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Lesson 5

1. Lesson 5

1.2. Explore

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Explore

 

In Share 1 you may have noticed you could develop different functions to model a given situation depending upon where you placed the origin. If developed correctly, each function will provide the same solution to the problem.

 

In Example 1: Water Stream, a quadratic function is developed to model the trajectory of water in a fountain. The strategy used in this example may also be applied to the problem in Try This 1. After reading Example 1: Water Stream, you may want to review your work from Try This 1.

 

Example 1: Water Stream

 

This is a photo of a woman putting her hand in water shooting from a fountain.

iStockphoto/Thinkstock

It’s a hot day. You and a friend are at a park with a cooling fountain. The stream of water leaves the nozzle 10 cm above the fountain’s water surface. The trajectory of the water stream reaches a height of 200 cm above the water when it is 300 cm from the nozzle. Your friend reaches out to touch the stream when she is 70 cm from the nozzle. What is the height of the stream of water when she touches it?

 

Assume that the trajectory of the water in the stream is a parabola.

 

Step 1: Draw a sketch of the situation. Include all given measurements and place the origin at the highest point of the trajectory.
 
 

This is a sketch that shows that the parabola that models the water stream opens downward from the origin. The point on the parabola farthest to the left is 200 centimetres below the origin and 300 centimetres to the left of the y-axis. The point farthest to the right is also 200 centimetres below the origin but is 300 centimetres to the right of the y-axis. The unknown height, h, lies 70 centimetres from the point on the left.

 

Step 2: With the origin at the vertex of the parabola, the y-intercept is at (0, 0). So, the quadratic function y = a(x − p)2 + q reduces to y = ax2.

 

 

 

Step 3: Notice that the nozzle is 200 cm − 10 cm, or 190 cm, lower than the vertex. To solve for a, take the coordinates of the nozzle located on the left at (−300, −190) measured in centimetres and insert the coordinates into the function.

 

 

 

The function can be written as or as the approximation y ≈ −0.002 111x2.

 

Step 4: Find the horizontal distance from where your friend touches the water to the vertex.

 

 
−300 cm + 70 cm = −230 cm

 

 

This is a sketch that shows that the parabola that models the water stream opens downward from the origin. The point on the parabola farthest to the left is 200 centimetres below the origin and 300 centimetres to the left of the y-axis. The point farthest to the right is also 200 centimetres below the origin but is 300 centimetres to the right of the y-axis. The unknown height, h, lies 70 centimetres from the point on the left. A hand touches the stream at this point for which the height must be determined.

 

Step 5: The horizontal distance at the point where the water is touched is −230 cm. Substitute this in for x so you can find the height at this point, y.

 

 

 

 

 

Step 6: The stream of water is about 112 cm below the vertex at this point. That means the stream will be approximately 88 cm above the water when it is touched.

 

 

This is a sketch that shows the parabola that models the water stream. The parabola opens downward from the origin. The point on the parabola farthest to the left is 200 centimetres below the origin and 300 centimetres to the left of the y-axis. The point farthest to the right is also 200 centimetres below the origin but is 300 centimetres to the right of the y-axis. The unknown height, h, lies 70 centimetres from the point on the left. A hand touches the stream at this point for which the height must be determined. The height at the hand’s position is shown as 200 centimetres minus 112 centimetres, or 88 centimetres.