Lines

1. 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.

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

 Slope Intercept Equation of a Line           y - y1 = m(x - x1)

3.  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

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

4. 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.