Lesson 2

Lesson 2: Graphs of Quadratic Functions

 

Focus

 

A batter focuses intently on the path of a baseball once it leaves the pitcher’s hand. The batter has only a fraction of a second to predict the ball’s speed and path and to decide to swing at the ball. If the batter decides to swing, the player must predict where the ball will be so that the bat can connect with the ball at the proper place and angle. The batter’s brain is predicting the path of the pitched fastball. This path can be accurately described mathematically using a quadratic function.

 

 

In this lesson you will look at how to analyze the characteristics of a quadratic function using its graph. The graph of a quadratic function can be useful in predicting the path of a baseball or other projectiles.

 

Lesson Questions

 

This lesson will help you answer the following inquiry questions:

• What characteristics of a quadratic function can be identified from the graph of the function?
  
  
• How can you tell the number zeros of a quadratic function given the coordinates of the vertex and the value of a in the quadratic function?

Assessment

• Lesson 2 Assignment

All assessment items you encounter need to be placed in your course folder.

 

Save a copy of the Lesson 2 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

Materials and Equipment

• graph paper

You may find it helpful to use 1-cm Grid Paper.

Launch

 

This section presents questions to help you determine if you have the skills and knowledge to complete this lesson successfully.

 

Before beginning this lesson, you should

• understand how changing a, b, and c affects the graph of y = ax2 + bx + c
  
  
• find the coordinates of points on the x-y plane
  
  
• locate a point on the x-y plane given its coordinates

Are You Ready?

1. In Lesson 1 you looked at how changing the values of a, b, and c affect the graph of a function written in standard form (y = ax2 + bx + c). Use the interactive Do You Know Your ABCs? to ensure you know your ABCs of the quadratic function written in standard form.
   
   
    
   

    

   On the right side of the applet, you will see sliders that can be used to change the values of a, b, and c. Before moving the sliders, consider the following:

    

   • Each graph has the same shape, so what does that tell you about the value of a?
     
     
   • Each graph has a different y-intercept. What does that tell you about the values of c for the graphs?
     
     
   • You may find it helpful to choose one of the coloured graphs to determine the values of a and b first.
     
     
   Find the values of the coefficients and constants that make the black dotted curve disappear behind the coloured curves. You can use a chart like the following to record your results.
   

    

   Values that Hide the Dashed Curve
   Curve Colour	Light Blue	Blue	Purple	Red	Orange	Green	Dark Green	Dark Blue	Dark Purple
   a-value
   b-value
   c-value

   
   Answer

Complete the following Are You Ready? questions to be sure you have mastered the prerequisite skill of identifying the coordinates of any point on a grid.

2. Use the following graph to answer questions 2.a. to 2.c.
   
   

    

   1. What are the coordinates of the red dot in the graph?
      
      
   2. What are the coordinates of the point where the graph crosses the x-axis? (This point is called the x-intercept.)
      
      
   3. What are the coordinates of the point where the graph crosses the y-axis? (This point is called the y-intercept.)
      
      

   Answer

   
   
3. Use the following graph to answer questions 3.a. to 3.c.
   
   

    

   

    

   1. What are the coordinates of the red dot in the graph of the parabolic function?
      
      
   2. What are the coordinates of the point where the parabolic graph crosses the x-axis? (This point is called the x-intercept.)
      
      
   3. What are the coordinates of the point where the parabolic graph crosses the y-axis? (This point is called the y-intercept.)
      
      

   Answer
   
   

4. Use the following graph to answer questions 4.a. to 4.c.

   
   
   

    

   1. What are the coordinates of the red dot in the graph of the parabolic function?
      
      
   2. What are the coordinates of the point where the parabolic graph crosses the x-axis? (This point is called the x-intercept.)
      
      
   3. What are the coordinates of the point where the parabolic graph crosses the y-axis? (This point is called the y-intercept.)
      
      

   Answer

If you successfully completed the Are You Ready? questions, move on to the Discover section of the lesson.

 

If you experienced difficulties with the Are You Ready? questions, use the resources in the Refresher section to review the concepts before continuing through the lesson.

Refresher

 

If you struggled with changing the value of a, b, and c to match the coefficients and constants of the standard form of a quadratic function to its graph, you may find it helpful return to Lesson 1 and review the Explore section.

