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Lesson 2

1. Lesson 2

1.9. Explore 5

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Example 2: A Baseball’s Path

 

A centre fielder in baseball makes a throw to the catcher at home plate, a distance of 126 m. The path of the ball is a parabola in which the ball rises 22 m above the hands of the centre fielder and the catcher.    

  1. What quadratic function in vertex form describes the path of the baseball?

  2. Determine the height on the parabola that is above and 10 m horizontally from the catcher. Express your answer to the nearest tenth of a metre.

Solution

  1. Draw a diagram of the situation and put the information given in the problem onto the diagram.

    Step 1: Let the vertex of the parabolic shape be at the maximum height of the ball and the origin (0, 0). Draw a set of axes. Let x and y represent the horizontal and vertical distances from the maximum height of the ball, respectively.

     

    This illustration shows the path of the baseball from the fielder to the catcher. The apex of the ball’s path is used as the origin of a coordinate axis.


    Step 2: The ball rises 22 m above the players. The ball is thrown 126 m, so half the throw is 63 m. The coordinates of the fielder will be (63, −22).

    This illustration shows the path of the baseball. It shows the total distance from fielder to catcher as 126 metres. The fielder is 63 metres horizontally from the apex of the ball’s path, as is the catcher. They are both 23 metres below the apex of the ball’s path. The illustration shows a coordinate axis set with the vertex of the ball’s parabolic path at the origin.
     

    Step 3: The vertex was chosen to be at (0, 0), so the values of p and q are both 0. Substitute these values into the quadratic function. To calculate the value of a, substitute the coordinates of the centre fielder’s position (63, −22) into the function.

     

     



    Step 4: Substitute a into the quadratic function. A quadratic function describing the path of the baseball, when the maximum height is chosen as the origin, is .

     

  1. Step 1: To determine the height of a point on the parabola that is above and 10 m horizontally from the catcher, you need to express 10 m as a distance from the origin.

     
    63 m − 10 m = 53 m (from the origin)

    Therefore, x = −53.

    Step 2: Substitute the value of x into the quadratic function.

     


    The baseball is 15.6 m below the origin.

    Step 3: The height of the origin is 22 m above the catcher, and the catcher is 22 m below the origin. The height of the ball above the catcher is the difference.

    The baseball is 6.4 m above the catcher when it is 10 m horizontally from the catcher.

     
    22 m − 15.6 m = 6.4 m