We have been given the following table that shows the relationship between ounces and calories.

Ounces	$8$	$12$	$16$	$20$
Calories	$100$	$151$	$202$	$250$

Using this table, we will write an equation of a trend line. To do so, let's begin by making a scatter plot of the data points in the form (ounces, calories).

Now, we can draw a trend line considering the fact that the numbers of calories and ounces cannot be negative. Note that this is a line drawn on the scatter plot that is as close to as many data points as possible.

Next, we will write an equation for the trend line in slope-intercept form.

y = m x + b

In this form,

m

is the slope and

b

is the

y -

intercept. First, let's find the slope by using the Slope Formula.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Here,

(x_{1}, y_{1})

and

(x_{2}, y_{2})

represent points that are on the line. There are many points that appear to be on the line, but we can choose the points

(8, 100)

and

(20, 250)

and substitute them into the formula.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(8, 100)$ & $(20, 250)$

m = \frac{2 5 0 - 1 0 0}{2 0 - 8}

SubTerms

Subtract terms

m = \frac{1 5 0}{1 2}

CalcQuot

Calculate quotient

m = 12.5

The slope of our trend line is

12.5 .

y = 12.5 x + b

Next, we will find the

y -

intercept by substituting the point

(8, 100)

into this equation.

y = 12.5 x + b

SubstituteII

$x = 8$ , $y = 100$

100 = 12.5 (8) + b

Multiply

100 = 100 + b

SubEqn

$LHS - 100 = RHS - 100$

0 = b

RearrangeEqn

Rearrange equation

b = 0

From here, we can write the complete equation.

y = 12.5 x

Alternative Solution

Finding the line of best fit

The line of best fit is a trend line that shows the relationship between two sets of data most accurately. It can be found by using a graphing calculator. To do so, we first make a scatter plot of the data. We push $STAT,$ choose Edit, and then enter the values in the first two columns.

Having entered the values, we can plot in a scatter plot them by pushing $2nd$ and $Y = .$ Then, we will choose one of the plots in the list. Make sure you turn the Plot1 ON, choose the type to be a scatter plot, and assign L1 and L2 as XList and Ylist. We can pick whatever type of mark we want.

Pushing the $GRAPH$ button will tell the calculator to plot the data set. If we are using a standard viewing window, we will need to change the settings so that we can see all of the data points.

Finally, to view the linear regression analysis of the data set, we press $STAT$ , scroll to right to view the CALC options, and then choose the fourth option in the list, LinReg.

From here, we can write the equation of the line of best fit as follows.

y = 12.525 x + 0.4

Exercise 43 Page 350

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