 

Review the definition of Coordinates. Then click on the tiny green square and drag it to different positions to see how the coordinates change. Be sure to see it in all four quadrants.

 

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

Discover

 

Objects thrown or projected through the air tend to follow a parabolic path that can be described using a quadratic function. The path is called a trajectory. In your Module 3 Project, you will analyze the trajectory of projectiles based on data about the paths of the projectile.

 

 

The shape and other characteristics of a parabola depend on the situation the quadratic function is modelling. For instance, some situations may have a parabola that opens upwards while other situations may have a parabola that opens downwards.

 

In Lesson 1 you identified certain characteristics of a quadratic function written in standard form y = ax2 + bx + c. You noticed that changing the value of a changes the shape of the graph while changing only b and c translates the parabola horizontally or vertically. From the graph of quadratic functions, you were able to see that the

• highest or lowest point on the graph of the quadratic function lies on the parabola’s vertical line of symmetry
  
  
• value of the parabola’s y-intercept is the same as the constant term, c, in the standard form of the quadratic

There are other characteristics of quadratic functions that can be used to help solve problems. For example, it is useful to know where the graph crosses the x-axis or what the maximum value or minimum value is for the graph.

 

Try This 1

 

Use the interactive applet Quadratic Graph Properties to determine additional characteristics of graphs of quadratic functions.

 

 

Use the a, b, and c sliders to create the following graphs. In your solutions, include a sketch or screenshot of the graph; the function that determines the graph; and the values of a, b, and c.

1. Create a graph that opens up and crosses the x-axis two times.
   
   
2. Create a graph that opens up and touches the x-axis once.
   
   
3. Create a graph that opens up and does not touch the x-axis.
   
   
4. Create a graph that opens down and crosses the x-axis two times.
   
   
5. Create a graph that opens down and touches the x-axis once.
   
   
6. Create a graph that opens down and does not touch the x-axis.

Place a copy of your graphs in your course folder.

 

Share 1

 

With a partner, compare the graphs that you created for Try This 1. Discuss how the shape of the graph is affected by the positive or negative value of a.

 

Place a copy of your Share 1 answer in your course folder for reference.

Explore

 

You may have noticed when you were creating different graphs in the Discover section that the number of times the parabola touches or crosses the x-axis depends on whether the graph opens up or down and the location of the vertex (minimum or maximum value) of the graph.

 

 

If the value of a is positive, the graph will open upwards. If the value of a is negative, the graph will open downwards. In each case, the position of the vertex will determine whether there are two, one, or zero x-intercepts.

 

If a is positive, the graph opens upwards and the following statements are true:

• If the vertex is below the x-axis, there will be two x-intercepts because both arms of the parabola will cross the x-axis.
  
  
  
  
• If the vertex is on the x-axis, there will be one x-intercept because the parabola touches the x-axis at one point.
  
  
  
  
• If the vertex is above the x-axis, there will be zero x-intercepts because the parabola will never cross the x-axis.
  
  

If a is negative, the graph opens downwards and the following statements are true:

 

• If the vertex is below the x-axis, there will be zero x-intercepts because the parabola will never cross the x-axis.
  
  
  
  
• If the vertex is on the x-axis, there will be one x-intercept because the parabola touches the x-axis at one point.
  
  
  
  
• If the vertex is above the x-axis, there will be two x-intercepts because both arms of the parabola will cross the x-axis.
  
  

If you are given the vertex and the direction of opening, can you decide how many x-intercepts the graph of the quadratic function will have?

Self-Check 1

 

Study the following six graphs. Note the vertex location, the direction of opening, and the number of x-intercepts for each graph.

 

Fill in the blanks to make each of the following statements true.

• The first blank can be replaced with the words below, above, or on.
  
  
• The second blank can be replaced with the words up or down.
  
  
• The third blank can be filled in with the numbers zero, one, or two.

