(chapter 3) quadratic function

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Chapter 4: Quadratic Function

Chapter 4: Quadratic Function 9090

4.1

4.1 : INTRODUCTION TO QUADRATIC FUNTION

: INTRODUCTION TO QUADRATIC FUNTION

•• _{Quadratic function was described as polynomial function}_{Quadratic function was described as polynomial function of degree 2.}_{of degree 2.} •• _{A function}_{A function f}_{f is a quadratic function if and only if}_{is a quadratic function if and only if f}_f (( x_{x) can be written in}_{) can be written in}

the form of: the form of:

where a

where a ≠≠ _{0 and a, b and c are constant}_{0 and a, b and c are constant}

•• _{The graph of the quactratic function is called}_{The graph of the quactratic function is called parabola}_parabola..

•• _{If the value of}_{If the value of}

_a

_{for a quadratic function is}_{for a quadratic function is positive}_{positive, therefore the graph}_{, therefore the graph}

(parabola) will open upward (

(parabola) will open upward ( concave upconcave up  U) – U) – minimumminimum

•• _{Meanwhile, if the value for}_{Meanwhile, if the value for}

_a

_is_{is negative}_{negative, therefore the graph}_{, therefore the graph}

(parabola) will open downward (

(parabola) will open downward (concave downconcave down -- ∩∩_{) -}_{) - maximum}_maximum..

ƒ(x) =

a

x

2

+

b

x +

cc

y y x-intercept x-intercept cc y-intercept y-intercept x x vertex(min) vertex(min)

eg:

eg: y = x

y = x

22

+ 7x + 10

-- a

a

is positive

is positive ((a

a = 1)

= 1)

--

concave up

- minimum vertex point

y

y vertex (max)vertex (max) y-intercept y-intercept cc x x x-intercept x-intercept

eg:

eg: y =-3 x

y =-3 x

22

+ 6x + 9

-- a

a

is negative ((a

is negative

a = 1)

= 1)

--

concave down

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•• _{The lowest (minimum) or the highest point (maximum) of a}_{The lowest (minimum) or the highest point (maximum) of a}

quadratic function is called the “

quadratic function is called the “vertexvertex”.”. Determine whether each given function below is

Determine whether each given function below is a quadratic functiona quadratic function or not. If it is, then s

or not. If it is, then state the value oftate the value of

a, b

andand

cc

and the shape of theand the shape of the graph (parabola) – concave up/down?

graph (parabola) – concave up/down? a)

a)

g(x) = 5x

22 b)b)

f(x) = 7x-2

c)

y = 2x

33

+ 4x

22

– 2x + 5

d)d)

f(v) = -10v

22

– 6

4.1.1:

Vertex point

•• _{If the value of}_{If the value of}

_a

_is_{is greater}_{greater than}_{than 0}_{0 (positive), then the quadratic function}_{(positive), then the quadratic function}

will have a

will have a minimum vertex point minimum vertex point ..

•• _{Meanwhile, if the value of}_{Meanwhile, if the value of a is less than 0}_{a is less than 0 (negative) then the quadratic}_{(negative) then the quadratic}

function will have a

function will have a maximum vertexmaximum vertex..

y = ax

22

+bx

+bx +c

+c

; a

; a >

> 0

0 y =

y

= -ax2

-ax2 +

+ bx

bx +

+ c

c ;

;

a

a <0

<0

Maximum vertex Maximum vertex

Minimum vertex Minimum vertex

Formula used to obtain the coordinate for

Formula used to obtain the coordinate for the vertex point

the vertex point

(x,y)

(x,y) of

of the

the function

function y

y =

= ƒ(x)

ƒ(x) =

= ax

ax

22

_{+ bx + c}

x-coordinate

=

= - b

- b

2a

y-coordinate

=

= 4ac

4ac –

– b

b

22

4a

or simply finds, or simply finds,           −− a a b b f f 2 2

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Find the vertex point for the given functions, and then determine Find the vertex point for the given functions, and then determine whether it is a maximum or minimum point:

whether it is a maximum or minimum point:

a) a) f(x) = xf(x) = x22 ₊_{+ x}_{x -12}_-12 _b)_{b) f(x)}_{f(x) =}_{= xx}22 _{+ x}_{+ x} c) c) f(x) = -3xf(x) = -3x22 ₊_{+ 2x}_{2x +}_{+ 8}₈ _d)_{d) f(x)}_{f(x) =}_{= 2x}_2x22 _{+ 5x – 3}_{+ 5x – 3} e) e) f(x) = xf(x) = x22 _{+ 4x + 6}_{+ 4x + 6}

4.1.2: y-intercept (0, c)

•• _{The y-intercept (0,c) is the point where the parabola pass through}_{The y-intercept (0,c) is the point where the parabola pass through}

the y-axis (or when x = 0 the y-axis (or when x = 0 ).).

