Exercise 18 Page 593

Check the first differences of the values of the attendance. What do your results mean?

Model: Linear
Equation: y=59x+189

Practice makes perfect

We will start by determining the type of function that best models the data. Then we will be able to write an exact equation that models the data.

Finding a Model

We want to find the most appropriate model for the given data set. Note that the numbers of the game have a common difference of 1. Therefore, we can check if the values of the attendance have a common difference, a common ratio, or constant second differences. It will tell us which model is most appropriate for the data set.

The values of the attendance have:	The model is:
A common difference	Linear
A common ratio	Exponential
Constant second differences	Quadratic

Let's start by checking our data set for a common difference!

Game	Attendance	First differences
1	248
2	307	+59 ↩
3	366	+59 ↩
4	425	+59 ↩
5	484	+59 ↩

The first differences of the values of the attendance are all 59, so a linear model fits the data.

Writing an Equation

We know that a linear function best models the attendance. y= mx+ b To write an equation to model the data we have to determine the values of the slope m and the y-intercept b. Let's start with m! We will use the Slope Formula and the points ( 1,248) and ( 2,307).

m = y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 1,248) & ( 2,307)

m=307- 248/2- 1

SubTerms

Subtract terms

m=59/1

DivByOne

a/1=a

m= 59

We have that m= 59. Note that the value of m is equal to the common difference! We could have used that information instead of the Slope Formula. Let's write our partial equation! y= 59x+ b Since none of the points in the data set have an x-coordinate equal to 0, using our partial equation we have to write an equation that can be solved for b. To do so, let's substitute the point ( 1, 248) into the partial equation.

y=59x+b

SubstituteII

x= 1, y= 248

248=59( 1)+b

▼

Solve for b

MultByOne

a * 1=a

248=59+b

SubEqn

LHS-59=RHS-59

189=b