Chapter 1 Functions and Linear Models Sections 1.3 and 1.4. Linear Function. A linear function can be expressed in the form. Function notation. Equation notation. where m and b are fixed numbers. Graph of a Linear Function. The graph of a linear function is a straight line.
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Linear Function
A linear function can be expressed in the form
Function notation
Equation notation
where m and b are fixed numbers.
Graph of a Linear Function
The graph of a linear function is a straight line.
This means that we need only two points to completely determine its graph.
m is called the slope of the line and
b is the y-intercept of the line.
Example: Sketch the graph of f (x) = 3x – 1
y-axis
Arbitrary point
(1,2)
(0,-1)
x-axis
y-intercept
Role of m and b in f (x) = mx + b
The Role of m (slope)
f changes m units for each one-unit change in x.
The Role of b (y-intercept)
When x = 0, f (0) = b
Role of m and b in f (x) = mx + b
To see how f changes, consider a unit change in x.
Then, the change in f is given by
Role of m and b in f (x) = mx + b
Role of m and b in f (x) = mx + b
Role of m and b in f (x) = mx + b
Slope = 3/1
y-intercept
Example:Sketch the graph of f (x) = 3x – 1
y-axis
(1,2)
x-axis
Example:Sketch 3x + 2y = 6
y-axis
x-intercept (y = 0)
x-axis
y-intercept (x = 0)
If aquantity q changes from q1 to q2 , the change in q is denoted by q and it is computed as
Example: If x is changed from 2 to 5, we write
Delta Notation
Example:the slope of a non-vertical line that passes through the points (x1,y1) and (x2,y2) is given by:
Example: Find the slope of the line that passes through the points (4,0) and (6, -3)
Delta Notation
Zero Slope and Undefined Slope
Example: Find the slope of the line that passes through the points (4,5) and (2, 5).
This is a horizontal line
Example:Find the slope of the line that passes through the points (4,1) and (4, 3).
This is a vertical line
Undefined
Examples
Estimate the slope of all line segments in the figure
An equation of a line that passes through the point (x1,y1) with slope m is given by:
Example: Find an equation of the line that passes through (3,1) and has slope m = 4
Can be expressed in the form y = b
y = 2
Can be expressed in the form x = a
x = 3
C(x)= “variable costs” + “fixed costs”
Dollars
Dollars
Cost
Cost
Units
Units
Dollars
Dollars
Cost
Cost
Units
Units
Dollars
Revenue
Units
Dollars
Dollars
Revenue
Revenue
Units
Units
Profit = Revenue – Cost
Dollars
Profit
Units
Dollars
Dollars
Profit
Profit
Units
Units
The Linear Models are
Cost Function:
** m is the marginal cost (cost per item), b is fixed cost.
Revenue Function:
** m is the marginal revenue.
Profit Function:
where x = number of items (produced and sold)
The break-even point is the level of production that results in no profit and no loss.
To find the break-even point we set the profit function equal to zero and solve for x.
The break-even point is the level of production that results in no profit and no loss.
Profit = 0 means Revenue = Cost
Dollars
Revenue
profit
loss
Break-even Revenue
Cost
Units
Break-even point
Example: A shirt producer has a fixed monthly cost of $3600. If each shirt has a cost of $3 and sells for $12 find:
a.The cost function
C (x) = 3x + 3600 where x is the number of shirts produced.
b.The revenue function
R (x) = 12x where x is the number of shirts sold.
c.The profit from 900 shirts
P (x) = R(x) – C(x)
P (x) = 12x – (3x + 3600) = 9x – 3600
P(900) = 9(900) – 3600 = $4500
Example: A shirt producer has a fixed monthly cost of $3600. If each shirt has a cost of $3 and sells for $12 find the break-even point.
The break even point is the solution of the equation
C (x) = R (x)
Therefore, at 400 units the break-even revenue is $4800
q = items demanded
Price p
q = items demanded
q = items demanded
Price p
Price p
q = items supplied
Price p
q = items supplied
q = items supplied
Price p
Price p
Market Equilibrium occurs when the quantity produced is equal to the quantity demanded.
supply curve
q
surplus
shortage
demand curve
p
Equilibrium Point
Market Equilibrium occurs when the quantity produced is equal to the quantity demanded.
q
supply curve
surplus
shortage
demand curve
Equilibrium demand
p
Equilibrium price
Example of Linear Demand
The quantity demanded of a particular computer game is 5000 games when the unit price is $6. At $10 per unit the quantity demanded drops to 3400 games.
Find a linear demand equation relating the price p, and the quantity demanded, q (in units of 100).
Example: The maker of a plastic container has determined that the demand for its product is 400 units if the unit price is $3 and 900 units if the unit price is $2.50.
The manufacturer will not supply any containers for less than $1 but for each $0.30 increase in unit price above the $1, the manufacturer will market an additional 200 units. Assume that the supply and demand functions are linear. Let p be the price in dollars, q be in units of 100 and find:
a. The demand function
b. The supply function
c. The equilibrium price and equilibrium demand
a. The demand function
b. The supply function
c. The equilibrium price and equilibrium demand
The equilibrium demand is 960 units at a price of $2.44 per unit.
Rate of change of q
Quantity at time t = 0
Linear Change over Time
A quantity q, as a linear function of time t:
If q represents the position of a moving object, then the rate of change is velocity.
We have seen how to find a linear model given two data points. We find the equation of the line passing through them.
However, we usually have more than two data points, and they will rarely all lie on a single straight line, but may often come close to doing so.
The problem is to find the line coming closest to passing through all of the points.
We use the method of least squares to determine a straight line that bestfits a set of data points when the points are scattered about a straight line.
least squares line
Given the following n data points:
The least-squares (regression) line for the data is given by y = mx + b, where m and b satisfy:
and
Example: Find the equation of least-squares for the data (1,2), (2,3), (3,7).
The scatter plot of the points is
Solution: We complete the following table
Sum: 6 12 29 14
Example: Find the equation of least-squares for the data (1,2), (2,3), (3,7).
The scatter plot of the points and the least squares line is
Coefficient of Correlation
A measurement of the closeness of fit of the least squares line. Denoted r, it is between –1 and 1, the better the fit, the closer it is to 1 or –1.
Example: Find the correlation coefficient for the least-squares line from the last example.
Points: (1 , 2), (2,3), (3,7)
= 0.9449