Download - MATH Chapter 4 Part 1
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What is a function? (not in your book)
Linear functions
Quadratic functions
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Section 4.1
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What Is A Function?
Definition:
A function is a relation between two sets of elements
such that to each element in the 1st set (theDomain,
typically called x, the independent variable) there
corresponds exactly one element in the second set (theRange, typically called y, the dependent variable).
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Definition: The domain of a function is theset of numbers that can replace theindependent variable, x, in a functionwithout the function doing somethingillegal, like dividing by 0.
Definition: The range of a function is theset of all possible output values of thefunction
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Definition: A linear function is a function of a
single variable with no exponent greaterthan one and whose graph is a straight line.
Linear functions are often written as
=+ ,
Definition: The slope (m) tells us how muchthe value of y changes for a one-unit
increase in xDefinition: The y intercept (b) is the point(0,b) where the line crosses the vertical
axis.
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(Mathematical) Definition: The slope ofa line is defined as the change in ydivided by the change in x, or,
=
for some collection of points (x1,y1)and (x2,y2)
To find the y-intercept, solve theequation of a line for b using either ofthe (x,y) points:
y = m*x + b b = y m*x
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1). Find the equation of a line using the points (2,4) and (3,6)Find the slope:
Find the y intercept:
Assemble the pieces:
2). Find the equation of a line using the points (1,3) and (3,2):
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Business Application:
Definition: A linear demand equation
describes how the quantity demanded of agood changes with an increase in price
Now instead of calling the points (x,y), we
use (p,q), where p = price and q = quantity Example: Find the linear demand equation
using the price information below.
Price ($p) Quantity Sold (q)
$2.00 5
$1.50 10
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A. =10 25B. =10+ 25
C. = 0.1+ 2.5
D. = 25E. = 10+ 2.5
Price ($p) Quantity Sold (q)
$2.00 5
$1.50 10
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Two car rental companies have the followingprice structure:
When is it cheaper to rent from A?
Strategy: Set the two equations equal to one
another and solve for the variable. Thesolution will be the indifference pointbecause at this point, it makes no differencecost-wise what you do
Company A Company B
$20 per day + $0.10 per
mile
$10 per day + $0.20 per
mile
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20 0.10( )10 0.20( )
A
B
Cost mileCost mile
= += +
The indifference
point is when x
(mileage) is 100. Ifyou drive more than
100 miles, you should
rent from A (blue
line) because it has
the lowest cost in
this range.
If you drive 0 to 100
miles, you should
rent from B, because
it has the lowest cost
in this range.
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When the amount of a good
that is supplied (S) matches
the amount of a good that is
demanded (D), the market is
said to be in equilibrium.
If we think of supply and
demand as linear
functions of quantity, then
the equilibrium point is the
point at which the supplyand demand lines
intersect.
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Definition: A quadratic function is a functionof a single variable with no exponentialpower greater than 2. It can be written as
y = ax2 + bx + c. The graph of such afunction is called a parabola.
If a > 0, the graph opens upward
If a < 0, the graph opens downward
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Definition: The vertex of a parabolais the point where the graph shiftsfrom decreasing to increasing
(minimum point), or from increasingto decreasing (maximum point)
The (x,y) coordinates of vertex pointare
The x coordinate of the vertex is theaxis of symmetry
,2 2
b bf
a a
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A. = 4 +7 9
B. =7 +7 9
C. = 14 12
D. =
E. = 4(0.5)
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Definition: A root or x-intercept of aquadratic equation is a value of theindependent variable that makes thequadratic equation equal 0.
Some quadratics, such
as these, have
complex roots
involving the number
i,which is the squareroot of -1.
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Theorem (math fact): The roots of aquadratic function whose variable is xcan be found using the following formula
(the quadratic formula): =
4
2
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Find the roots of the equation
y = x2-6x+8
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Here we see the two
values of x that result in
the function beingequal to 0. When we
plug in 2 or 4, we get
f(2) = f(4) = 0.
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1). x2 -3x+4
2). -x2 +2x+3
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A. {-4,1}
B. {0,3}
C. {1.5,1.75}D. {4}
E. No real roots
{-4,1}
{0,3}
{1.5,1.75} {4
}
Noreal
root
s
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The demand for a product is related to the price
charged.
Usually, we write quantity as a function of price:
= + ,
where q = quantity and p = price.
But we can write price as a function of quantity as wellby solving the above equation for p:
=1
,
Then we have what is called the inverse demand or
price curve
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Earlier, we found that the demandequation for the soda vendor was
q = -10p + 25.
Find the inverse demand equation
(i.e., solve for p).
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Definition: Revenue = price per unit x quantity.
We have solved for p (price), using the inverse demandcurve
=1
So revenue , call it r(q), will be p x q: =
1
=1
which is a quadratic equation.
Find the revenue equation for the soda vendor
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Definition: Profit = Revenue Total CostsDefinition: the break-even quantities are theamounts that you must sell to exactly cover allof your costs and give you 0 profit.
In the context of quadratic equations, thebreak-even quantities are the roots of theprofit equation:
We can then obtain a profit function of theform profit = (), where ()represents a cost function
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A simple cost equation is =+
Definition: A fixed cost (F) does not changewith quantity sold (ex: machines, salaries,
utilities)
Definition: A variable cost (V) changes withquantity sold (ex: ingredients, hourly wages)
Suppose the vendors total cost equation is() = 5 + 0.25
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Recall that Profit = Revenue Costs.The vendors profit equation istherefore:
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The zeros of the profit equation give thebreak-even point:
The x-coordinate of the vertex of thisparabola gives the quantity thatmaximizes the vendors profit
The y-coordinate of the vertex of thisparabola gives the maximum profit
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A. 2.5
B. {2.5,20}
C. {20}D. {11.25,2.5}
E. {} (no breakeven
points)2.5
{2.5,20}
{20}
{11.25
,2.5}
{}(n
obr
eake
venpoints)
0% 0% 0%0%0%
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