Lines

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Lines

Lines (definitions)

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

Definition

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

(y - b)/(x - a) = k.

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

Slope Intercept Equation of a Line

y - y₁ = m(x - x₁)

Piecewise Linear Functions

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

Example

f (x) =

{

x + 4	x < -2
2x - 1	-2 < x < 1
-2x	1 < x

$Graph of a piecewise function. On the right of x = -2 is a line from the left to (-2,0). In the middle is a line segment from (-2,-5) to (1,1). On the right is a line from (1,-2) going down.$

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

Applications

Example

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 -1/5. Hence the equation is:

        y - 140 = -1/5 (x - 80)

or

        y   = -1/5 x + 16 + 140

or

        y = -1/5 x + 156

Exercises
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)
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?
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?
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.

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