Lesson 1

Lesson 1: Investigating Quadratic Functions

Focus

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In this lesson you will learn about the vertex form of the quadratic function, y = a(x − p)2 + q. You will notice patterns between the values of a, p, and q in the function and the shape and position of the graph. You will then be able to predict and graph the path of a quad in a jump, a thrown football, a flying hockey puck, or any real-world projectile example.

Outcomes

• explain why y = a(x − p)2 + q is a quadratic function
  
  
• compare the graph of y = ax2 to the graph of y = x2, and state rules about the effect of a
  
  
• compare the graph of y = x2 + q to the graph of y = x2, and state a rule about the effect of q
  
  
• compare the graph y = a(x − p)2 to the graph of y = x2, and state a rule about the effect of p

Lesson Questions

You will investigate the following questions:

• How do the values of a, p, and q affect the appearance and placement of the graph of the function y = a(x − p)2 + q?
  
  
• What are the characteristics of quadratic functions and their graphs?

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 Module 3 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 are having difficulty with concepts or calculations, contact your teacher.

Materials and Equipment

You will need a graphing calculator and graph paper.

Launch

Do you have the background knowledge and skills you need to complete this lesson successfully? This section, which includes Are You Ready? and Refresher, will help you find out.

Before beginning this lesson, you should be able to

• complete a data table for a given function
  
  
• draw a graph from a table of values
  
  
• identify domain and range
  
  
• identify the x- and y-intercepts of a graph

Are You Ready?

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

1.  
   1. Complete a data table like the one shown for the quadratic function y = x2 + 2.
      
      
       

      x	0	1	2	3	−1	−2	0.5
      y

      
      
      Answer
      
      
   2. Draw a graph from the table of values in question 1.a. Join the data points in a smooth curve. You can use the graph paper provided, or you can search the Internet for a graphing tool. Answer
      
      

   3. Identify the x- and y-intercepts, if any, by looking at the graph. Answer

      
      
2.  
   
   1. Fill in a data table like the one shown for the quadratic function y = −5(x − 1)2 + 5, which describes the path of a quad during a jump.
      
      
       

      x	0	1	2	0.5	1.5
      y

      
      
      Answer
      
      
   2. Draw a graph from the table of values in question 2.a. Join the data points in a smooth curve. Answer
      
      

   3. Identify the x- and y-intercepts, if any, by looking at the graph. Answer

      
      

3. Define domain and range and give an example of each. Answer

How did the questions go? If you feel comfortable with the concepts covered in the questions, skip forward to Discover. If you experienced difficulties, use the resources in Refresher to review these important concepts before continuing through the lesson.

Refresher

Use Graph to review how to complete a data table of values and draw a graph for a given function.

Review how to identify domain and range. Work through Concept Development Domain and Range, which offers information, practice questions, and feedback. You will need to click on Domain and Range on two different screens.

Practise identifying the x- and y-intercepts of a graph. An x-intercept of a graph is the x-coordinate of a point where the graph crosses the x-axis. A y-intercept of a graph is the y-coordinate of a point where the graph crosses the y-axis.

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

Share 1

Based on your observations from Try This 1, discuss the following questions with a partner or group.

1. How do the width and orientation of the graph change as the value of a changes? Describe the pattern you saw.
   
   
2. How does the position of the graph change as the value of q changes? Describe the pattern you saw.
   
   
3. How does the position of the graph change as the value of p changes? Describe the pattern you saw.
   
   
4. Summarize your discussion by creating three general rules about the effects of a, p, and q on y = a(x − p)2 + q. You might find the chart shown provides a convenient way to summarize your understandings.
   
   
    

   Variable Changed in y = a(x − p)2 + q	What You Noticed (sketch, patterns)	General Rule
   a
   p
   q

Save your responses in your course folder.

Explore

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In Try This 1 you had a chance to view many different graphs of quadratic functions. You may have noticed that the graph of a quadratic function is always a particular shape, called a parabola. In the photo shown, the thickest blue line can be described as having a parabolic shape. Can you see the similarity between the thick blue line and the graphs of quadratic functions you have observed so far in this lesson?

