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Lesson 5

1. Lesson 5

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