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.

Back to Math 103A Home Page

Back to the Math Department Home

e-mail Questions and Suggestions