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Lesson 5

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Table of contents

1. Lesson 5

Mathematics 20-2 M3 Lesson 5

Module 3: Quadratics

 

Lesson 5: Factored Form of Quadratic Functions

 
Focus
 

This is a photo of a football player kicking a football.

© denyskuvaiev/25646996/Fotolia

A field goal may be the deciding factor in a football game. The shape of the path that the football travels after it has been kicked is a parabola. The direction of the kick is important so that the ball goes between the uprights of the goal post, but the kicker must also be sure that the height of the ball will be above the horizontal crossbar of the goal post.

 

Since the football will travel the path of a parabola, you know that you can use a quadratic function to model this situation. If you consider the goal post to be the y-axis, what the kicker needs to be concerned about is the y-intercept of the parabola.

 

factored form: a quadratic function written in the form
y = a(x − r)(x − s)

While the vertex form of a quadratic function allows you to quickly and easily find the vertex of the function, it is not always easy to quickly determine the x- and y-intercepts. However, you can rewrite the quadratic function in another form, called the factored form. This form is excellent for situations where the x- and y-intercepts are needed to solve a problem.

 

Lesson Questions
 

This lesson will help you answer the following inquiry questions:

  • How is the graph of a quadratic function related to the constants r and s in the factored form, y = a(x − r)(x − s)?

  • How do you sketch the graph of a quadratic function in the factored form?

  • How do you solve problems using a quadratic function in the factored form?
Assessment

All assessment items you encounter need to be placed in your course folder.

 

course folder

Save a copy of the Lesson 5 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

 

Materials and Equipment
  • graphing calculator

 

1.1. Launch

Mathematics 20-2 M3 Lesson 5

Module 3: Quadratics

 
Launch
 

Before beginning this lesson, you should be able to

  • write the trinomial form of a binomial squared

  • factor a trinomial that is a perfect square

1.2. Are You Ready?

Mathematics 20-2 M3 Lesson 5

Module 3: Quadratics

 
Are You Ready?
  1. Factor each of the following perfect-square trinomials by showing them as a binomial squared.

    1. x2 + 14x + 49 Answer
      2.  
    2. x2 − 6x + 9 Answer
  1. Factor each of the following trinomials.

    1. x2 + 7x + 10 Answer
      2.  
    2. 3x2 − 6x − 24 Answer
  1. Use the gizmo “Factoring Special Products (ExploreLearning)” to factor polynomials involving perfect-square binomials, difference of squares, or constant factors.

     

    This is a play button that opens “Factoring Special Products.”

    Screenshot reprinted with permission of ExploreLearning.


If you successfully completed the Are You Ready? questions, move on to the Discover section of the lesson.

 

If you experienced difficulties with the questions, use the resources in the Refresher section to review the concepts before continuing through the lesson.


 

1.3. Refresher

Mathematics 20-2 M3 Lesson 5

Module 3: Quadratics

 
Refresher
 

For information on factoring quadratic expressions, watch the “Factoring Trinomials with a Leading 1 Coefficient” video from the Khan Academy.

 

 

This is a play button that opens “Factoring Trinomials with a Leading 1 Coefficient.”

Khan Academy (CC BY-NC-SA 3.0)



For information on factoring quadratic expressions, watch the “Factoring Special Products” video from the Khan Academy.

 

 

This is a play button that opens “Factoring Special Products.”

Khan Academy (CC BY-NC-SA 3.0)



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 20-2 M3 Lesson 5

Module 3: Quadratics

 
Discover
 

The intercepts of a quadratic function usually provide information that is useful in solving a problem. The factored form allows you to identify the x- and y-intercepts of the quadratic function.

 

By now, you will recognize that a quadratic function in standard form, y = ax2 + bx + c, is made up of a trinomial, ax2 + bx + c. It follows, then, that you can factor a quadratic function using the same methods as you would use to factor a trinomial. The factored form of a quadratic function is written as y = a(x − r)(x − s).

 

In an earlier lesson you learned that the y-intercept is equal to the value of c in the standard form of a quadratic function. What if the function is in factored form? How can the y-intercept be found?

 

Substitute 0 for x in the factored form of the function and solve for y.

 

 

 

Recall that when you multiply two negative numbers together, you get a positive number. This means that y = a(−r)(−s) can also be written as y = a(r)(s). So, the y-intercept can be calculated from the factored form of a quadratic function using the formula y = ars.

