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Mathematics LibreTexts

2.1: Lines

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1. Lines (definitions)

Everyone knows what a line is, but providing a rigorous definition proves to be a challenge.

Definition: Line

A line with slope m through a point P=(a,b) is the set of all points (x,y) such that

ybxa=m.

2. The Slope Intercept Form of the equation of a Line

Given a point (x1,y1) and a slope m, the equation of the line is

Definition: Slope Intercept Equation of a Line

yy1=m(xx1)

3. Piecewise Linear Functions

A function is piecewise linear if it is made up of parts of lines

Example 1

f(x)={x+4if x22x1if 2<x<12xif x1

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We graph this line by sketching the appropriate parts of each line on the same graph.

4. Applications

Example 2

Suppose you own a hotel that has 150 rooms. At $80 per room, you have 140 rooms occupied and for every $5 increase in price you expect to have two additional vacancies. Come up with an equation that gives rooms occupied as a function of price.

Solution

Let x be the price of a room and y be the number of rooms occupied. Then we have an equation of a line that passes through the point (80,140) and has slope 15. Hence the equation is:

y140=15(x80)

or

y=15x+16+140

or

y=15x+156.

Exercise 1

What should you do if your two year old daughter has a 40 degree C temperature?

Hint: We have the two points: (0,32) and (100,212).

Exercise 2

Suppose that your company earned $30,000 five years ago and $35,000 three years ago. Assuming a linear growth model, how much will it earn this year?

Exercise 3

My rental was bought for $204,000 three years ago. Depreciation is set so that the house depreciates linearly to zero in twenty years from the purchase of the house. If I plan to sell the house in twelve years for $250,000 and capital gains taxes are 28% of the difference between the purchase price and the depreciated value, what will my taxes be?

Exercise 4

Wasabi restaurant must pay either a flat rate of $400 for rent or 5% of the revenue, whichever is larger. Come up with the equation of the function that relates rent as a function of revenue

Larry Green (Lake Tahoe Community College)


This page titled 2.1: Lines is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

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