Lesson 5

Lesson 5: Modelling Problems with Quadratic Functions

 

Focus

 

Hemera/Thinkstock

 

A high school football player catches a perfectly thrown pass near the end zone. The quarterback knows physically how to throw the football right to that spot because of careful and consistent practice over an extended period of time.

 

The parabolic path of the football, the trajectory, can be modelled and predicted using quadratic functions. Your skill at modelling, predicting, and analyzing situations using quadratic functions can also be enhanced by careful and consistent practice. That is what this lesson is all about.

 

Outcomes

 

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

• write a quadratic function that models a given situation and explain any assumptions made
  
  
• solve a problem, with or without technology, by analyzing a quadratic function

Lesson Questions

 

You will investigate the following questions:

• How are quadratic functions developed and used to model real-life situations?
  
  
• What assumptions are appropriate to model a situation using quadratic functions?

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 3 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

Discover

 

You will analyze a real-world situation and model the situation by writing a quadratic function in the standard form, y = ax2 + bx + c , or in the vertex form, y = a(x − p)2 + q.

 

Try This 1

 

iStockphoto/Thinkstock

 

Assume that the football pass in the picture went a distance of 32 m and was 10 m higher than the thrower and the receiver at its highest point. Assume that both the thrower and the receiver are 1.8 m tall.

1. Sketch a parabola that shows this situation approximately to scale, and label the distances you know.
   
   
2. Would you rather put the origin, (0, 0), at one of the endpoints of the trajectory, at the height of the thrower or receiver, at the highest point of the throw, or at ground level? Explain your choice.
   
   
3. Choose a position for the origin and put in the x- and y-axes. Use the characteristics of your graph to find values to replace any of the parameters a, b, c, p, or q in y = ax2 + bx + c or in y = a(x − p)2 + q.
   
   
4. Use a known point to help solve for other parameters and develop a function describing the trajectory of the football.
   
   
5. How high is the football when it is 25 m from the thrower?

Save your responses in your course folder.

 

Share 1

 

Share your answers to Try This 1 with a partner or group, and then discuss the following questions.

1. What help can you give your group or partner to help everyone develop a function to describe the pass?
   
   
2. Are there different functions possible to describe this situation? Will each function supply the same answer to the problem?
   
   
3. What were the differences and similarities between the assumptions each person had to make about the situation?

Save your responses in your course folder.

Explore

 

In Share 1 you may have noticed you could develop different functions to model a given situation depending upon where you placed the origin. If developed correctly, each function will provide the same solution to the problem.

 

In Example 1: Water Stream, a quadratic function is developed to model the trajectory of water in a fountain. The strategy used in this example may also be applied to the problem in Try This 1. After reading Example 1: Water Stream, you may want to review your work from Try This 1.

 

Example 1: Water Stream

 

It’s a hot day. You and a friend are at a park with a cooling fountain. The stream of water leaves the nozzle 10 cm above the fountain’s water surface. The trajectory of the water stream reaches a height of 200 cm above the water when it is 300 cm from the nozzle. Your friend reaches out to touch the stream when she is 70 cm from the nozzle. What is the height of the stream of water when she touches it?

 

Assume that the trajectory of the water in the stream is a parabola.

 

Step 1: Draw a sketch of the situation. Include all given measurements and place the origin at the highest point of the trajectory.

 

 

Step 2: With the origin at the vertex of the parabola, the y-intercept is at (0, 0). So, the quadratic function y = a(x − p)2 + q reduces to y = ax2.

 

 

Step 3: Notice that the nozzle is 200 cm − 10 cm, or 190 cm, lower than the vertex. To solve for a, take the coordinates of the nozzle located on the left at (−300, −190) measured in centimetres and insert the coordinates into the function.

 

 

The function can be written as or as the approximation y ≈ −0.002 111x2.

 

Step 4: Find the horizontal distance from where your friend touches the water to the vertex.

 

−300 cm + 70 cm = −230 cm

 

 

Step 5: The horizontal distance at the point where the water is touched is −230 cm. Substitute this in for x so you can find the height at this point, y.

 

 

 

Step 6: The stream of water is about 112 cm below the vertex at this point. That means the stream will be approximately 88 cm above the water when it is touched.

 

Try This 2

 

Suppose the tip of the nozzle had been chosen as the point of origin in Example 1: Water Stream. The sketch would be a similar shape; however, the vertex would be at (300, 190).

1. Draw and label a graph of the problem in Example 1: Water Stream, but place the origin at the tip of the nozzle.
   
   
2. Write a quadratic function in the vertex form to model the situation. Why is the vertex form an excellent choice here?
   
   
3. Calculate the value of the coefficient a by substituting a known point that lies on the function.
   
   
4. What is the height of the stream of water when it is a horizontal distance of 70 cm from the nozzle? Compare your answer to the answer found in Example 1: Water Stream. The answers should be the same. Which method do you prefer?

Save your responses in your course folder.

 

Placing the Origin

 

When modelling a problem, you choose a position for the origin that allows you to make a quadratic equation with a minimum number of computations. Putting the origin at the vertex, a point under the vertex, or at one of the endpoints of the parabola usually works best.

 

No matter where you place the origin or whether you use the standard form or vertex form to develop the quadratic equation, the final solution should be the same.

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Self-Check 1

 

Model the following situation by developing a quadratic function, and then solve the problem using your function.

 

 

An old bridge support is a parabolic arch made out of stone. The arch is 4 m high at the vertex and 5 m wide at the base. A family needs to drive a rented van under the arch to reach their new home. The van is 3 m wide at the top.

1. Draw a sketch of the situation with origin at the vertex. Answer
   
   
2. Write a quadratic function that models the situation. Answer
   
   
3. What is the maximum height the van can be and still pass through the arch? Explain any assumptions you make in your solution. Answer
   
   
4. Would you arrive at the same answer if you put the origin in another place, like at road level on the left side of the arch? What function would be used to model the situation then? Answer

Try This 3

 

Sometimes you will run into a question where you don’t know how to solve the quadratic function. In these cases, your graphing calculator can be a great help. Try This 3 is a problem of this type.

 

A hockey player shoots the puck. If her shot had been taken on level ice with no obstructions, the puck would have risen from the ice surface in a parabolic arc to a height of 4.9 m and touched down again 34.8 m from the starting point.

 

Hemera/Thinkstock

1. Draw a diagram of the situation with the origin at the point where the puck is shot.

2. Write a quadratic function that models this situation.

3. To score in the situation described, the shot must clear the goalie’s left shoulder and enter the net at a height of 1.18 m. How far from the goal line should the puck lift off the ice so the shooter scores?

Lesson 5 Summary

 

In this lesson you investigated the following questions:

• How are quadratic functions developed and used to model real-life situations?
  
  
• What assumptions are appropriate to model a situation using quadratic functions?

You encountered a variety of situations that you modelled by developing quadratic functions. For many of the situations, including parabolic trajectories and parabolic structures, drawing a sketch of the situation and putting in axes helped to write the quadratic function.

 

You learned that the axes could be placed at various positions on the sketch. Some of the more useful positions were at the vertex, directly under the vertex, and at either end of the parabolic path. The position chosen for the vertex influenced how easily a solution could be found.

 

To write a quadratic function from a sketch of the graph, the coordinates of the vertex and at least one other point on the graph should be known. Then you can write the complete function.

 

You expanded your skill in using a graphing calculator. You used a graphing calculator to refine your estimates of the maximum, minimum, and intercept points of a function.

 

In the next module you will investigate how to factor and solve quadratic equations that, in this lesson, you could only estimate using a graphing calculator. You will also learn to solve quadratic inequalities.