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
y−bx−a=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
y−y1=m(x−x1)
3. Piecewise Linear Functions
A function is piecewise linear if it is made up of parts of lines
Example 1
f(x)={x+4if x≤−22x−1if −2<x<1−2xif x≥1
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:
y−140=−15(x−80)
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)