Print bookPrint book

Lesson 4

Site: MoodleHUB.ca 🍁
Course: Math 20-1 SS
Book: Lesson 4
Printed by: Guest user
Date: Monday, 15 September 2025, 2:20 PM

Description

Created by IMSreader

Table of contents

1. Lesson 4

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Lesson 4: Properties of y = ax2 + bx + c

 
Focus

 

This is a photo collage of water squirting from fountains and equipment at a spray park.

Hemera/Thinkstock

 

From your work on Module 3 Project: Spray Park, you know that when water is spayed in a single stream, the path the water follows is in the shape of a parabola. The size and shape of that parabola depend on the angle and speed of the water as it leaves the spray nozzle.

 

The path can be modelled with a quadratic function, expressed either in the standard form, y = ax2 + bx + c, or in the vertex form, y = a(x − p)2 + q. How can you use the constants in the standard form of a quadratic function to figure out the distance the water will go?

 

In this lesson you will learn how to determine the characteristics of a function, including the vertex, axis of symmetry, y-intercept, roots, and width of the parabola when given a quadratic function in the standard form, y = ax2 + bx + c.

 

Outcomes

 

At the end of this lesson you will be able to

  • describe the characteristics of a function expressed in the form y = ax2 + bx + c and explain the strategy used to arrive at those characteristics

  • sketch the graph of a function given in the form y = ax2 + bx + c

  • verify that a function given in the form y = ax2 + bx + c represents the same function as one given in the form y = a(x − p)2 + q

Lesson Questions

 

You will investigate the following questions:

  • How do you determine the characteristics of a quadratic function in the form y = ax2 + bx + c?

  • How can you determine whether two functions written in the forms y = ax2 + bx + c and y = a(x − p)2 + q represent the same function?

Assessment


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

  • completion of the Lesson 4 Assignment (Download the Lesson 4 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

Materials and Equipment

 

You will need graph paper.


1.1. Discover

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Discover
 

From studying the vertex form in Lessons 1 and 2, you know that the parameter a influences the parabolic graph’s direction of opening (upwards or downwards) and the width of the curve.

 

Parameter a is also found in the standard from of the quadratic function. How do the other parameters in the standard form, b and c, influence the characteristics of the curve?

 

Try This 1

 

Use Quadratics in Polynomial Form - Activity B to graph the functions in the two charts below. hint In order for this application to function properly, sign into LearnAlberta.ca first and then click on the play button provided below.

 

This is a play button that opens Quadratics in Polynomial Form - Activity B.

Screenshot reprinted with permission of ExploreLearning.

In Part A you will examine the effect of changing the value of b. In Part B you will examine the effect changing c has on the graph. Record your observations in charts like the ones shown. The charts will help you view emerging patterns.

 



Part A

  1. Complete a chart like the one shown and observe the effect of b on the parabola.

     
    Equation a b c Vertex y-Intercept Graph
    y = −1x2 + (−3)x − 1            
    y = −1x2 + (3)x − 1            
    y = −1x2 + (1)x − 1            
    y = −1x2 + (−1)x − 1            
    y = −2x2 + (−1)x − 1            


  2. Do you see any significant pattern? If you think you may have a generalization, try several other values for b to see if your generalization seems to be correct.

Part B

  1. Complete a chart like the one shown and observe the effect of c on the parabola.

     
    Equation a b c Vertex y-Intercept Graph
    y = −1x2 + (−3)x − 3            
    y = −1x2 + (−3)x − 2            
    y = −1x2 + (−3)x + 1            
    y = −1x2 + (−3)x + 3            


  2. Do you see any significant pattern? If you think you may have a generalization, try several other values for c and observe if your generalization seems to be correct.

course folder Save your responses in your course folder.

 

Share 1

 

Share your observations and generalizations about the effect of changing the value of b and c with a classmate. Be sure to address how the graph of the quadratic function is affected by a change in the value of b or c.

 

course folder Save your responses in your course folder.

Try these hints when using Quadratics in Polynomial Form - Activity B:

  • The “copy graph to clipboard” button can be used to capture the graph and record it in your chart.
  • Select the “show vertex/intercept data” to easily observe this information.


1.2. Explore

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Explore

 

In Lessons 1 and 2 you explored the effects of parameters a, p, and q in the vertex form. In Discover you investigated the standard form and the information you can obtain about the graphs of quadratic functions from parameters b and c. The following tables summarize the effect of all of these parameters on the graph.