Try This 2

In Axis of Symmetry you will explore the shape of the parabola and its special features.

Maximum and Minimum Points

In Axis of Symmetry you may have noticed the pattern between a and the existence of a maximum or minimum point. The following chart summarizes the pattern.

a	Sketch	Orientation	Maximum or Minimum
3		opens upward	minimum
−3		opens downward	maximum

You may have found that the value of a in y = a(x − p)2 + q has an effect on the parabola. The value of a in the equation influences the sharpness of the curve of the parabola. The table shown summarizes this effect.

Value of a	Effect on Parabola
As the value of a increases above 0 . . .	. . . the curve goes from very wide to more and more narrow.
As the value of a becomes increasingly negative . . .	. . . the downward-opening curve becomes more and more narrow.

You also explored the value of p and q in the vertex form of a quadratic function.

q	The value of q in the equation gives the position of the vertex of the parabola relative to the y-axis (the q-value moves the vertex up and down). That is one reason this form of the equation is called the vertex form.
p	The value of p in the equation gives the position of the vertex of the parabola relative to the x-axis (the p-value moves the vertex left and right). The value of p is also the x-coordinate of the axis of symmetry of the parabola.

Self-Check 1

In Share 1, you created a table to organize your observed patterns and the three general rules you developed to describe the effect of a, p, and q on y = a(x − p)2 + q. Use your table and the graphs provided below to help answer the following questions.

1. Match each equation with its graph.
   
   
   1. y = x2 Answer
      
      
   2. y = 3x2 Answer
      
      
   3. y = −4x2 Answer
      
      
   4. y = 0.3x2 Answer
      
      
   5. y = x2 + 5 Answer
      
      
   6. y = x2 − 4 Answer
      
      
   7. y = (x − 4)2 Answer
      
      
   8. y = (x + 5)2 Answer
      
      
2. Which of the graphs has a maximum value? What is that value? Answer
   
   
3. What is the minimum value of graph 2? Answer
   
   
4. What is the domain and range of graph 7? Answer
   
   
5. Why are all of the functions in Try This 1 considered to be quadratic functions? Answer
   
   
6. Look at the equations in the chart. Without graphing, predict what each graph would look like.
   
   
    

   Equation y = a(x − p)2 + q	What Does a Tell You About This Graph?	What Does p Tell You About This Graph?	What Does q Tell You About This Graph?
   y = 6(x − 3)2 + 5
   y = 6(x − 3)2 − 4
   y = 6x2 + 1
   y = −6(x + 3)2 + 1
   y = −0.5(x − 3)2 + 1

   
   
   Answer

Connect

Lesson 1 Assignment

Lesson 1 Summary

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In this lesson you investigated the following questions:

• How do the values of a, p, and q affect the appearance and placement of the graph of the function y = a(x − p)2 + q?
  
  
• What are the characteristics of quadratic functions and their graphs?

You studied a specific form of the quadratic function, the vertex form. A quadratic function has a variable raised to the second power—in other words, squared. You learned that the graph of the function a(x − p)2 + q forms a parabolic shape. You discovered the effect of the variables a, p, and q on the shape of the parabola.

You saw that a affects the shape and orientation of the parabola. The farther a is from 0, the narrower the parabola. The orientation of the parabola is also affected by a in the way summarized in the table.

Direction of Opening	If a > 0, the parabola opens upward.
	If a < 0, the parabola opens downward.

The variable p determines where the parabola is on the x-axis, and q determines where the parabola is on the y-axis.

The parabolic shape and other terms described during this lesson are summarized in Summary of Common Characteristics of Quadratic Functions.

In the next lesson you will investigate how to determine the coordinates of the vertex of a quadratic formula and how to accurately sketch the graph. You will learn how to determine the x- and y-intercepts of the graph and write a quadratic equation from a graph.