Lesson 9
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| Course: | Math 20-2 SS |
| Book: | Lesson 9 |
| Printed by: | Guest user |
| Date: | Saturday, 6 September 2025, 2:35 AM |
Description
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1. Lesson 9
Module 3: Quadratics
Lesson 9: Solving Problems with Quadratic Equations
Focus
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A high school football player catches a perfectly thrown pass near the end zone. The quarterback knows 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.
Up to this point, you have worked with quadratic functions that modelled real-world situations, like snowboard jumps and football kicks. Now you will learn to write quadratic functions that describe situations in the world around you. You will also analyze the functions you create to find answers to questions related to those situations.
Lesson Questions
This lesson will help you answer the following inquiry questions:
- How do you write a quadratic function that models a given situation and explain any assumptions made?
- How do you solve a problem, with or without technology, by analyzing a quadratic function?
Materials and Equipment
- graphing calculator
Assessment
- Lesson 9 Assignment
- Module 3 Project
All assessment items you encounter need to be placed in your course folder.
Save a copy of the Lesson 9 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.
1.1. Explore
Module 3: Quadratics
Explore
In the following textbook example you will see how to find parameters of a quadratic function from the function’s graph.
Study “Example 3: Determining the equation of a quadratic function, given its graph” on pages 341 and 342 of the textbook. Pay particular attention to how the value for a is found.
If possible, work with a partner or small group on Try This 1. Different people could decide to use different functions.
Try This 1
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The following question will have you analyze a real-world situation and write a quadratic function that models the situation. Your function might be in the standard form, y = ax2 + bx + c, or in the vertex form, y = a(x − h)2 + k.
- Assume that a football pass went a distance of 32 m and was 10 m higher than the passer and the receiver at its highest point. Assume the passer and the receiver are both 1.8 m tall. Sketch a parabola that shows this situation approximately to scale and label the distances you know.
- Would you rather put the origin of the x- and y-axes at one of the ends of the trajectory, at the highest point, or at ground level? Why?
- Choose a position for the origin and put in the x- and y-axes. Can you use the characteristics of your graph to put values in for any of the constants in y = ax2 + bx + c or y = a(x − h)2 + k?
- Work with what you know to develop a function describing the trajectory of the football.
- How high was the football when it is 25 m from the thrower?
Share 1
Based on your graph and development of the quadratic function in Try This 1, discuss the following questions with another student or appropriate partner.
- Are there different functions possible to describe the same situation?
- Can different functions still supply the same answer to the problem?
- What assumptions did your partner make about the situation? In what ways does this differ from the assumptions you made?
- To what extent are the assumptions reasonable?
- How did you arrive at the quadratic function you developed?
Summarize your discussion.
1.2. Explore 2
Module 3: Quadratics
Quadratic equations can be used to model situations in many contexts. You have looked at trajectories of projectiles and business situations. You will now have a chance to hone your modelling and solving skills by studying an example. Then you get to try a few problems yourself.
iStockphoto/Thinkstock
Example 1: Stream of Water
It is a hot day and you and a friend are at a park with a cooling fountain. The stream of water leaves the nozzle 10 cm above the water’s 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. What is the height of the stream of water when it is a horizontal distance of 70 cm from the nozzle, which is as far as she can safely reach?
To answer the question, place the origin at the highest point. Draw a sketch of the situation that includes all of the given measurements.
Assume that the trajectory of the water in the stream is a perfect parabola. Also assume that all measurements are to the nearest centimetre and that the water is perfectly level.
With the origin at the vertex of the parabola, the y-intercept is at (0, 0), so the quadratic function y = ax2 + bx + c reduces to y = ax2.
To solve for a, take the coordinates of the nozzle, (−300, −190) measured in centimetres, and insert the coordinates into the function.
The function can be written as 
or as the approximation −0.002 11x2.
The horizontal distance of 70 cm from the nozzle, as far as your friend can safely reach, will be −300 + 70, or −230 cm, from the vertex. You want the height of the stream of water at that point, which is related to the value of y. So, substitute −230 in for x and solve for y.
