Lesson 2

Lesson 2: Quadratic Function in Vertex Form

 

Focus

 

© Paul Yates/1096096/Fotolia

 

A batter really focuses on the path of a baseball once the ball leaves the pitcher’s hand. The batter has only a fraction of a second to predict the ball’s speed and path and must decide whether or not to swing. If the batter decides to swing, she or he must predict where the ball will be so the ball can be connected with the bat at the right place and angle. The ball’s path can be accurately described mathematically using a quadratic function.

 

In this lesson you will learn how to describe the coordinates of the vertex of a quadratic function and find the x- and y-intercepts of the graph. You will not only learn how to sketch a graph from the equation, but you will also learn how to determine the equation from the characteristics of the graph. Doing so will allow you to comfortably predict the path of a baseball or other projectile, such as the water needed for your Module 3 Project.

 

Outcomes

 

At the end of this lesson you will be able to

• identify and verify the coordinates of the vertex for a quadratic function of the form y = a(x − p)2 + q and write a rule describing how to do so
  
  
• sketch the graph of y = a(x − p)2 + q using transformations; and identify the vertex, domain and range, direction of opening, axis of symmetry, and x- and y-intercepts
  
  
• explain, using examples, how the values of a and q may be used to determine whether a quadratic function has zero, one, or two x-intercepts
  
  
• write a quadratic function in the form y = a(x − p)2 + q for a given graph or a set of characteristics of a graph

Lesson Questions

 

You will investigate the following questions:

• How can you sketch a graph of a quadratic function knowing the a-, p-, and q-values?
  
  
• How can you write an equation for a quadratic function from its 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 Module 3 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

Materials and Equipment

 

You will need 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

• identify and write the coordinates of any point on a grid
  
  
• define the meaning of domain and range as it relates to an equation

Are You Ready?

 

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

1.  
   1. What are the coordinates of the red dot in the graph shown? Answer
      
      
   2. What is the domain and range of the section of the graph shown? Answer
      
      
       
      
      
      
2.  
   1. What are the coordinates of the red dot in the graph line shown? Answer
      
      
   2. What are the domain and range of only the section of the parabolic graph shown? Answer
      
      
   3. If this graph were extended indefinitely, how would this change the domain and range?
      
      
       
      
      
      
      Answer
      
      
3.  
   1. What are the coordinates of the red dot in the graph shown? Answer
      
      

   2. What is the domain and range of only the section of the graph shown?
      
      

       
      
      
      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

 

Remind yourself how to identify and write the coordinates of any point on a grid. Read the definition and work with the applet at Coordinates. In the applet, select “SHOW GRID LINES.” Then drag the small green square to different positions to see how the coordinates change. Be sure to move the square to all four quadrants.

 

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Review how to identify the domain and range of an equation. Go to Domain and Range, and select Domain and Range from Graphs. Review the 11-page lesson.

 

 

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

Discover

 

© lightpoet/27428791/Fotolia

 

It is time to investigate how the coordinates of the vertex of a quadratic function are related to the variables a, p, and q in the function y = a(x − p)2 + q.

 

Try This 1

 

In Quadratic Function (Vertex Form), move the vertex (marked as a red dot) of the quadratic function to a variety of different locations on the grid. Note how the coordinates of the vertex (in red) compare to the values of the variables a, q, and p of the quadratic function (in black).

1. Record your observations in a table like the one shown.
   
   
    

   y = a(x − p)2 + q	Vertex Coordinates
   	( , )
   	( , )
   	( , )
   	( , )
   	( , )

   
   
   
2. What relationship do you see between the coordinates of the vertex and the variables in the quadratic function y = a(x − p)2 + q?
   

Save your responses in your course folder.

 

Share 1

 

With a partner or group, compare and discuss your answers to Try This 1.

 

Summarize your discussion by creating a rule that describes how to identify the coordinates of the vertex when given a quadratic function in the form of y = a(x − p)2 + q.

 

Save your responses in your course folder.

Explore

 

In Lesson 1 you created three general rules about the effects of a, p, and q on y = a(x − p)2 + q. In Try This 1, just now, you identified a rule that relates the vertex to y = a(x − p)2 + q. Go to your course folder and add your new rule to the three rules you identified in Lesson 1, Share 1, question 4.
 
You are now going to create one more rule, your fifth rule in total. This rule will help you create a sketch of a quadratic function.

 

Try This 2

1. Use Quadratic Function (Vertex Form) to explore how the variables a and q can be used to determine the number of x-intercepts in a given quadratic function. You may use a table like the one shown to record your observations.
   
   
    
   
   
   
    

   a	q	Sketch	Number of x-Intercepts
   1	5
   1	1
   1	0
   1	−1
   1	−5
   −1	5
   −1	1
   −1	0
   −1	−1
   −1	−5

   
   
   

2. What relationship do you see between the number of x-intercepts and the variables a and q in the quadratic function y = a(x − p)2 + q?

Save your responses in your course folder.

 

Share 2

 

Share your responses from Try This 2 with a partner or group.

 

Summarize your discussion by creating a rule that describes how to determine the number of x-intercepts in a quadratic function in the form y = a(x − p)2 + q.

 

Save your responses in your course folder.

 

Add your new rule that relates the variables a and q to the number of x-intercepts to the list of rules you started in Lesson 1, Share 1, question 4. You should now have five rules.

Self-Check 1

1. For the quadratic function y = 4(x − 1)2, identify the following values:
   
   
   1. vertex Answer
      
      
   2. direction of opening Answer
      
      
   3. axis of symmetry Answer
      
      
   4. domain Answer
      
      
   5. range Answer
      
      
   6. number of x-intercepts and what the x- and y-intercepts are, if any Answer

Try This 3

 

Use the five rules you saved in your course folder to help identify useful pieces of information about the quadratic function . Then use Sketch the Graph to create a sketch of the quadratic function .

