Linear Function:

Form: y = mx +b

The values of m and b are to be specified.  They are called the slope and the vertical axis intercept respectively.   Geometrically, y=mx+b is indeed a line.  The value of m measures the orientation of the line relative to the horizontal axis.  It can be found if the "rise" and the "run" of the line are known.  It is known that one can construct a straight line on the plane connecting two distinct points if locations or coordinates of those two points are known.  The coordinates of those two points generate the "rise" and the "run".  The ratio, , is the slope m.  For definiteness, let suppose and are the coordinates of the first and the second points, respectively.  By rise we mean and by run we mean so that the slope m is given by .  Once the slope m is constructed, then the value of   b  can be determined.  We replace y, x, m in the formula y=mx+b  by the values of and the value of the slope respectively.  Solve the resulting equation for b.  Of course we could have use and instead of and

Example 1:

Construct a line, more precisely, the equation of a line that passes through points (1,1) and (2,0).
The equation is of course y = mx+b.  We are to determine the values of m and b.
In this example one can declare that =1, =1 and =2, =0 so that  m = == - 1.  In the equation y=mx+b replace y and x by and respectively.  Also replace m by its values namely -1.  We have 1=(-1) 1+b.  Therefore b=2.  Use these values of m=-1 and b=2 in the equation y=mx+b to identify the equation asked by the problem.  That is y =(-1)x+2= -x+2 is the the equation of the line that passes through points (1,1) and (2,0).  

There is no other line, or equivalently, there is no other equation of a line that can describe the line that passes through (1,1) and (2,0).  Of course there are infinitely many lines that goes through (1,1).  Likewise there are infinitely many lines that goes through  (2,0).  However, there is only one line that can go through (1,1) and (2,0) simultaneously, and that line is described algebraically by y = - x+2.  Below is part of the graph of : y = -x +2.

Not all linear equation is given in the form of y = mx + b.  For instance,  3x + 3y = 6, is a linear equation ( that is an equation whose geometric representation is a line ).  Nevertheless,  we can algebraically manipulate any linear equation so that the form y = mx + b ultimately is constructed.  Subtracting 3x from 3x + 3y = 6 will give us 3y = 6 - 3x.  Divide the last equality by 3 so that y = = 2 - x.  Equivalently, y = -x + 2.  So we have shown that 3x + 3y = 6 is mathematically equivalent to y = -x +2.  Consequently the line 3x + 3y = 6 has the same slope and y-intercept as y = -x + 2, namely  -1 and 2 respectively.  The fact that any linear equation can be brought to the slope-intercept form, y = mx + b, is very important.  

Since the slope measures the deviation of a line from a horizontal line then any two lines with the same slopes must be parallel.  Thus the lines 3x+3y = 6  and y = - x + 27 are parallel for the slopes of each line is  -1.  Yet the lines 6x + 6y = 12 and y = 2x +1 are not parallel.  The slope of 6x + 6y = 12 is -1 while the slope of y = 2x + 1is 2.

If the product of the slopes of two lines is -1 then the two lines are perpendicular to one another.  (The exception to this rule is that a horizontal line is perpendicular to a vertical line.  The product of their slopes is not -1.)   Thus any line of slope 1 will be perpendicular to the line 3x + 3y = 6.  In particular, the line 3x - 3y = 6 is indeed perpendicular to 3x + 3y = 6.  Recall that from example 1, the slope of 3x + 3y = 6 is -1 and its graph is given above.  The reader should  be able to verify that the slope of 3x - 3y = 6 is 1.   Here is the graphs of the two lines.

Applications:

(Linear) Supply and Demand

Linear functions of the form y = f(x) = mx + b can be used to model some simple approach to supply and demand analysis.  Instead of letters x and y,  we will use letters q (quantity) and p (price) respectively.  (We will be interested in the case of non negative price and non negative quantity).  Thus whenever we plot a graph in the supply and demand analysis, the vertical axis is reserved for p while the horizontal axis is reserved for q.  Altough the price determines the demand of the consumers and the producers supply, we will still write p as a function of q.  More precisely, we will write p = mq + b.  This p will in turn represent demand, D(q)  or supply, S(q).  The equilibrium price occurs whenever D(q) = S(q).  Geometrically, that is the intersection of lines D(q) and S(q).  The corresponding value of q for which D(q) = S(q) is called the equilibrium quantity.

Once this equilibrium price is determined then we can establish if setting a particular price will cause a shortage or surplus of the quantity, q.

Example 2:

Suppose the price p determines the supply and demand functions for butter pecan ice cream as follows.
p = S(q) = q  and  that  p = D(q) = 100 - q.  Of course p is measured in dollars and q is the number of 10 gallon tubs.
Find the equilibrium quantity and price.   Would a price of  $60 generate a shortage or surplus of 10 gallon tubs of butter pecan ice cream ?

