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Lesson 3

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Course: Math 20-1 SS
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Date: Monday, 15 September 2025, 2:38 PM

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Table of contents

1. Lesson 3

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Lesson 3: Completing the Square

 
Focus

 

This is a photo of a snowmobile with rider jumped into the air.

Hemera/Thinkstock

 

Riding a snowmobile is a great winter activity. Running your snowmobile up a slope, jumping the sled into the air, and landing in soft snow can be exhilarating.

 

Competitions are sometimes held to see who can jump the farthest or the highest. The path of the snowmobile can be modelled and predicted using quadratic functions. What is learned from these functions can guide riders to greater length or height in a jump.

 

Up to this point in the module you have only worked with the quadratic function in vertex form, y = a(x − p)2 + q. This form is excellent for modelling situations in which there is a maximum or minimum value. This form enables you to know what the maximum height of a jump might be by looking at the coordinates of the vertex, which can be read directly from the p- and q-values of the function.

 

In this lesson you will encounter the standard form of the quadratic function.The standard form is written as f(x) = ax2 + bx + c or y = ax2 + bx + c.

 

The standard form of the quadratic function is more useful in predicting distance of jumps and modelling some other real-life situations. You will learn to model situations in the standard form and convert the standard form to the vertex form.

 

Outcomes

 

At the end of this lesson you will be able to

  • write the quadratic function given in the standard form, y = ax2 + bx + c, in the vertex form, y = a(x − p)2 + q, by completing the square

  • explain the reasoning for the process of completing the square as shown in a given example

  • identify, explain, and correct errors in an example of completing the square

Lesson Questions

 

You will investigate the following question:

  • How do you complete the square to convert a quadratic function in the standard form, y = ax2 + bx + c, into the vertex form, y = a(x − p)2 + q?

Assessment


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

  • completion of the Lesson 3 Assignment (Download the Lesson 3 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

1.1. Launch

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

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

  • square binomials
  • factor perfect-square trinomials


1.2. Are You Ready?

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Are You Ready?

 

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

  1. Write the expanded form of each of the following:

    1. y = (x + 4)2 Answer

    2. f(x) = (x − 5)2 Answer

    3. y = 4(x + 3)2 Answer
  1. Factor each of the following perfect-square trinomials by showing the trinomials as a binomial squared.

    1. x2 + 14x + 49 Answer

    2. x2 − 12x + 36 Answer

    3. x2 − 6x + 9 Answer

    4. x2 − 24x + 144 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.



1.3. Refresher

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Refresher

 

This is a play button that opens “Square a Binomial.”

Khan Academy (CC BY-NC-SA 3.0)

Review examples, common errors, and the procedure for squaring binomials in “Square a Binomial.”

 


This is a play button that opens “Factoring Perfect Square Trinomials.”

Khan Academy (CC BY-NC-SA 3.0)

Work through factoring examples and a general case in “Factoring Perfect-Square Trinomials.”



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



1.4. Discover

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Discover

 

Try This 1

 

Complete the square using algebra tiles.

 

Step 1: Open Trinomial. Read the definition and examples and then scroll down to Demonstration Applet. In the applet the simplified form of the trinomial at the bottom represents the algebra tiles shown. Your task is to drag tiles into the workspace and form a perfect square.

 

 
This is a play button that opens Trinomial.



Step 2: Begin with the trinomial that is presented in Trinomial Demonstration Applet, which you accessed in Step 1. The trinomial is x2 + 2x − 8. Hold down the “r” key and click on an x-tile to rotate the tile. Move the x-tiles to form the partial grey square as shown.

 

 

This shows a partial grey square consisting of one x2-tile with a vertical x-tile on the right side and a horizontal x-tile along the base. There are 8 red negative-one tiles, which have not yet been used.

 

Step 3: To turn the grey x2 and x’s into a square, you will have to drag a +1 tile to the corner. When you are completing a square, every time you drag in a positive tile, you have to drag in a corresponding negative tile. This ensures that the value of the simplified trinomial remains the same. To keep your equation balanced, drag in a −1 tile now.

 

 

The partial grey square from Step 2 has been completed by adding a plus-one tile on the lower right. A red negative-one tile has been added to the eight red negative-one tiles.

 

Note: x2 + 2x + 1 is a perfect square since it has one factor that is squared, (x + 1)2.

 

 

This illustration shows a square of side (x + 1), and 9 negative-1 tiles. The lengths of the sides of the square are labelled (x + 1). (x + 1) squared is shown as a perfect square or x squared + 2x + 1.

 

Therefore, the original given equation, x2 + 2x − 8 may be written as (ax2 + bx + k) − 9. This is similar to the unsimplified form at the bottom of the applet with the zero-coefficient terms deleted.

 

 

This illustration shows a square of side (x + 1) and 9 negative-1 tiles. The lengths of the sides of the square are labelled (x + 1). (x + 1) squared − 9 is shown as x squared + 2x + 1 − 9.