1. Graph 1 has a vertex ________ the x-axis, is opening ______, and has ____ x-intercept(s). Answer
   
   
2. Graph 2 has a vertex ________ the x-axis, is opening ______, and has ____ x-intercept(s). Answer
   
   
3. Graph 3 has a vertex ________ the x-axis, is opening ______, and has ____ x-intercept(s). Answer
   
   
4. Graph 4 has a vertex ________ the x-axis, is opening ______, and has_____ x-intercept(s). Answer
   
   
5. Graph 5 has a vertex ________ the x-axis, is opening ______, and has ____ x-intercept(s). Answer
   
   
6. Graph 6 has a vertex ________ the x-axis, is opening ______, and has ____ x-intercept(s). Answer

As you have seen, quadratic functions can be used to model different situations. The characteristics of graphs of quadratic functions can be used to gather information about a situation. This information can be helpful in solving problems.

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Marko used the data he collected to sketch a graph of the height of the volleyball against time. The graph was shaped like a parabola. From this graph, Marko was able to determine when the volleyball reached its greatest height. Marko noticed that the maximum height of the graph corresponds with the maximum value, or vertex, of the graph. He used points that have the same y-value to determine the equation of the axis of symmetry. The axis of symmetry is halfway between the two points with the same y-values.

 

 

 

Use the applet Axis of Symmetry to explore the axis of symmetry in connection with the vertex and x-intercepts for different quadratic functions. Use the sliders for a, b, and c to create different parabolas.

 

Another way to describe a quadratic function and its parabolic graph is by its domain and range. Recall that the domain is all possible x-coordinates on the graph. The range is all possible y-coordinates on the graph.

 

In the quadratic function y = ax2 + bx + c, the domain is all possible values of x and the range is all possible values of y.

 

If you were to project a “shadow” of a graph onto the y-axis, the resulting interval corresponds to the range of the relation. If you project a “shadow” of a graph onto the x-axis, the resulting interval corresponds to the domain of the relation.

 

Try This 2

 

The Domain and Range applet allows you to see the “shadow” that the domain of the relation casts on the horizontal axis and the “shadow” that the range of the relation casts on the vertical axis.

 

 

Use the applet to answer the following questions. Note: The shadows for the domain and the range are different colours. The shadow tells you the domain and the range of each relation.

1. Select the “Circle” graph. Slide the domain slider down to the bottom to see the shadow.
   
   
   1. How would you express the domain of the circle in words?
      
      
   2. How would you express the domain of the circle in symbols?
      
      
   3. Express the range of the circle in words and in symbols. (You can move the “Range Shadow” slider all the way to the left to see the range shadow.)
      
      
   Answer
   
   
2. Now de-select the “Circle” graph, and select the “Quadratic 1” graph.
   
   
   1. When the quadratic 1 function is written in standard form, would the value of a be greater than or less than zero? How do you know?
      
      
   2. What is the domain of the Quadratic 1 function?
      
      
   3. What is the range of the Quadratic 1 function?
      
      
   Answer
   
   
3. Now de-select the “Quadratic 1” graph, and select the “Quadratic 2” graph.
   
   
   1. When the Quadratic 2 function is written in standard form, would the value of a be greater than or less than zero? How do you know?
      
      
   2. What is the domain of the Quadratic 2 function?
      
      
   3. What is the range of the Quadratic 2 function?
      
      
   Answer

For this module on quadratics, it is important to know the following:

• The domain of a quadratic function is always a member of the real numbers.
  
  
• The range is always greater than or equal to the minimum value if the parabola opens upwards.
  
  
• The range is less than or equal to the maximum value if the parabola opens downwards.

When determining domain and range, or any characteristics of the graph of a quadratic function, it is important to think about what situation is being modelled by the data. For instance, in Self-Check 2, you looked at a graph of the height of a volleyball versus time. In this situation, a negative height is not possible—it would mean that the volleyball went through the floor! If you were to give the range for this graph, it could not be less than zero.

 

 

 

It is important to always keep the context of the problem in mind. This helps you to ensure that the values you determine for the characteristics of the graph make sense or are reasonable for the given situation.

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Suppose you are given the vertex of a parabola (6, 8) and one of the x-intercepts (2, 0). From this information, you can determine the following:

• the coordinates of the other x-intercept
  
  
• whether the value of a in the quadratic function is negative or positive
  
  
• the maximum or minimum value of the function

This information can be determined without drawing the graph by using the axis of symmetry and some reasoning. Since the vertex is at (6, 8), the axis of symmetry is at x = 6. The given x-intercept (2, 0) is 4 units to the left of the axis of symmetry (6 − 2 = 4). Therefore, the other x-intercept will be 4 units to the right side of the axis of symmetry at x = (6 + 4), or x = 10.