The stream of water will be about 112 cm below the vertex. That means the stream will be 200 − 112, or 88 cm, above the water at a horizontal distance of 70 cm from the nozzle.
Self-Check 1
Suppose the tip of the nozzle had been chosen as the point of origin in Example 1. The sketch would be a similar shape. However, the vertex would be at (300, 190) and the axes would be in a different place.
- Draw a sketch of the graph with the origin at the tip of the nozzle. Answer
- Write a quadratic function in the vertex form to model the situation, and calculate the value of the coefficient a. Why would the vertex form be an excellent choice here? Answer
- What is the height of the stream of water when the stream is a horizontal distance of 70 cm from the nozzle? Answer
1.3. Explore 3
Module 3: Quadratics
The answer you obtained in Self-Check 1 is the same as the answer obtained by placing the origin at the vertex. Which method do you prefer?
When modelling a problem, choose a position for the origin that simplifies the amount of computation required. Putting the origin at the vertex, at a point under the vertex, or at one of the endpoints of the parabola often works well.
The choice of whether to use the standard form or the vertex form of the quadratic function depends on the values you know from the situation. As you have seen, either form will give you the same final answer.
Hemera/Thinkstock
Self-Check 2
In this activity you will model a situation using a quadratic function; then you will use a graphing calculator or the quadratic formula to find the answer to a related problem.
The water stream in the photo leaves the sea serpent’s mouth at a height of 1.4 m above the water and rises another 4 m to its maximum height at a horizontal distance of 8 m from the spout. How far from the serpent will the stream hit the water’s surface? Give your answer to the nearest 0.1 m. What assumptions are you making?
If you get stuck, consider the following hints:
- If you choose the origin to be at the serpent’s mouth, the coordinates of the vertex are (8, 4).
- If you choose the origin to be at the serpent’s mouth, the y-value where the stream hits the water’s surface will be y = −1.4.
When inputting negative quantities into your graphing calculator, be sure to press the negative key, (−), located directly left of the ENTER key. The negative key on the calculator is a subtraction symbol enclosed in a parentheses. Use the subtraction key, −, located two spaces above the ENTER key only for the subtraction operation of finding the difference between two terms.
Take a look at modelling a different type of problem. In your textbook, study “Example 3: Solving a problem by creating a quadratic model” and “Example 4: Visualizing a quadratic relationship” on pages 428 and 429. You will see how to set up and solve a problem with and without a graphing calculator.
If you haven’t done so already, you may want to add what you’ve learned about creating equations to help solve problems to your notes organizer.
If you feel you need a bit more practice, complete Self-Check 3 before moving to Connect.
Self-Check 3
Complete all or parts of questions 3, 5, 7, and 8 from on pages 430 and 431 of the textbook. When you finish a question, check your work using the shortened answers given on page 566 of the textbook. If you are still unclear about how to answer some questions, ask your teacher for help.
1.4. Connect
Module 3: Quadratics
Connect
Lesson Assignment
Complete the Lesson 9 Assignment that you saved to your course folder.
Project Connection
Go to the Module 3 Project and complete Part 4. Submit your Module 3 Project work to your teacher for assessment.
1.5. Lesson 9 Summary
Module 3: Quadratics
Lesson 9 Summary
In this lesson you investigated the following questions:
- How do you write a quadratic function that models a given situation and explain any assumptions made?
- How do you solve a problem, with or without technology, by analyzing a quadratic function?
You encountered a variety of situations that were modelled using quadratic functions. For many 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 readily the model provided the requested answer.
© EtiAmmos/18850087/Fotolia
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. With the vertex form, the coordinates of the vertex tell you the values of the coefficients h and k.
Substituting the coordinates of the origin into a partially completed vertex form of the function, or into the standard form of the function when the origin is at the vertex, enables you to solve for the coefficient a, which will be identical in both forms of the function for that particular situation. Then you can write the complete function.
Refer now to the Module 3 Summary for an overview of concepts you have discovered in this module.