 

 


1.5. Discover 2

Mathematics 20-2 M3 Lesson 5

Module 3: Quadratics

 
Try This 1
 

Use Factored Quadratic to draw graphs of the following quadratic functions and answer the associated questions.

 

 

This is a play button that opens Factored Quadratic.

  1. In the applet, draw the quadratic function f(x) = 4x2 + 4x − 24 by moving the slider to change the values for a, b, and c.

    1. What are the x-intercepts for the graph? Answer
      2.  
    2. How many x-intercepts are there? Answer
  1. The factored form of the function in question 1 is f(x) = 4(x + 3)(x − 2).

    1. What are the values of r and s? Answer
      2.  
    2. How do the values of r and s relate to the x-intercepts for the graph? Answer
      3.  
    3. From the graph, what is the y-intercept of the graph? Answer
      4.  
    4. Calculate the y-intercept using the formula y = ars. Answer
  1. Draw the quadratic function f(x) = −3x2 + 6x − 3.

    1. What are the x-intercepts for the graph? Answer
      2.  
    2. How many x-intercepts are there? Answer
  1. The factored form of the function in question 3 is f(x) = −3(x − 1)(x − 1), which can also be written as f(x) = −3(x − 1)2.

    1. What are the values of r and s? Are they different numbers in this case? Answer
      2.  
    2. How do the values of r and s relate to the x-intercepts for the graph? Answer
      3.  
    3. What is the y-intercept of the graph? Answer
      4.  
    4. Calculate the y-intercept using the formula y = ars. Answer
  1. A quadratic function is written as f(x) = 2x2 + 6x + 8.

    1. What are the x-intercepts for the graph? Answer
      2.  
    2. How many x-intercepts are there? Answer
  1. The function from question 5 cannot be written in factored form. The partially factored form of this function is f(x) = 2(x2 + 3x + 4).

    1. Why can’t the function be written in fully factored form? Answer
      2.  
    2. Can values for r and s be found? Answer
      3.  
    3. What is the y-intercept of the graph? Answer
      4.  
    4. If you didn’t have the graph, which form of the function can you use to find the y-intercept? Answer

 


1.6. Explore

Mathematics 20-2 M3 Lesson 5

Module 3: Quadratics

 
Explore
 

In question 2.b. of Try This 1, you saw that the values of r and s are equal to the values of the x-intercepts in the factored form of a quadratic function.

 

This shows a graph under the heading y = a(x – r)(x – s) with a > 0. The parabola opens upwards with x-intercepts at r and s and a vertex below the x-axis. This shows a graph under the heading y = – a(x – r)(x – s) with a < 0. The parabola opens downwards with x-intercepts at r and s and a vertex above the x-axis.

 

Recall that quadratic functions can have zero, one, or two x-intercepts. By looking at the factored form of a quadratic function, you can determine the number of x-intercepts.

  • If r and s are different values, there are two x-intercepts.

  • If r and s are equal, there is only one x-intercept and the vertex is on the x-axis.

If there are no x-intercepts, the function cannot be written in factored form.

 

This graphic shows three parabolas. One has two x-intercepts at r and s, which are distinct values. The second has one x-intercept at r, and r and s are equal. The third has no x-intercepts and cannot be written in factored form.

Source: CANAVAN-MCGRATH ET AL. Principles of Mathematics 11, © 2012 Nelson Education Limited. p. 362. Reproduced by permission.

 

zero: a value x in the domain of a function, f, that satisfies the equation f(x) = 0

The x-intercepts correspond to the zeros of the quadratic function. A zero is the value you get for x when you make the function equal to zero.

 

The zeros can be used to determine more information about the function and its graph.

 


textbook

Read “Example 1: Graphing a quadratic function given in standard form” on pages 338 and 339 of the textbook. As you read, consider how the zeros are used to determine the equation of the axis of symmetry and the vertex of the parabola.

Self-Check 1
  1. Complete “Check Your Understanding” question 1 on page 346 of your textbook. Answer

  2. Complete “Check Your Understanding” question 2 on page 346 of your textbook. Answer

  3. Complete “Practising” question 4 on page 346 of your textbook. Answer

  4. Complete “Practising” question 7 on page 347 of your textbook. Answer

1.7. Explore 2

Mathematics 20-2 M3 Lesson 5

Module 3: Quadratics


Quadratic functions represent the trajectory or path of many projectiles and can be used to answer questions and solve problems related to those projectiles. As you have seen, the factored form of a quadratic function allows you to easily identify the x- and y-intercepts of the graph. From these values, you can find the axis of symmetry and the coordinates of the vertex. You can then use these values to sketch the graph of a function that models the trajectory of a projectile—like a piece of paper being tossed into the trash.

