Math 141-Summer 2014
Chapter 1 – Linear Functions and Math. Models
Lines : A Review
Slope of a line passing through two points (x1, y1) and (x2, y2) is given by
m=
Equation of a line: can be expressed in two forms
a) Slope – intercept form:
b) Point-slope form :
Example. Find the equation of the line passing through the points (2, – 3) and (1, 4).
If x increases by 4, what is the corresponding change (increase or decrease) in y.
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Math 141-Summer 2014
Chapter 1 – Linear Functions and Math. Models
Linear Functions and Mathematical Models
What is a function?
A function f is a rule that assigns each value of x, one value of y.
e.g. y = f(x) = x + 2
Linear Depreciation
Value of an asset decreases linearly over time. The function is given by ( t, V )
Example 1. A car was purchased in 2000 for $15000. It has a scrap value of $3000 after 12 years. The
value of the car depreciates linearly.
a) Find a linear equation that represents the value of the car in t years.
b) What was the value of the car in 2007?
c) What is the rate of depreciation of the car?
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Math 141-Summer 2014
Chapter 1 – Linear Functions and Math. Models
Linear Cost, Revenue, and Profit Functions
Cost function : C(x) = cx + F
Revenue function : R(x) = sx
Profit function : R(x) – C(x)
Break-even point : When R(x) = C(x) , the point (x, y) is said to be the break even point.
Geometrically, break- even point is the point of intersection of the cost and revenue functions.
Example 2. A corporation sells packed sets of office supplies. Each set sells for $8.60, and the
variable cost of producing each set is 40% of the selling price. The company incurs a fixed cost of
$50,000 every month.
a) Find the cost, revenue, and profit functions, if they are known to be linear.
b) How many sets must the company sell every month, to have a profit of $25000?
c) How many sets must the company produce to break even? What would be the break-even revenue?
Example 3. A company sells a pen for $0.95. It costs the company $2500 to produce 2000 pens, and
incurs a fixed cost of $1000 every month. If the cost, revenue, and profit functions are know to be
linear, how many pens should the company produce to break even?
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Math 141-Summer 2014
Chapter 1 – Linear Functions and Math. Models
Linear Demand and Supple Functions
A demand equation gives the relationship between unit price and quantity demanded. It's given by
(x, p).
A supply equation gives the relationship between unit price and quantity supplied. It's given by
(x, p).
Market Equilibrium occurs when demand = supply. It is the point at which the demand and supply
equations intersect.
Equilibrium point is given by (x0, p0).
Example 4. The demand and supply equations for a certain type of food is given by 2x + 4p – 20 = 0
and 4x – 3p – 7 = 0, respectively, where x represents the quantity in 100's and p is the unit price of the
food item. What is the market equilibrium?
Example 5. For a price of $450, the quantity demanded of a certain brand of music player is 3000 per
week. For each decrease in the unit price of $20 below $450, the quantity demanded increases by 200
units. The suppliers will not market any music players if the unit price is $250 or lower. But they will
make 2500 units available at a unit price of $550. The demand and supply equations are known to be
linear.
a) Find the demand and supply equations.
b) Find the equilibrium price and quantity.
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Math 141-Summer 2014
Chapter 1 – Linear Functions and Math. Models
1.Q Quadratic Functions
a) Standard form:
Vertex :
b) Vertex form:
Vertex:
Example 6. Consider the quadratic equation 2x2 + 10x – 5 = 0.
a) Does the parabola have a maximum or a minimum?
b) What is the vertex of the parabola?
c) Find the x – coordinates of the x – intercepts.
Example 7. Consider the quadratic equation – 3(x – 4)2 + 10 = 0.
a) Does the parabola open upward or downward?
b) What is the vertex of the parabola?
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Math 141-Summer 2014
Chapter 1 – Linear Functions and Math. Models
We know that Revenue function is R(x) = sx, where s is the selling price per unit.
We have also learned the demand function, known as the price-demand function, which implies that the
price of an item varies with the number of items demanded by the consumer. Hence, R(x) = px, where
p is the price-demand function.
Combining the two concepts, we can say that the revenue functions are not linear.
If the price-demand function is linear, then the revenue function will be a quadratic function.
Example 8. If the price-demand function of a product is given by p = 25 – 4x, where x is the number of
units sold, and price is in dollars; how many items should be sold for the revenue to be maximum?
What will be the maximum revenue?
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Math 141-Summer 2014
Chapter 1 – Linear Functions and Math. Models
Example 9. A company has determined that if the price of an item is $40, then 150 units will be
demanded by the consumer. 100 unites will be demanded if the price is $50. The company has a fixed
cost of $3500, and a variable cost of $10 per item. The company's cost and price-demand functions are
linear.
a) Find the price-demand equation.
b) What is the revenue function?
c) Find the number of items sold to maximize the revenue. What is the maximum revenue?
d) What is the company's profit function?
e) How many items should be sold to get maximum profit? To break even?
f) what is the maximum profit?
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