 

 
Parameter Effect on Graph  
c

The y-intercept will always be the value of c in the function

y = ax2 + bx + c.

 This is the graph of y = 2x squared + 4x − 2. The y-intercept (0, −2) is marked with a red dot.

a

 This value influences the width of the curve of the parabola. This is the graph of y = 2x squared + 4x − 2 and y = 20x.
If a is negative, the parabola opens downward. This is the graph of y = −2x squared + 4x − 2.
If a is positive, the parabola opens upward. This is the graph of y = 2x squared + 4x − 2.
p The axis of symmetry is at x = p. This is the graph ofy = (x − 2)squared + 1 showing the parabolic graph, the equation, and the axis of symmetry at x = 2, which is the value of p.
q q is the minimum value if a > 0 or the maximum value if a < 0. The minimum or maximum occurs at y = q.

q is the minimum value if a > 0.

 

This is the graph of y = 2x2 + 4x – 2 and y = 2(x + 1)2 –4. The graph has an a value greater than 0. Therefore, the graph has a minimum value.

 

q is the maximum value if a < 0.

 

This is the graph of y = –2x2 + 4x – 2. The graph has an a-value less than 0. Therefore, the graph has a maximum value.
p and q The vertex is at (p, q).

This shows the graph of y = (x–2)2 + 1. The parabolic graph, the equation, and the vertex at (2, 1) are shown.


You may have noticed that parameter b was not included in the table. The effect of b on the graph is not an obvious pattern that will be studied in this course.

 

Try This 2

 

Begin with the quadratic function in standard form, y = 3x2 − 6x − 3.

  1. What information can you obtain by looking at the parameters a, b, and c?

  2. Graph this quadratic function using your graphing calculator or Quadratics in Polynomial Form - Activity B.

  3. Compare your answers from question 1 to the graph of the function you created in question 2. Do your predictions match what you see in the graph?

  4. Change the function from standard form to vertex form by completing the square.

  5. Graph the vertex form of this function using either your calculator or Quadratic Function (Vertex Form). How do these two graphs compare? What does this tell you?

  6. What other information can you find about the graph by looking at the parameters p and q in the vertex form? Do your predictions match your graphs?

    You may need to log in to LearnAlberta.ca first in order to get Quadratic Function (Vertex Form) to run.

     
    This is a play button that opens Quadratics in Polynomial Form - Actvity B.

    Screenshot reprinted with permission of ExploreLearning.

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

course folder Save your response in your course folder.



1.3. Explore 2

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

In Try This 2 you obtained information about the graph of a quadratic function by looking at the parameters in the standard form and vertex form. You then verified the information gathered algebraically by graphing both forms of the function. If both the standard form and vertex form represent the same quadratic function, their graphs will be identical.

 

Self-Check 1

  1. Consider the following pairs of equations. Does each pair represent the same function? Without graphing these functions, what information can you obtain from the parameters found in the standard and vertex forms? Graph your answer to confirm.

 
 
 
a. (i) y = x2 − 6x − 3 (ii) y = (x − 3)2 − 12 Answer
b. (i) y = x2 − 2x − 8 (ii) y = (x − 2)2 − 8 Answer
c. (i) y = −2x2 − 2x + 2.5 (ii) y = −2(x + 2)2 + 3 Answer

This is a play button that opens Quadratics in Vertex Form - Activity A.

Screenshot reprinted with permission of ExploreLearning.

You may find Quadratics in Vertex Form - Activity A useful. This applet lets you graph a function in vertex form and see the standard form (labelled polynomial). (You may need to log in to LearnAlberta.ca first in order to run Quadratics in Vertex Form - Activity A.)

 

How do you see this applet being useful in Module 3? How could you have used the applet in Self-Check 1?

 



Example: Sketching the Graph of a Function in Standard Form

 

This example shows steps that could be used to sketch a graph of a quadratic function given in standard form.

 

Sketch the graph of y = 2x2 + 2x − 1.

 

Step 1: Look at parameter a in y = 2x2 + 2x − 1. You know that a is positive, so the parabola opens upwards. The value of a is greater than 1, so the parabola will be narrower than the graph of y = x2.

 

 

This illustration shows an upward-opening parabola with arrows from the sides of the parabola going inward.

 

Step 2: Look at parameter c in y = 2x2 + 2x − 1. c is −1. Therefore, the y-intercept will be −1.

 

Step 3: Rearrange to vertex form, y = a(x − p)2 + q, by completing the square.

