Lesson 5

Lesson 5: Solving Quadratic Systems Graphically

Focus

The illustration to the right shows what the Walterdale Bridge, which spans the North Saskatchewan River in Edmonton, might look like when the existing bridge is replaced. Notice that the arches resemble parabolas and that the supports resemble lines. Do the arches intersect? Where do the supports and arches intersect?

Previously in this module you used the graphs of quadratic functions to solve quadratic equations. In this lesson you will study systems of linear-quadratic and quadratic-quadratic equations. Just as you did with the illustration, you will focus on where lines and parabolas intersect. You will also discover the use of multiple parabolas in the design of buildings and other structures.

Outcomes

At the end of this lesson you will be able to

• explain the meaning of the points of intersection of a system of linear-quadratic or quadratic-quadratic equations
  
  
• explain, using examples, why a system of linear-quadratic or quadratic-quadratic equations may have zero, one, two, or an infinite number of solutions
  
  
• determine and verify the solution of a system of linear-quadratic or quadratic-quadratic equations graphically, with technology
  
  
• relate a system of linear-quadratic or quadratic-quadratic equations to the context of a given problem
  
  
• model a situation using a system of linear-quadratic or quadratic-quadratic equations

Lesson Questions

You will investigate the following questions:

• Why are the points of intersection of a system of equations related to its solution?
  
  
• How does a graphical representation facilitate mathematical understanding?

Assessment

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

• completion of the Lesson 5 Assignment (Download the Lesson 5 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Try This and Share activities
  
  
• additions to Module 4 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

Materials and Equipment

You will need the following to complete the optional Math Lab in the Going Beyond section:

• two small objects of similar size and weight (e.g., tennis balls, toy cars, whiteboard brushes)
  
  
• a flat elevated surface that is at least 1 m in length (e.g., table top, whiteboard ledge)

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

• solve linear systems using graphs
  
  
• find the equation of a line
  
  
• find the equation for a parabola

Are You Ready?

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

1. What is a system of equations? Give an example. Answer
   
   
2. What is the significance of the points of intersection of the graph of a system of equations? Answer
   
   
3.  
   1. Solve the following linear system using a graphical method.
      
      x + y = 7
      2x − y = 5
      
      Answer
      
      
   2. Verify your solutions. Answer
      
      
4. Given the information for each function, determine the equation of a
   
   
   1. line that passes through the points (0, 3) and (4, 6) Answer
      
      
   2. parabola with vertex (2, 10) that passes through the point (−3, −15) Answer

Refresher

Use the Solving Linear Systems by Graphing tutorial to review the following:

• Define a system of equations.
  
  
• Review the significance of the point of intersection as it relates to a system of equations.
  
  
• Determine the solution to a system of linear equations by graphing.
  
  
• Review how to verify the solution to a system of equations.

If you encounter difficulties working with the mini-explorer, you may want to use your own graphing technology instead.

Linear Systems Graph can be used to help graph and solve systems of linear equations.

Use the interactive lesson entitled Given Two Points on the Graph to practise writing equations of linear functions. Select Determining the Equation of a Line. Then choose Given the Slope and a Point. Work through slides 1 to 9 only.

Use Quadratic Function (Vertex Form) to find the equation of a parabola.

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

Discover

The following graphs show the different ways two linear functions can be sketched on the same set of axes.

In Try This 1 you will construct all the ways that quadratic functions and linear functions can be sketched. You will need to use graph paper. You can use Four Grids or use your own graph paper.

Try This 1

Part A: Linear-Quadratic System

1. Think about the number of ways that a parabola and a line can intersect on the coordinate plane. For each of the ways this can happen, sketch a possible graph.
   
   
2. Consider the following system of equations:
   
   
    
   y = x2 + 4x − 1
   
    
   y = x + 1
   
   

   1. Why do you think this is called a linear-quadratic system?
      
      

   2. How many solutions are possible for this system?
      
      

   3. How could you determine the solutions?

Part B: Quadratic-Quadratic System

3. Think about the number of ways that two parabolas can intersect on the coordinate plane. For each of the ways this can happen, sketch a possible graph.
   
