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Math 141, Spring 2014, Benjamin
Aurispa
Review of Lines
If (x1 , y1 ) and (x2 , y2 ) are two points on a line L, we define the slope of the line, m by:
Example: Find the slope of the line passing through the points (1, 3) and (−2, 5).
If x decreases by 9, what is the corresponding change in y?
Point-Slope Form: The equation of the line with slope m that passes through the point (x1 , y1 ) is:
Slope-Intercept Form: The equation of the line with with slope m and y-intercept b is:
Example: Find the equation of the line that passes through the points (−4, 3) and (−2, −1).
The x-intercept of a line is the value of x where the line crosses the x-axis.
and solve for x.
To find the x-intercept, plug in
The y-intercept of a line is the value of y where the line crosses the y-axis.
To find the y-intercept, plug in
and solve for y.
If given an x-int. of a and a y-int. of b, use the points (a, 0) and (0, b) to find the equation of the line.
Find the x and y-intercepts of the line 3x − 4y = 16.
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Math 141, Spring 2014, Benjamin
Aurispa
Horizontal and Vertical Lines
The slope of a horizontal line is
.
.
The equation of the horizontal line through (a, b) is:
The slope of a vertical line is
.
The equation of the vertical line throught (a, b) is:
.
1.3 Linear Functions and Mathematical Models
A function y = f (x) is a rule that assigns to each value of x one value of y. We input a value for x, and the
function spits out a value for y. We say that y is a function of x. (x is the independent variable. y is the dependent variable.) Sometimes other letters are used instead of x and y, but the idea of a function is the same.
Depreciation:
Depreciation is when the value of an item decreases over time. For depreciation, we write the value, V , as
a function of time, t.
Example: Suppose I buy a car with an original value of $30,000 and it is to be depreciated linearly over its
useful life of 7 years with a scrap value of $2,000. (The scrap value of an item is its value at the end of its
useful life.) Find the depreciation equation, where value, V , is written as a function of time, t.
What will the value of my car be at the end of 5 years?
The rate of depreciation is the absolute value of the slope of the depreciation equation: |m|.
What is the rate of depreciation for my car?
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Math 141, Spring 2014, Benjamin
Aurispa
Cost, Revenue, and Profit Functions:
The cost function, C(x) is the total cost of manufacturing x units of a product.
The cost function is made up of both fixed costs and variable costs. Fixed costs do not depend on how many
units are manufactured, variable costs do.
If we let F stand for the fixed costs, and c stand for the production cost per item, then C(x) =
The revenue function, R(x) is the total revenue gained from the sale of x units of the product. If we let s
.
stand for the selling price of the item, then R(x) =
The profit function is the net amount of money gained: P (x) =
Example: Suppose Aggie Pizza Co. has total costs of $2040 when 100 pizzas are made and total costs of
$2290 when 150 pizzas are made. Pizzas are sold for $15 each. Find the cost, revenue, and profit functions
for Aggie Pizza Co.
What are the fixed costs?
What is the production cost per item?
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Math 141, Spring 2014, Benjamin
Aurispa
Supply and Demand
A demand equation gives the relationship between the price of the product and the quantity demanded
by consumers. Usually, if the price of an item increases, the demand for the item goes down and vice versa.
We write the unit price p as a function of x, (p = f (x)), where x is the quantity demanded.
A supply equation gives the relationship between the price of the product and the quantity supplied or
produced by the company. Usually, if the price of a product increases, the company’s supply will increase
and vice versa. Here, we also write p as a function of x, (p = f (x)), where x is the quantity supplied.
The slope of a demand curve is:
The slope of a supply curve is:
Example: MacSoft sells CDs. Suppose MacSoft provides us with the following data. At a unit price of $10,
3000 CDs are demanded. When the unit price decreases by $3, the quantity demanded increases by 1000
units. The supplier will supply 1000 CDs at a unit price of $5, but will not market any CDs at a unit price
of $4 or lower. Find the demand and supply equations for MacSoft.
At what price will no consumers buy a CD?
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Math 141, Spring 2014, Benjamin
Aurispa
1.4 Finding the Point of Intersection
Break-Even Point for Cost/Revenue/Profit Models
The break-even point is where you will start to make money. It is where your revenues “catch up” with
your costs.
The break-even point consists of the break-even quantity and break-even revenue.
The break-even point occurs where:
(or
).
