain and range of any radical function using the ideas of transformations.

In Lesson 2 you will study how the graphs of a function and the square root of the same function are related.