 

Example 1: Formulating a Quadratic Equation from a Graph

 

In Try This 3 you sketched a graph from an equation. Now you will do the opposite—you will formulate a quadratic equation from a graph.

 

Question

 

What quadratic function describes the parabola on the grid below?

 

 

Solution

 

Step 1: Find the variables p and q. Since the vertex coordinates are (p, q), you can read the coordinates of the vertex of the parabola from the graph as (−5, −4).

 

p = −5
q = −4

 

Step 2: Substitute p and q into the quadratic function equation.

 

 

Step 3: Find the value of a. The graph opens upward, so you know the value of a is positive. You can pick another point on the graph and substitute its coordinates into the function to solve for a. An easy point is the y-intercept, which is at (0, 1).

 

 

Step 4: Substitute in the variables. Put the found values of a, p, and q into the equation for a quadratic function.

 

y = 0.2(x + 5)2 − 4

 

Step 5: Verify the equation. Verify the equation by substituting the coordinates of a second point into the equation and see if the equation is true. For example, a second point shown on the parabola is (−1, −0.8). The left side equals the right side, so the equation has been verified.

Left Side	Right Side
(4)2y (4)2−0.8 (4)2−0.8 (4)2−0.8 −0.8 −0.8	0.2(x + 5)2 − 4 0.2((−1) + 5)2 − 4 0.2(4)2 − 4 0.2(16) − 4 3.2 − 4 −0.8

 

Self-Check 3

1. What quadratic function describes the parabola on the grid shown?
   
   
    
   
   Answer
   

    

2. What quadratic function describes a parabola with the following characteristics?
   
   

   • The parabola opens downward.
   • The vertex of the parabola is at (4, 6).
   • There are two x-intercepts. One is at (6, 0) and the other is at (2, 0).
   • The y-intercept is at (0, −18).
     
     

   Answer

Example 2: A Baseball’s Path

 

A centre fielder in baseball makes a throw to the catcher at home plate, a distance of 126 m. The path of the ball is a parabola in which the ball rises 22 m above the hands of the centre fielder and the catcher.    

1. What quadratic function in vertex form describes the path of the baseball?
   
   
2. Determine the height on the parabola that is above and 10 m horizontally from the catcher. Express your answer to the nearest tenth of a metre.

Solution

1. Draw a diagram of the situation and put the information given in the problem onto the diagram.
   
   

   Step 1: Let the vertex of the parabolic shape be at the maximum height of the ball and the origin (0, 0). Draw a set of axes. Let x and y represent the horizontal and vertical distances from the maximum height of the ball, respectively.
   
   

    

   

   
   Step 2: The ball rises 22 m above the players. The ball is thrown 126 m, so half the throw is 63 m. The coordinates of the fielder will be (63, −22).
   
   

   
    
   
   Step 3: The vertex was chosen to be at (0, 0), so the values of p and q are both 0. Substitute these values into the quadratic function. To calculate the value of a, substitute the coordinates of the centre fielder’s position (63, −22) into the function.

    

    

   
   
   Step 4: Substitute a into the quadratic function. A quadratic function describing the path of the baseball, when the maximum height is chosen as the origin, is .
   

    

   

2. Step 1: To determine the height of a point on the parabola that is above and 10 m horizontally from the catcher, you need to express 10 m as a distance from the origin.
   
   
    
   63 m − 10 m = 53 m (from the origin)
   
   Therefore, x = −53.
   
   Step 2: Substitute the value of x into the quadratic function.
   
   
    
   
   
   The baseball is 15.6 m below the origin.
   
   Step 3: The height of the origin is 22 m above the catcher, and the catcher is 22 m below the origin. The height of the ball above the catcher is the difference.
   
   The baseball is 6.4 m above the catcher when it is 10 m horizontally from the catcher.
   
   
    
   22 m − 15.6 m = 6.4 m

Self-Check 4

1. What quadratic function describes the parabola formed by the path of a football that is kicked from ground level 60 m downfield and that reaches a maximum height of 10 m? Answer
   
   
2. What is the farthest distance the kicker could be from the goal post to clear the crossbar, a height of 3.1 m above the playing field? Answer
   
   
    
   
   Hemera/Thinkstock
   
   

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If you feel you have a solid understanding of how to describe and sketch the parabola from the vertex form of the quadratic function and how to write the quadratic function from a graph or the characteristics of a graph, go to Connect.

 

Self-Check 5

Lesson 2 Summary

 

© Amy Walters/851003/Fotolia

 

In this lesson you investigated the following questions:

• How can you sketch a graph of a quadratic function knowing the a-, p-, and q-values?
  
  
• How can you write an equation for a quadratic function from its graph?

You found that the quadratic function can provide information about the vertex:

p is the x-coordinate of the vertex. q is the y-coordinate of the vertex. The vertex coordinate = (p, q).

You also developed a rule to identify the number of x-intercepts. Your rule was similar to this:

 

If q = 0, only one x-intercept is present. If q ≠ 0, a and q are the same sign and there are no x-intercepts a and q are different signs and there are two x-intercepts

 

You used this information to graph a function and write an equation to represent a graph.

 

If you required the value of a, you could substitute the values of another point on the parabola into the function containing the known p- and q-values. You could then calculate the value of a as the unknown. It is convenient to choose the y-intercept as that other point because x = 0 for the y-intercept. Therefore, solving for a becomes simpler.

 

In the next lesson you will investigate how to solve quadratic equations in vertex form by completing the square. This skill will be a very powerful tool for you. You will also be introduced to another form of the quadratic function, and you will learn to convert back and forth between the two forms. This will enable you to model real-world situations even more effectively.