 

Watch the animation Trash Throw to see how the factored form of a quadratic can be used to determine how far away a wastebasket is from a desk.

 

 

This is a play button that opens Trash Throw.

Comstock/Thinkstock



textbook

Read “Example 4: Solving a problem modelled by a quadratic function in factored form” on pages 343 and 344 of the textbook. As you read, consider both Krystina’s and Jennifer’s solutions to the problem.

Self-Check 2

  1. This is a photo of a man dressed in traditional leather clothing holding a bow and arrow.

    iStockphoto/Thinkstock

    A bow hunter shoots across a level meadow at a distant target. The quadratic function in factored form describing the trajectory of the arrow is y = −0.0054(x + 3)(x − 95). The horizontal distance, in metres, from the hunter is x, and y is the height, in metres, above the level of the meadow. From the equation given, answer the following questions.
 
  1. What are the x-intercepts? Answer

  2. How far did the arrow go? Answer

  3. From what height was the arrow released? Answer

  4. What is the axis of symmetry for the flight of the arrow? Answer

  5. How high did the arrow go? Answer

  6. What would a sketch of the graph look like? Answer

  7. What domain and range of the function are reasonable? Answer

This is a photo of a baseball player who has just hit a baseball.

© Kuvien/8985948/Fotolia

  1. A baseball is hit for a home run according to the function h = −0.005 27(d + 1)(d − 159). In the function, h is the height, in metres, of the ball above the field and d is the distance, in metres, of the ball from the batter.

    1. What are the d-intercepts? Answer

    2. How far would the ball go on a level field? Answer

    3. At what height was the ball hit? Answer

    4. How high was the ball above the field when it crossed the outfield wall 125 m from the batter? Answer



textbook
  1. Complete “Practising” question 9 on page 347 of the textbook. Answer

Did You Know?


This is a photo of a baseball being hit by a batter with the catcher and umpire watching closely from behind the batter.

© itsallgood/2985382/Fotolia

 

There is controversy over who has the record for the longest home run hit in the major baseball leagues. The legendary Babe Ruth hit more long balls than anyone. Air resistance slows down a baseball, so it doesn’t follow a perfectly parabolic path. Projections made using a perfect parabolic path are about 100 ft (30 m) too long. Claims of hits over 600 ft (183 m) are an exaggeration, but a few individuals have hit a baseball over 500 ft (153 m). Besides Babe Ruth, some of the players who have accomplished this truly remarkable achievement numerous times are Mickey Mantle, Jimmie Foxx, and Frank Howard.



textbook

Study “Example 2: Using a partial factoring strategy to sketch the graph of a quadratic function” from pages 339 and 340 of the textbook. Pay particular attention to how the vertex can be found even if you cannot factor the quadratic function.

Self-Check 3
 

Now, take your turn at sketching the graph of a partially factored quadratic function.

  1. This is an illustration of an arch with a road going beneath the arch.
    An old bridge support is a parabolic arch constructed out of stone. A quadratic function describing the height, in metres, of the arch is f(x) = −0.64x2 + 3.2x + 4. Movers want to bring a 4-m wide mobile home through the arch.
 
  1. Sketch a graph of the function without using electronic graphing technology or a table of values. Answer

  2. What are the approximate roots (the x-intercepts) of the function? How can you verify your estimate? Answer

  3. What is the maximum height of a 4-m wide structure that can pass through the arch? Answer

  1. This is a photo of an igloo.

    Hemera/Thinkstock

    The interior profile of a particular igloo is modelled by the function h = −d2 + 3d − 0.2. The height, in metres, of the interior surface compared to the snow outside is h and the distance from the entrance is d.
 
  1. Factor the function as much as possible. Answer

  2. Sketch a graph of the partially factored function without using electronic graphing technology or a table of values. Answer

  3. How much higher than the snow outside is the highest place inside the igloo? Answer

  4. How much lower is the entrance than the snow outside? Answer

  5. An Inuit hunter bumps his head on the inside wall when he stands 0.82 m inside the igloo at the level of the entrance. How tall is he? Answer