Set S(q) = D(q) to determine the equilibrium price and quantity.
That is q = 100 - q.  Solve this equation for q.  We then obtain q = 125.  This is the equilibrium quantity.  If we use this value of q in either  expression  of S(q) or D(q) we will obtain the equilibrium price.  Let us use q=125 in S(q).  Then p = S ( 125) = (125) =50.  If we use q =125 in D(q) we would arrive at the same value for p, namely 50.  Thus we have obtained that the equilibrium price is $50 while 125 is the equilibrium quantity of 10 gallon tubs of butter pecan ice cream.

Now set p to be 60 and let us use such value of p in the equation for supply, p = S(q) = q.  So we have 60 = q.  Solving for q will give us,  q = 150.   This means for a price of $60, the producers will supply 150 tubs of butter pecan ice cream.
Next, use such a value of p=60 in the equation for demand, p = D(q) = 100 - q.  That is 60 = 100 - q.  This will give us q = 100.  This means for a price of $60, the consumers will demand only 100 tubs of butter pecan ice cream.

Since the producers supply is more than the consumers demand at the price of $60, then there will be a surplus in the market at such a price.  In fact any price above the equilibrium will trigger a surplus in the market in this case, while any price below the equilibrium price will cause a shortage.  Here is the graphs of the suppply and demand functions.

The red and the blue lines in the first plot represent the demand and supply functions respectively.  The green and gray area represent the shortage and the surplus area.

(Linear) Cost  and Break Even Analysis

Linear functions, y = f(x) = mx +b  can also be used to model a cost analysis of manufacturing certain items.  Instead of  f(x), the function is identified by C(x).  Thus the form to be analyzed is C(x) = mx + b, where C(x) represents the cost to produce x many items.  The slope, m, represents the marginal cost (per item) and b represents the fixed cost.  The marginal cost is the rate of change of cost, C(x),  at x.  In this linear case it is constant and equals to m.  But in the future, particularly when we are discussing the derivatives of functions, we will see that the marginal cost is not necessarily constant.  It depends the amount of production, x.   This is due to the fact that the functions involved are not linear any longer.  Examples for fixed cost are the cost to set up a factory, to train workers, etc.  This is assumed to be constant regardless of the linearity or non-linearity of the cost function.

The revenue, R(x), from selling x units of a product, is equal to the product of the price per unit, p, and the number of units sold, x.  That is R(x) = p·x
Profit, P(x),  is defined as the difference between the revenue, R(x),  and cost, C(x).  That is  P(x) = R(x) - C(x).  Break even is a case where P(x) is identically zero.  No profit (nor loss) occurs.  That is the revenue is the same as the cost.

Example 3:

You sell T-shirts at a craft fair for $9.50 each.  The marginal cost to produce one T-shirt is $3.50.  The total cost to produce 60 tshirts is $ 300.  How many T-shirts should be produce and sell to make a profit of $510 ?  To break even, how many T shirts are to be produced and sold?

The equation P(x)=R(x)-C(x) is to be used with P(x) being set to 510 (that is the profit you desire) and R(x) = px with p equals to 9.  Thus so far we have the following result: 510 = 9.5x - C(x).  We are to solve this for x.  

Well, the expression for C(x) does not exist yet.  We know however that the form for C(x) can be assumed to be linear, so that C(x) = mx + b.  We know m.  The slope is equal to the marginal cost, 3.50.  Furthermore if you produce 60 T-shirts,  (that is set x=60) then the cost, C(x),  is set at 300.  Thus we have 300 = 3.5 ( 60 ) + b .

From this b can be computed.  In fact b = 300 - 3.5 (60) = 90.  Having identified m = 3.5 and b = 90, then the cost function is given by, C(x) = 3.5 x + 90.  Use this result in the statement: 510 = 9.5x - C(x) so that we have 510 = 9.5x - (3.5 x + 90).  Solve this for x.  We then find: 600 = 6x,  so that x  =100.
Thus you need to produce 100 T shirts to enjoy a profit of $510.

To break even set P(x) = 0  That is  0 = R(x) - C(x)  or equivalently, 0 = 9.5 x - (3.5 x + 90)  Thus 0 = 6 x - 90, or x = 15.  You therefore need to produce 15 T-shirts to break even.  Here are some relevant graphs

The blue and the red lines in the first plot represents the revenue and the cost functions respectively.  They intersect at the "break even" point.  The break even point "splits" the blue region in the second plot.  The smaller blue region corresponds to the loss region, while the larger one corresponds to the profit region.  Notice that in the loss region, the line of cost is "higher" than the line of revenue, while in the profit region the line of cost is "lower" than the line of revenue.  The third plot shows that non negative profit occurs after x=15