Lesson 3

Explore

 

What you may have concluded in Share 1 is that linear regression is not appropriate for the bouncing ball data for two reasons:

• Graphical: The data is clearly not linear as the y-coordinates increase and then decrease.
• Contextual: After the ball goes up, it must come down—a process that can’t be modelled with a straight line.

The curve you created in Try This 1 is quadratic. As with linear data, spreadsheets and graphing calculators can perform quadratic regressions.

 

 

 

The quadratic regression equation for the bouncing ball data calculated using a graphing calculator is y = −9.79x2 + 10.01x, where x is time in seconds and y is height in metres.

 

Notice that function matches trend of the data.

 

 

A regression equation that approximates the data can be used to determine how long the ball is in the air. Since the ball starts on the ground at time zero, the time it lands for the second bounce will be equal to the time it is in the air.

 

The ground is at height y = 0, so substituting 0 into the equation results in the following equation to be solved:

 

0 = −9.79x2 + 10.01x

 

The right side of this equation can factored:

 

 

By the zero product factor property:

 

x = 0

 

or 

 

 

Sometimes a quadratic equation cannot be solved by factoring. In these cases, the quadratic formula can be used:

 

 

The first answer, 0 s, corresponds to the first bounce. The second answer, 1.02 s, corresponds to the second bounce and the length of time that the ball is in the air.

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