 

Remember that the value of a in the quadratic function y = ax2 + bx + c indicates whether the parabolic graph of the quadratic function opens upwards or downwards. If a is positive, the graph opens upwards and the vertex is the minimum value of the function. If a is negative, the graph opens downwards and the vertex is the maximum value of the function. The given x-intercept (2, 0) is below the vertex (6, 8); so the graph must open downwards. This tells us the value of a in the quadratic function is negative and the vertex is a maximum.

 

Therefore, the maximum value of the function is 8, the value of y at the vertex.

 

Self-Check 3

1. Use the following graph to complete this question.
   
   
   

   For the graph, describe the
   
   

   1. vertex
      
      
   2. direction of opening
      
      
   3. axis of symmetry
      
      
   4. domain
      
      
   5. range
      
      
   6. x-intercepts
      
      
   7. y-intercept
      
      
   8. maximum or minimum value of the function
   
   
   Answer
   
   
2. Use the following graph to complete this question.
   
   
   
    
   For the graph, describe the
   
   

   1. vertex
      
      
   2. direction of opening
      
      
   3. axis of symmetry
      
      
   4. domain
      
      
   5. range
      
      
   6. x-intercepts
      
      
   7. y-intercept
      
      
   8. maximum or minimum value of the function
   
   
   Answer
   
   
3. Using the principles of symmetry, sketch the graph of y = −2x2 +16x − 16. Use the following information to sketch your graph:
   
   
   • The vertex is at (4, 16).
   • One point on the graph is (2, 8).
   • Another point on the graph is (0, −16).
     
     
   If you need some graph paper to complete this question, you can print out a grid at 1-cm Grid Paper. If you want to sketch the graph electronically and know how to use a drawing program, feel free to do it that way.
   
   Answer

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Did you notice that the y-intercept is the same as the value of c in the standard form of the quadratic function y = ax2 + bx + c? Look back at the graphs in Self-Check 3. Observe how the constant at the end of the function is always the y-intercept. That will always be true.

 

To see why, put the value of 0 in for x in the standard form of the quadratic function.

 

y = ax2 + bx + c

y = a(0)2 + b(0) + c

 

Can you see that for a quadratic function in standard form, the y-intercept will be y = c?

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Connect

 

Lesson Assignment

Lesson 2 Summary

 

In this lesson you found that the coordinates of the vertex of the parabola are for the point of maximum or minimum value of the quadratic function. The coordinates of this point can be determined from the graph. The axis of symmetry is the x-value of the vertex.

 

The domain will be all values of x that are possible for the function. It is symbolically written as when it includes all real numbers. The range is all the values of y that are possible for the function. It is symbolically written as when it includes all real numbers greater than 7, for example. When determining domain and range, it is important to consider the situation that is being modelled. Is it possible for a value to be below zero?

 

If the value of a is positive, the graph will open upwards. If the value of a is negative, the graph will open downwards. In each case, the position of the vertex will determine whether there are two, one, or zero x-intercepts.

 

If a is positive, the graph opens upwards and the following statements are true:

• If the vertex is below the x-axis, there will be two x-intercepts because both arms of the parabola will cross the x-axis.
  
  
• If the vertex is on the x-axis, there will be one x-intercept because the parabola touches the x-axis at one point.
  
  
• If the vertex is above the x-axis, there will be zero x-intercepts because the parabola will never cross the x-axis.

If a is negative, the graph opens downwards and the following statements are true:

• If the vertex is below the x-axis, there will be zero x-intercepts because the parabola will never cross the x-axis.
  
  
• If the vertex is on the x-axis, there will be one x-intercept because the parabola touches the x-axis at one point.
  
  
• If the vertex is above the x-axis, there will be two x-intercepts because both arms of the parabola will cross the x-axis.

In Lesson 3 you will investigate how to use a graphing technology to find the maximum or minimum value, the axis of symmetry, the x- and y-intercepts, and the coordinates of the vertex.