•• _{To find the value of the y-intercept, simply replace x = 0 into}_{To find the value of the y-intercept, simply replace x = 0 into the}_the

function. function. y/f(x) y/f(x) x x y-intercept y-intercept y-intercept y-intercept x x

Given the function f(x) = ax

Given the function f(x) = ax22_{+bx+c, then the y-intercept is c.}_{+bx+c, then the y-intercept is c.}

Example 2:

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x

==

a

ac

b

2

4

2 2

−

±

−

Find the y-intercept for the

Find the y-intercept for the following quadratic functions:following quadratic functions: a) a) f(x) = xf(x) = x22 ₊_{+ x}_{x -12}_-12 _b)_{b) f(x)}_{f(x) =}_{= xx}22 _{+ x}_{+ x} c) c) f(x) f(x) = = -3x-3x22 ₊_{+ 2x}_{2x +}_{+ 8}₈ _d)_{d) f(x)}_{f(x) =}_{= 2x}_2x22 _{+ 5x – 3}_{+ 5x – 3} e) e) f(x) f(x) = = xx22 _{+ 4x + 6}_{+ 4x + 6}

4.1.3 : x-intercept

•• _{x-intercept (root/s) is the}_{x-intercept (root/s) is the point where the pa}_{point where the parabola}_{rabola pass through x-axis}_{pass through x-axis}

(or when y=0). (or when y=0).

•• _{when y}_{when y = 0,}_{= 0, : ax}_{: ax}22 _{+ bx + c = 0}_{+ bx + c = 0}

•• _{There are 3 conditions for the root of a}_{There are 3 conditions for the root of a given quadratic function:}_{given quadratic function:}

•• _{The value/s of the x-intercept/s can be gain in 2 ways:}_{The value/s of the x-intercept/s can be gain in 2 ways:} -- quadratic formulaquadratic formula

-- factorizationfactorization a)

a) Quadratic Formula :Quadratic Formula :

Given the quadratic equation:

ax

22

+ bx + c = 0,

The value for x can be determine using the formula; The value for x can be determine using the formula;

Attention

Attention !!! : If!!! : If bb22 _{– 4ac < 0}_{– 4ac < 0 ; therefore they do not intercept x-axis.}_{; therefore they do not intercept x-axis.}

Example 3:

y y x x

•• _{The parabola pass through}_{The parabola pass through}

TWO

TWOpoints at the x-axis.points at the x-axis.

••

_b

22

-4ac > 0

y y x x •• _{The parabola touch the x-}_{The parabola touch the}

x-axis at

axis atONEONEpoint.point.

••

_b

22

-4ac = 0

y y x x •• _{The parabola}_{The parabola} _{DO NOT}_{DO NOT}

pass/touch the x-axis. pass/touch the x-axis.

••

_b

22

-4ac < 0

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Solve the following quadratic equation (to find the value/s of x) : Solve the following quadratic equation (to find the value/s of x) : a)

a) 0 = x0 = x22 _{+ x -12}_{+ x -12}

Solution: Solution:

i. Determine the value of a, b and c: i. Determine the value of a, b and c:

a

a = = 1, 1, b b = = 1 1 and and c c = = -12-12

ii. Replace the value into the formula: ii. Replace the value into the formula:

x = x = )) 1 1 (( 2 2 )) 12 12 )( )( 1 1 (( 4 4 1 1 1 1±± 22 −− −− − − x = x = 2 2 49 49 1 1±± − − x = x = 2 2 7 7 1 1++ − − or or x x == 2 2 7 7 1 1−− − − x x = = 3 3 x x = = -4-4

therefore the function intercept the x-axis at (3,0) and (-4,0) therefore the function intercept the x-axis at (3,0) and (-4,0)

b) b) 0 = x0 = x22 _{+ x}_{+ x} c) c) 0 = -3x0 = -3x22 _{+ 2x + 8}_{+ 2x + 8} d) d) 0 = 2x0 = 2x22 _{+ 5x – 3}_{+ 5x – 3} e) e) 0 = x0 = x22 _{+ 4x + 6}_{+ 4x + 6}

Example 4:

x

==

a

ac

b

2

4

2 2

−

±

−

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Chapter

Chapter 4: 4: Quadratic Quadratic Function Function 9595

b) Solving a quadratic equation using factorization : b) Solving a quadratic equation using factorization :

Solve the following quadratic equation (to find the value/s of x ): Solve the following quadratic equation (to find the value/s of x ): a) a) 0 = x0 = x22 _{+ x -12}_{+ x -12}

x

22

x

xx

(x+4)

(x-3)

(x+4)

(x-3)

4

4 -3

-3

-12

4x

-3x

xx

therefore x therefore x22 ₊_{+ x}_{x -12}_{-12 =}_{= 0}₀ (x+4)(x-3) (x+4)(x-3) = = 00 x x + + 4 4 = = 0 0 or or x x – – 3 3 = = 00 thus, thus, x x = = -4 -4 or or x x = = 33 b) b) 0 = x0 = x22 _{+ x}_{+ x} c) c) 0 = -3x0 = -3x22 _{+ 2x + 8}_{+ 2x + 8} d) d) 0 = 2x0 = 2x22 _{+ 5x – 3}_{+ 5x – 3}

