8.5 Class Activity

The Incentive

Challenge Get into groups of 2 or 3. Come up with an equation that models (generally) the shape of the following graph. Whichever group that comes up with a logical equation first wins a prize.

Scenario Jack and Jill went up a hill to make it to a basketball game in Basketball City. To go to Basketball City, they must cross Inconvenient Hill, which can be modelled by the function: f(x)= x 2 +20

When Jack and Jill reach the highest point of the hill, Jill playfully pushes Jack, causing the basketball he was holding to fall out of his hands. Oh no! The basketball’s path can be modelled by the function: g(x)=1.75(0.9) x |cosx|

Question 1 O Two functions are used to express the path of the ball. Identify the two functions and explain their impact on the function (explain what each function does). O To model this function, would we use +/ -- / X / ÷ to combine the 2 functions?

Answer Cosine function x Exponential Function |cos x|  1.75(0.9) x

O The cosine function is the bounce factor and the exponential function is the consistent decay factor. O Cosine function instead of sine function because when Jack drops the ball, it falls from a height above zero. O The exponential function is multiplied by 1.75 because that is the height at which Jack drops the ball.

To model the basketball’s path as it travels down the hill, would you combine the two functions using addition, subtraction, multiplication, or division? Explain your choice. Question 2

Answer Addition The total height of the ball is needed to be found, which depends on the height of the hill (the hill’s function) and the height of the bounce (the ball’s function).

Question 3 Combine the two functions and express the path of the basketball as it travels down the hill as a single composite function.

Answer h(x) = f(x) + g(x) h(x) = (- 0.05x 2 +20) + (1.75(0.9) x |cosx| h(x) = x (0.9) x |cosx|

f(x) + g(x)=h(x)

Challenge O Come up with an alternative real life example that makes use of composite functions and draw a simple sketch. O You have 30 seconds to discuss with your partner.

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