### Introduction

Linear relations are everywhere even though you may not notice them. The linear relation/snap i have chose is a gym membership at a gym called Fit4Less. This relation is partial because it has an initial value. Direct variations do not have initial values and start at the origin.

### Learning goals does it relate?

#### learning goals

Linear Relations

• Using Data Management to Investigate Relationships • Understanding Characteristics of Linear Relations • Connecting Various Representations of Linear Relations

Analytic Geometry

• Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph • Investigating the Properties of Slope • Using the Properties of Linear Relations to Solve Problems

Studying this linear relation reflects on my learning goals because i am investigating relationships, showing that i understand the characteristics of linear relationships, connecting various lines, showing that i know the properties of slopes and showing that i can solve linear relations.

### PHASE 1

### What is the cost and equation?

Since it costs $4.99 every 2 weeks and has a initial fee of 44, the equation would be Y=2.5X+44. X being the amount of weeks you go to the gym and Y being the total cost. The reason it is 2.5X instead of 4.99X is because 4.99 is for two weeks and to find the cost for one week you have to divide 4.99 by 2 which is 2.49. 2.49 rounded is 2.5. This equation has a positive slope which makes it a positive line. X is the independent variable and Y is the dependent variable. Y is the dependent variable because it "depends" on X. in this equation X is the amount of weeks you visit the gym (time) and Y is the cost ($money)

M=2.5 B=44 M is the cost per week while b is the initial cost of a membership.

### Scenario

Lets pretend i went to this gym for 3 weeks how much would it cost? If i insert 3 into the equation it will be Y=2.5(3)+44. Y=2.5(3)+44 simplified is Y=51.5. Therefore if i went to the gym for 3 weeks it would cost me $51.5.

### T.O.V

Amount of weeks(X)and(time) Price(Y) and (money in $)

### Converting the equation into standard form

Standard form is the equation of a line in the form of Ax+By=C.

To convert Y=2.5x+44 you have to move a few numbers.

Y=2.5X+44~2.5X-y+44+0

Standard Form of a gym membership: 0=2.5X-y+44

### Calculating Change in Y and Change in X

To calculate the change in Y you have to set X to 0, to get the change in X you have to set X to 0.

Change in Y: 0=2.5(0)-y+44~0=-y+44~y=44

Change in X: 0=2.5X-y+44~-44=-2.5X~-44÷2.5=x~-17.6=X

Therefore, the change in y is 44 and the change in X is -17.6. The change in X is the coordinate of the X-Int and the change in Y is the Y-int.

#### What is the x-int?

The x-int is the point on the line where y is 0. In this case it would mean the point where it cost nothing. The x-int can also be used to find the slope.

### Graph of the line using desmos

Hold shift and click to open^ (in case it does not work https://www.desmos.com/calculator/7daiakcaig)

### PHASE 2

### Changing the Equation

Lets say i changed the equation form Y=2.5X+44 to Y=3.5X+33. Making the slope higher and decreasing the value of the y intercept changes the equation a lot. When you increase the slope the gym membership will cost more per week and the line will be steeper. When you decrease the y-int the initial value will be cheaper and the line will intercept the y axis lower. Lets try this with a lower slope and a higher y-int. For example Y=1.5X+55. When the slope is lower the price will be cheaper per week and the line will be less steep. When the y-int is higher the initial fee will cost more and the line will start higher.

### Which line is better in different scenarios?

PlanA:Y=2.5X+44 PlanB:Y=3.5X+33 PlanC:Y=1.5X+55

All of these plans can be better in different cases. In the case that you are just trying out the gym you may want to start with plan b so you dont have to pay alot all at once. if you are going to the gym for long term you should chose plan C because the price per week is lower and you only have to pay the fee once. Plan a is good if you are visiting the gym every so often but plan a and b are more beneficial.

### P.O.I

The three lines(Y=2.5X=44,Y=1.5+55 and Y=3,5+33) all intersect at the same coordinates. the coordinates are (11,71.5). You may see all the lines intersect on this desmos graph

These lines intersect at the same point because when the 1.5 is added the slope the y-int is subtracted by 10. When 1.5 is subtracted from the slope 10 is added to the y-int.

Credits:

Created with images by valkovav - "sports indoor cycling fitness" • alantankenghoe - "Cruise Gym" • MartialArtsNomad.com - "Sprint track"