 

 

 

Step 4: Find the vertex from the parameters p and q. The vertex is found from (−p, q) and is .

 

Step 5: Look at parameter q.

 

 

 

From the value of q you know the minimum value of the function will be .

 

Step 6: Look at parameter p again.

 

 

 

This tells you that the axis of symmetry is at .

 

Step 7: Look at the axis of symmetry and the y-intercept. The axis of symmetry is at , so you know a second point will be the mirror image of the y-intercept (0, −1), at (−1, −1).

 

The y-intercept is very close to the vertex, so estimating the width of the graph could be tricky. Get a point further up the arm of the parabola. Try substituting 1 for x and find the point (1, 3), which is also on the graph.

 

Step 8: Sketch the graph.

 

 

This illustration shows the graph of y = 2x squared + 2x − 1 along with the points (1, 3) and (−2, 3) on the graph. The axis of symmetry is shown at x = −0.5.

 

Self-Check 2
  1. Practise what you have learned about the properties of quadratic functions. Describe the characteristics of each function and, from those characteristics, sketch the graphs of the functions.

    1. y = 2x2 − 8x + 10 Answer

    2. Answer

1.4. Explore 3

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Working Backwards: Expanding and Then Simplifying the Vertex Form

 

You have been verifying whether two functions written in the forms y = ax2 + bx + c and y = a(x − p)2 + q represent the same function by graphing the two functions to see whether their graphs are identical.

 

Another method to verify your work is to work backwards by expanding and then simplifying the vertex form of the quadratic function to see if the vertex form matches the standard form.

 

Try This 3
  1. Take the vertex form you derived in Self-Check 2 question 1.a., y = 2(x − 2)2 + 2. Expand the function by squaring the binomial term. Then simplify the polynomial on the right side of the function to a trinomial. This will show that the function is equivalent to the original form, y = 2x2 − 8x + 10. hint hint

  2. Now show that y = 4(x −3)2 + 6 is not equivalent to y = 4x2 + 24x + 30.

    1. Square the binomial term and simplify the polynomial to a trinomial.

    2. Complete the square of the quadratic function in standard form.

If you feel you have a solid understanding of how to determine the characteristics of a quadratic function in the form y = ax2 + bx + c and how to determine whether two functions written in the forms y = ax2 + bx + c and y = a(x − p)2 + q represent the same function, go to Connect. If you need a bit more practice, complete Self-Check 3.



textbook
Self-Check 3

 

Complete questions 4, 5, 6, 7, 8, and 11 on page 193 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.



Simplifying the polynomial to a trinomial means

 

Squaring the binomial means

 

1.5. Connect

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Connect
 
Lesson 4 Assignment


assessment

Open your copy of Lesson 4 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.



assessment

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

 

course folder Save all your work in your course folder.



1.6. Lesson 4 Summary

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Lesson 4 Summary
 

In this lesson you investigated the following questions:

  • How do you determine the characteristics of a quadratic function in the form y = ax2 + bx + c?

  • How can you determine whether two functions written in the forms y = ax2 + bx + c and y = a(x − p)2 + q represent the same function?

This image shows a parabolic radar dish.

© sss78/31470934/Fotolia

 

You learned that some characteristics of the graph of a quadratic function can be determined directly from the standard form of the quadratic function.

  • The y-intercept will always be the value of c in the function y = ax2 + bx + c.

  • The value of a is the same in either form of the quadratic function. The value of a influences the width of the curve of the parabola and whether the curve will open downward or upward. If a is negative, the parabola opens downward; if a is positive, the parabola opens upward.

To determine the axis of symmetry and the coordinates of the vertex of the parabola, it is useful to convert the function to the vertex form by completing the square. In the vertex form, y = a(x − p)2 + q, the following are true:

  • The vertex is at (p, q).

  • The axis of symmetry is at x = p.

  • The minimum value if a > 0 or the maximum value if a < 0 occurs at y = q.

You learned three ways to check whether a function given in the form y = ax2 + bx + c represents the same function as one given in the form y = a(x − p)2 + q:

  • Method 1: Graph both forms of the quadratic function.

  • Method 2: Square the binomial in the vertex form, y = a(x − p)2 + q, and simplify to see if the function matches the standard form.

  • Method 3: Complete the square of standard form, y = ax2 + bx + c, to see if the function matches the vertex form of the function.

In the next lesson you will learn to model situations using quadratic functions. You will solve problems by analyzing the functions.