math chapter 4 part 1

8/10/2019 MATH Chapter 4 Part 1

1/40

What is a function? (not in your book)

Linear functions

Quadratic functions


2/40

Section 4.1


3/40

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).


4/40


5/40

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


6/40

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.


7/40


8/40

(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


9/40

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):


10/40

Rank Responses

1

2

34

5

6

1 2 3 4 5 6

0% 0% 0%0%0%0%


11/40


12/40

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


13/40

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


14/40

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


15/40

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.


16/40

Rank Responses

1

2

3

4

5

6

1 2 3 4 5 6

0% 0% 0%0%0%0%


17/40

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.


18/40


19/40

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


20/40


21/40

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


22/40


23/40

A. = 4 +7 9

B. =7 +7 9

C. = 14 12

D. =

E. = 4(0.5)


24/40

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.


25/40

Theorem (math fact): The roots of aquadratic function whose variable is xcan be found using the following formula

(the quadratic formula): =

4

2


26/40

Find the roots of the equation

y = x2-6x+8


27/40

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.


28/40

1). x2 -3x+4

2). -x2 +2x+3


29/40

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

0% 0% 0%0%0%


30/40


31/40

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


32/40

Earlier, we found that the demandequation for the soda vendor was

q = -10p + 25.

Find the inverse demand equation

(i.e., solve for p).


33/40

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


34/40

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


35/40

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


36/40

Recall that Profit = Revenue Costs.The vendors profit equation istherefore:


37/40

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


38/40

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%


39/40

Rank Responses

1

2

34

5

6

1 2 3 4 5 6

0% 0% 0%0%0%0%


40/40

math chapter 4 part 1

Documents