Exercise 28 Page 594

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 number of visitors.

Finding a Model

The given graph seems to not curve, so it might be modeled by a linear function. To make sure, we will check whether the y-values have a common difference. Let's identify the coordinates of the points on the graph.

Now we will organize the coordinate pairs (x,y) in a table.

Note that the x-values have a common difference of 1. Let's check the first differences of the y-values!

x	y	First differences
1	800
2	700	-100 ↩
3	600	-100 ↩
4	500	-100 ↩

The first differences of the y-values are all - 100, so our assumption is correct — a linear model fits the data.

Writing an Equation

We know that a linear function best models the number of visitors. 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 ( 1,800) and ( 2,700).

m = y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 1,800) & ( 2,700)

m=700- 800/2- 1

SubTerms

Subtract terms

m=- 100/1

DivByOne

a/1=a

m=- 100

We have that m=- 100. 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=- 100x+b Since none of the points in the graph 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, substitute the point ( 1, 800) into the partial equation.

y=- 100x+b

SubstituteII

x= 1, y= 800

800=- 100( 1)+b

▼

Solve for b

MultByOne

a * 1=a

800=- 100+b

AddEqn

LHS+100=RHS+100

900=b

RearrangeEqn

Rearrange equation

b=900

Let's finish writing the equation to model the number of visitors! y=- 100x+900 Our answer corresponds to option A. Below we have included a graph that shows how the equation models the given data.