 

Step 4: Complete the following tasks for each of the original trinomials shown in the chart and some trinomials that you make up:

  • Begin with the first two terms of the trinomial and complete the square by dragging in positive squares. Record the positive value added.
  • Bring in corresponding negative squares to balance the positive squares brought in. Record the total value left over.
  • Write the trinomial in unsimplified form.
  • Take screenshots or make sketches of the perfect square and the leftover value.
  1. Record your observations in a chart like the one shown here.

     
    Original Trinomial Positive Value Added Perfect-Square Trinomial

    Total Value

    Left Over

    		 Trinomial Written as the Unsimplified Form
    x2 + 2x − 8 1

    x2 + 2x + 1

     

    This illustration shows a square of side (x + 1).

    −9

     

    This illustration shows nine negative-one tiles.

    		 (x2 + −2x + 1) − 9
    x2 + 4x − 8        
    x2 + −8x + 5        
    x2 + 6x + 4        


  2. What relationship do you see between the coefficient of the second term in the original trinomial and the positive value added? hint hint

course folder Save your responses in your course folder.

 

Share 1

 

Based on your observations from Try This 1, discuss the following questions with a partner or group.

  1. What is the relationship between half of the coefficient of the second term in a trinomial and the number added to complete the square? Describe the patterns you found.

  2. Summarize your discussion by creating a general rule about the number added to complete a square and the value of b in y = ax2 + bx + c.

course folder Save your responses in your course folder.

Look at the relationships in all the trinomials to see the pattern. Does a pattern become clearer if you look at the relationship between half of the coefficient of the second term in the trinomial and what was added?
In the first original trinomial, x2 + 2x − 8, what is the relationship between the “2” in 2x and the “1” that was added.


1.5. Explore

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Explore

 

It is often easier to model problems using the standard form of the quadratic function, y = ax2 + bx + c; however, it is easier to draw a graph and find minimum or maximum values from the vertex form, y = a(x − p)2 + q. This is why you are learning to convert quadratic functions from the standard form to the vertex form.

course folder Save Module 3 Glossary Terms in your course folder now.



glossary

Here are some of the words you will want to define in Module 3 Glossary Terms in this lesson:

  • standard form of the quadratic equation
  • vertex form of the quadratic equation
  • binomial
  • trinomial
  • polynomial
  • coefficient
Try This 2

 

In Try This 1 you used algebra tiles to complete the square for a trinomial in the form ax2 + bx + c. The first quadratic function in the table, x2 + 2x − 8, was shown to equal the sum of the perfect-square trinomial and the leftover term.

 

 

Quadratic Function

Perfect-Square Trinomial

Leftover Term

x2 + 2x − 8

x2 + 2x + 1

 

This illustration shows a square of side (x + 1).

− 9

 

This illustration shows nine negative-one tiles.
  1. Factor the perfect-square trinomial. hint

  2. Write the quadratic in the new form: x2 + 2x − 8 = (square of a binomial) + (leftover term).

  3. How does this form compare with the vertex form, y = a(x − p)2 + q, of a quadratic function?

  4. Rewrite two more quadratics from the chart in Try This 1 in the vertex form using this method.

course folder Save your responses in your course folder.

This step is important to getting the next two questions right. Did you get this: x2 + 2x + 1 = (x − 1)2?


1.6. Explore 2

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Completing the Square Using Only Algebra

 

You are ready to learn to complete the square for a quadratic function in standard form using only algebra. While algebra tiles are your best bet if you need a good visual of what is happening to the equation as you complete the square, completing the square using only algebra is faster and easier.

 

Completing the square using algebra will enable you to convert the standard form of the quadratic function, y = ax2 + bx + c, to the vertex form, y = a(x − p)2 + q.

 

This is an illustration that shows that the standard form will be converted to the vertex form.

 

Example 1: Completing the Square

 

Complete the square to convert the following quadratic function to vertex form.

 

 



textbook
Sometimes there is a coefficient in front of the x2-term. Look at a couple of examples of how this is handled on page 186 of the textbook. Begin at “Method 2: Use an Algebraic Method.” Note how the square brackets are used to keep track of the values multiplied by the coefficient of the x2-term.


caution

Be careful when converting to the vertex form of a quadratic equation. One of the most common mistakes students make is that they forget to calculate the sign (+ or −) of the coefficients when working with brackets.

 

It is important that you always include the sign and the value of the coefficients in front of a bracket when you put numbers into a bracket or remove numbers from the bracket.

 

Watch Animated Example 1: Completing the Square and Animated Example 2: Completing the Square. Pay special attention to the signs and the use of brackets in each case. Note the subtle difference between these examples.

 

 
This is a play button that opens Animated Example 1: Completing the Square.
 
This is a play button that opens Animated Example 2: Completing the Square.


1.7. Explore 3

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Self-Check 1
  1. Complete the square and convert the following quadratic functions to vertex form.

    1. y = x2 + 10x + 20 Answer

    2. f(x) = x2 − 8x − 2 Answer

    3. y = 3x2 + 12x + 7 Answer

    4. f(x) = −4x2 + 24x − 30 Answer
  1. Identify two errors in the following completing-the-square sequence.