Example: Ben’s Office Supply store sells only pens. The store incurs production costs of $3 per pen. The
production of 50 pens in a month incurs a total cost of $650. When 300 pens are sold, the company earns
revenue of $1500. What is Ben’s break-even point?
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Math 141, Spring 2014, Benjamin
Aurispa
Finding Intersection Points using a Calculator
1. Graph both lines. Adjust the window so you can see the point of intersection on the screen.
2. Press 2nd TRACE (or CALC ) and choose 5:intersect.
3. Select the lines you want to intersect by moving the cursor up and down and pressing ENTER.
4. When it asks you to Guess, just press ENTER again. (It does this because other graphs may have
more than one intersection point and you would have to move the cursor closer to the one you want.)
Equilibrium Point for Supply/Demand Models
Market equilibrium (the equilibrium point) occurs when the quantity produced (supplied) equals the quantity
demanded. The economy tends toward market equilibrium.
The equilibrium point is made up of the equilibrium quantity and the equilibrium price.
Market equilibrium occurs where
.
In the MacSoft example from 1.3, we found the following equations for demand and supply.
3
Demand: p = − 1000
x + 19
1
Supply: p = 1000 x + 4
What is the equilibrium point for this computer monitor market?
Example (based on 1.4 #26, Tan):
Suppose a company that sells calculators has a demand equation of x + 4p − 800 = 0 and a supply equation
of x − 20p + 1000 = 0. What are the equilibrium quantity and price?
How many calculators would consumers take if they were free?
At what price will suppliers not market calculators?
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Math 141, Spring 2014, Benjamin
Aurispa
1.Q Quadratic Functions
A quadratic function is a function of the form y = ax2 + bx + c where a 6= 0 . This is called the standard
form of a quadratic. The graph of EVERY quadratic function is a parabola. Whether the parabola opens
upward or downward depends on the sign of a. The vertex of a parabola is the point on the graph that
has either the lowest y-value (when opening up) or the highest y-value (when opening down).
If a quadratic function is written in standard form, the x-coordinate of the vertex can be found using
b
x=−
. Then, to find the y-coordinate of the vertex, just plug in this x-value into the function.
2a
Quadratic functions can also be written in vertex form: y = a(x − h)2 + k where a 6= 0 . You still use a
to determine if the parabola opens up or down, but the vertex is very easy to idenify. In this form, the
vertex is just the point (h, k) .
The x-intercepts of a quadratic function are found by setting the quadratic equal to 0 and solving for x. If
you remember how to factor, you can do so, but the quadratic formula always works when the quadratic
function is in standard form.
The solution(s) to the equation ax2 + bx + c = 0 are given by
√
−b ± b2 − 4ac
x=
provided
2a
b2 − 4ac ≥ 0
Example: Consider the quadratic function y = 3x2 − 18x + 35.
• Find the vertex and determine if the parabola opens upward or downward.
• What is the minimum value? maximum value?
• What are the x-coordinates of the x-intercepts?
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Example: Consider the quadratic function y = −4x2 − 12x − 8.
• Find the vertex and determine if the parabola opens upward or downward.
• What is the minimum value? maximum value?
• What are the x-coordinates of the x-intercepts?
Examples: Find the vertex and state the maximum/minimum value for the following quadratic functions.
• y = 4(x + 5)2 − 3
• y = −3(x − 4)2 + 9
Applications
In the previous section, we saw that when the selling price s of an item is fixed, then the revenue function
is R(x) = sx. We also learned, however, that sometimes the price p of an object depends on the quantity
demanded by consumers x. So, in general, if you are given a price-demand equation p, then the revenue
function will be R(x) = px and will actually be a quadratic function.
Example: Suppose the price-demand equation for a product is given by p = 84 − 7x.
• What is the revenue function?
• How many items should be sold to maximize revenue? What is the maximum revenue?
• What is the price per unit when the revenue is maximized?
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Example: A company sells gadgets. They can sell 10 gadgets when the price is $170 and they can sell 20
gadgets when the price is $120. The company incurs production costs of $70 per gadget and the company
has fixed costs of $805.
• Find the price-demand equation, assuming it is linear.
• How many items should be sold to maximize revenue? What is the maximum revenue?
• How many items should be sold to maximize profit? What is the maximum profit?
• Determine the price of a gadget when the profit is maximized.
• How many gadgets does the company need to sell to break even?
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