B. Working with Word Problems

When solving a problem, pay close attention to what it is you are interested in finding. In order to solve a problem, you may need to determine an x-value, a y-value, a zero, a maximum, a minimum, the domain, the range, or a y-intercept. Understanding what is needed within a given context will be key to solving the problem.

All problems are different; however, some of the following strategies may help you solve problems.

  • draw a diagram that represents the scenario
  • write an equation or a function that represents the scenario
  • list the information given in the problem
  • assign variables to unknown values if they are not already stated
  • formulate a strategy to determine the unknown
  • verify and/or use reasoning to confirm the solution is reasonable

Example 1

A soccer kick can be modelled by the function , where h is the height of the ball, in metres, and t is the time, in seconds, that the ball is in the air.

a. How long will the ball be in the air?

Step 1: Assess the information given.

  • A function is given:
  • h is the height in metres
  • t is the time is seconds

Step 2: State what you are asked to find.

  • Time the ball was in the air.

Step 3: Decide how to determine this value.

  • The ball will be at a height of 0 m when it is kicked and when it lands. In between the time it is kicked and the time it lands, it will be in the air. Setting h(t) = 0 will allow you to determine the time(s) the ball is at a height of 0.

The two solutions to the equation are 0 and 6. The ball is kicked at 0 seconds and it lands at 6 seconds. As such, the ball will be in the air for 6 − 0 = 6 seconds.

b. Determine the maximum height reached by the ball, to the nearest tenth of a metre.

Step 1: Assess the information given.

  • A function is given:
  • h is the height in metres
  • t is the time in seconds

Step 2: State what you are asked to find.

  • Maximum height reached by the ball.

Step 3: Decide how to determine this value.

  • The maximum will occur at the vertex.
  • The coordinates of the vertex can be determined if the equation of the axis of symmetry is known.
  • The axis of symmetry is located at the midpoint between the x-intercepts.

The x-intercepts of the graph of the function are 0 and 6.

The equation of the axis of symmetry is t = 3. Use this information to determine the coordinates of the vertex.

The vertex is at (3, 45). The y-coordinate of the vertex corresponds to the maximum height reached by the ball. The maximum height reached by the ball will be 45 m.

c. When will the ball be at a height of 12 m?

when the ball is 12 m high.

The ball will be at a height of 12 m at 0.43 seconds and at 5.57 seconds.

d. State the restrictions, if any, on the domain and range for this scenario.

The ball is kicked at t = 0 and lands at t = 6 . The model is only useful when the ball is in the air, so the domain is . The height of the ball ranges between 0 m and 45 m, so the range is .