   
4. Consider the following system of equations:
   
   
    
   y = x2 − 6x + 13
   
    
   y = –2x2 + 12x − 14
   
   

   1. Why do you think this is called a quadratic-quadratic system?
      
      

   2. How many solutions are possible for this system?
      
      

   3. How could you determine the solutions?

Save your work in your folder.

The concepts you have explored here represent the basis of this lesson. You will work with these concepts further in the remainder of the lesson.

Explore

Parabolic designs represent a modern innovation in architecture. The aesthetic appeal of the parabola has been used for many structures, including the Los Angeles airport, and the J. S. Dorton Arena in North Carolina, both shown in the photo collage. In both cases, the architects based their designs on the intersection of two parabolas. The point of intersection of the parabolas in each structure is the focal point of the structure.

In Mathematics 10C Learn EveryWare you studied systems of linear equations. You learned that you can graphically determine the solution to a linear system by graphing the equations and then finding the points of intersection. In this lesson you will apply the same graphing techniques to linear-quadratic and quadratic-quadratic systems.

Solving a System of Equations by Graphing

Recall from your previous studies that to solve a system of equations means to find the values of the variables that will simultaneously satisfy all equations in the system. For example, consider the following system of linear equations:

x + y = 8

2x − y = 7

The solution to the equation is x = 5 and y = 3. This is because these values will satisfy each of the equations in the system. You can verify this by substituting the values into each equation as shown.

Equation 1: x + y = 8

Equation 2: 2x − y = 7

One way that you can solve a system of equation is by graphing. Recall that the points of intersection of the graphs of a system represent the solutions to the system.

Try This 2

1.  
   1. Solve the following quadratic systems by graphing. You may wish to do this using paper and pencil or using a graphing tool.
      
      
      1. 4x − y + 3 = 0
         2x2 + 8x − y + 3 = 0
         
         
      2. 2x2 − 16x − y = −35
         2x2 − 8x − y = −11
      
      Save a copy of your hand-sketched graphs or, if you used a graphing tool, by taking a screen capture of the completed graphs.
      
      
   2. Verify the solution by comparing the left side to the right side, as shown in the example before this Try This activity.
      
      
2. Retrieve your work from Try This 1. Compare your answers from the graphs in Try This 2 to those in Try This 1, Part A, question 1 and Try This 1, Part B, question 3. Do the graphs from this activity have a corresponding match in your earlier work? If not, make any necessary additions or revisions.

Save your work in your course folder.

Relating a System of Equations to a Given Context

Now that you have learned how to solve linear-quadratic and quadratic-quadratic systems by graphing, you are prepared to consider how these systems can be related to different contexts.

In the next section you will consider a classic physics problem in action. Watch Gravity Experiment (“Newton's Laws: Summary”) to get started.

Share 1

With a partner, consider the following systems.

Decide which system would best represent the scenario shown in Gravity Experiment (“Newton's Laws: Summary”). Give explanations to support your choice. Also give reasons to reject the other choices.

Example 1: Olympic Springboard Diving

Self-Check 2

Connect

Lesson 5 Assignment

Lesson 5 Summary

In this lesson you investigated the following questions:

• Why are the points of intersection of a system of equations related to its solution?
  
  
• How does a graphical representation facilitate mathematical understanding?

You studied graphical methods of solving systems of linear-quadratic and quadratic-quadratic equations. As in the case of linear systems that you studied previously, the solutions to quadratic systems are found at the points of intersection. This is due to the fact that such points lie on both graphs and therefore satisfy both equations simultaneously.

Throughout this lesson, you used graphs to represent the systems. You learned how to model problems using both linear-quadratic and quadratic-quadratic systems. In many cases, you used a sketch or drawing to help you visualize the problem. By doing so, you were able to better connect the nature of the problem to the math underlying the solution.

In the next lesson you will continue to work with quadratic systems. You will explore methods of solving these systems algebraically.