1. Lesson 5

1.4. Explore 3

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Self-Check 1

 

Model the following situation by developing a quadratic function, and then solve the problem using your function.

 

A sketch shows a parabolic bridge span 4 metres high and 5 metres wide.

 

An old bridge support is a parabolic arch made out of stone. The arch is 4 m high at the vertex and 5 m wide at the base. A family needs to drive a rented van under the arch to reach their new home. The van is 3 m wide at the top.

  1. Draw a sketch of the situation with origin at the vertex. Answer

  2. Write a quadratic function that models the situation. Answer

  3. What is the maximum height the van can be and still pass through the arch? Explain any assumptions you make in your solution. Answer

  4. Would you arrive at the same answer if you put the origin in another place, like at road level on the left side of the arch? What function would be used to model the situation then? Answer
Try This 3

 

Sometimes you will run into a question where you don’t know how to solve the quadratic function. In these cases, your graphing calculator can be a great help. Try This 3 is a problem of this type.

 

A hockey player shoots the puck. If her shot had been taken on level ice with no obstructions, the puck would have risen from the ice surface in a parabolic arc to a height of 4.9 m and touched down again 34.8 m from the starting point.

 

This is a photo of a hockey player shooting a puck at the net.

Hemera/Thinkstock

Did You Know?


The fastest hockey shot ever officially recorded was by Boston Bruins defenceman Zdeno Chara, at a blistering 170 km/h. There is controversy over that record because Bobby Hull is reported to have shot a puck at the staggering speed of 190 km/h while skating at high speed down the ice.

  1. Draw a diagram of the situation with the origin at the point where the puck is shot.
  1. Write a quadratic function that models this situation. hint
  1. To score in the situation described, the shot must clear the goalie’s left shoulder and enter the net at a height of 1.18 m. How far from the goal line should the puck lift off the ice so the shooter scores? hint
The function is . However, it will be easier to work with the approximation y ≈ −0.016 18(x − 17.4)2 + 4.9.

Substitute the height of 1.18 m into the function for y and solve for x.

 

 

 

The challenge at this point is solving an equation that looks like 0 ≈ −0.016 18x2 + 0.563x − 1.18. You will learn how this is done algebraically in Module 4. The great thing is that you can use your graphing calculator or another graphing tool to find the answer. Remember that a graphing tool was provided in Lesson 4.

 

Graph the function y = 0.016 18(x − 17.4)2 + 4.9.

 

Find a point on the graph where y = 1.18. What does x equal? If you are using a graphing calculator, you might use the Trace or Table functions to find x at this point. Refer to your calculator manual or contact your teacher for support.

 

When y = 1.18, x should be about 2.24. So, the puck should lift off the ice 2.24 m from the goal line for the hockey player to make her shot.