     


    Answer

  2. Convert the following functions to the vertex form, and then describe the graphs based on what you learned in Lessons 1 and 2. Explain how you know the characteristics of the graphs. Put the functions into a graphing calculator or use Quadratic Function (Vertex Form) to check your result.

     
    This is a play button that opens Quadratic Function (Vertex Form).


    1. y = 3x2 − 30x + 77 Answer

    2. y = −2x2 − 24x − 69 Answer
Try This 3

 

Graph the functions in both the vertex form and the standard form to check that your work is correct. If your graphs are exactly the same, you have completed the square correctly!

  1. Take the two forms of the quadratic function in Self-Check 1 question 3.a., y = 3x2 − 30x + 77 and y = 3(x − 5)2 + 2. Verify that both forms are the same function by graphing each form on a graphing calculator.

    Here’s how to use your graphing calculator:

    • Press the “Y=” key at the top and enter the standard form of the function in the “Y1=” field.

       
      This shows the display from a graphing calculator.

    • Press “ENTER” or the down arrow to go to the next line.

    • Enter the vertex form of the function in the “Y2=” field.

       
      This shows the display from a graphing calculator.

    • Press the “GRAPH” key. The calculator will draw the graphs of the two functions. If the two functions are the same, you will only see one parabola. If you see two parabolas, then you have not completed the square correctly.

       
      This shows the display from a graphing calculator. A parabola is displayed.

  2. Take the two forms of the quadratic function in Self-Check 1 question 3.b., y = −2x2 − 24x − 69 and y = −2(x + 6)2 + 3. Verify that both forms are the same function by graphing each form on a graphing calculator.

    If you have trouble inputting the equations, consult your teacher or another student who knows how to perform this sequence. You will use this graphing skill again and again.


    course folder If you have used an online graphing tool, save screenshots of your work in your course folder.

1.8. Explore 4

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Did You Know?

Today in the shot put event, elite athletes throw the shot much farther than anyone ever could when world records were first established. Training techniques and increased understanding of the physics and biophysics involved have enabled huge gains.

 

Men throw a 7.27-kg shot and women throw a 4-kg shot. Here are some interesting statistics:

 

 
  Year Athlete Country Distance (m)
Female 1924 Violette Gouraud-Morris Paris, France 10.15
1987 Natalya Lisovskaya Moscow, USSR 22.63
Male 1909 Ralph Rose San Francisco, USA 15.54
1987 Randy Barnes Los Angeles, USA 22.63

 

Dylan Armstrong from Kamloops, British Columbia, holds the Canadian shot put record of 21.58 m.



Self-Check 2

 

This is an image of a man preparing to throw a shot put.

Black 100/Photodisc/Thinkstock

 

The path of a shot put can be closely approximated by a quadratic function. Assume that a super-athlete starts to throw the shot from a height of 1.6 m. The standard form of the quadratic function describing the parabolic path is y = −0.0445x2 + 0.09434x + 1.6. The output distances will be in metres.

  1. Convert the quadratic function for the shot put from standard form to vertex form. Answer

  2. How high will the shot go? Answer

  3. How far from the thrower will the maximum height be? Answer

  4. Complete a graph to check your results using a graphing calculator, graphing program, or graph paper. Graph both forms of the function to verify that your conversion was accurate. hint Answer

  5. From the graph, explain approximately how far from the thrower the shot will land. Answer

If you feel you have a solid understanding of how to convert a quadratic equation in standard form to the vertex form, go to Connect. If you need a bit more practice, complete Self-Check 3.



textbook
Self-Check 3

 

Complete questions 3, 9, 12, 14, and 16 on pages 193 to 195 in the textbook. Check your work in the back of the textbook. If you are still unclear about how to answer some questions, contact your teacher.


formula
Add the standard and vertex forms of quadratic equations to your copy of Formula Sheet.
You may need to adjust your window settings to see the parabola on your calculator window. One quick method to start the window adjustment is to use zoom, and then select “zoomfit.”


1.9. Connect

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Connect
 
Lesson 3 Assignment


assessment

Open your copy of Lesson 3 Assignment, which you saved in your course folder at the beginning of this lesson. Complete the assignment.

 

course folder Save all your work in your course folder.

 

Project Connection


assessment

Go to Module 3 Project: Spray Park. Complete Activity 1: Part 2.

 

course folder Save all your work in your course folder.



1.10. Lesson 3 Summary

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Lesson 3 Summary
 

In this lesson you investigated the following question:

  • How do you complete the square to convert a quadratic function in the standard form, y = ax2 + bx + c, into the vertex form, y = a(x−p)2 + q?

You learned about the standard form of the quadratic function. You converted quadratic functions in the standard form to the vertex form by completing the square.

 

 

The values of p and q, which give the coordinates of the vertex of the parabola, can easily be identified in the vertex form, y = a(x−p)2 + q. The value of p is also the x-coordinate of the axis of symmetry of the parabola.

 

In the next lesson you will investigate how to determine the characteristics of a quadratic function in standard form and how to accurately sketch the graph. You will learn how to determine that a quadratic function in the standard form represents the same function as a given quadratic function in the vertex form.