+

x x x x x x xx

Example 5:

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Chapter

4.2 : SKETCHING THE GRAPH OF QUADRATIC FUNCTION

••

The graph of a quadratic function is in the form of

The graph of a quadratic function is in the form of parabola

parabola

••

_{Steps to sketch the quadratic function graph;}

y = ƒ(x) = ax

22

_{+ bx + c:}

1. 1. Determine

Determine shape of the graph

shape of the graph (concavity) :

(concavity) :

Look at the value of a:

a : positive



concave up (U)

a : negative



concave down (

∩∩

₎₎

2.

2. Find the

Find the vertex point

vertex point (x,y) using the formula

(x,y) using the formula

::

3. 3. Find the

Find the y-intercept

y-intercept : replace x = 0 into the function

: replace x = 0 into the function



y = (a x 0

22

_{) + (b x 0) + c}



y = c

4. 4. Find the

Find the x-intercept

x-intercept : replace y=0 into the function and

: replace y=0 into the function and

find the value of x using the

find the value of x using the quadratic formula

quadratic formula or the

or the

factorization

factorization method:

method:

5. 5. Draw the axis and tick all of the points (vertex, y-

Draw the axis and tick all of the points (vertex,

y-intercept, x-intercept/s)

intercept, x-intercept/s)

6. 6. Draw a

Draw a parabola

parabola that connects all of the points and

that connects all of the points and

label

label the graph.

the graph.

x

x =

= -

- b

b

y

y =

= 4ac

4ac –

– b

b

22

2a

4a

2a

4a

x

==

a

ac

b

2

4

2 2

−

±

−

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Chapter

Sketch the graph for each

Sketch the graph for each of the following quadratic functions:

of the following quadratic functions:

a)

a) f(x) = x

f(x) = x

22

₊

_{+ x}

_{x -12}

_-12

_b)

_{b) f(x)}

_{f(x) =}

_{= xx}

22

_{+ x}

c) f(x) = -3x

22

₊

_{+ 2x}

_{2x +}

_{+ 8}

₈

_d)

_{d) f(x)}

_{f(x) =}

_{= 2x}

_2x

22

_{+ 5x – 3}

e)

e) f(x) = x

f(x) = x

22

_{+ 4x + 6}

4.3

4.3 FORMING

FORMING

A

QUADRATIC

EQUATION

•• _{To form a quadratic equation, we need to know at least 3 points}_{To form a quadratic equation, we need to know at least 3 points that}_that

reside on the function/parabola. reside on the function/parabola.

•• _Steps:_Steps:

-- Substitute all three coordinates of x and y into the general form ofSubstitute all three coordinates of x and y into the general form of the quadratic equation;

the quadratic equation; y = axy = ax22 _{+ bx + c}_{+ bx + c..}

-- Therefore, we will have 3 equations in the mean of a, b and c.Therefore, we will have 3 equations in the mean of a, b and c.

-- Solve this three equations sSolve this three equations simultaneously (using either theimultaneously (using either the substitution, elimination,

substitution, elimination, or inverse matrix, or inverse matrix, Cramer’s rule method) Cramer’s rule method) toto find the value of a, b dan c that

find the value of a, b dan c that satisfy the three equations.satisfy the three equations.

-- Finally, Finally, rewrite the eqrewrite the equation by replacing uation by replacing the value of a, the value of a, b and c.b and c.

Example 6:

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Chapter

Form a quadratic equation that passes through the points (1,8), (3,20) Form a quadratic equation that passes through the points (1,8), (3,20) and (-2,5)

and (-2,5)

i.i. Substitude Substitude all three all three coordinates into the coordinates into the gerenal form gerenal form of of quadratic equation

quadratic equation y = axy = ax22 _{+ bx + c.}_{+ bx + c.}

ii.

ii. Solve all three equations simultaneously to find the value of a, bSolve all three equations simultaneously to find the value of a, b and c:

and c:

iii.

iii. Rewrite the equation y = axRewrite the equation y = ax 2 2 _{+ bx + c by replacing the value of a,}_{+ bx + c by replacing the value of a,}

b and c into the equation. b and c into the equation.

Form a quadratic equation that passes through the points (0,12) , (-6,0) Form a quadratic equation that passes through the points (0,12) , (-6,0) and (2,0). and (2,0).

Example 7:

Example 8:

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Chapter

4.4

4.4 APPLICATIONS

APPLICATIONS

-

DEMAND

AND

SUPPLY

FUNCTION,

EQUILIBRIUM

Many situations in economics can be described by using quadratic Many situations in economics can be described by using quadratic functions.

functions.

a)

Demand and Supply Function :Demand and Supply Function :

•• _{The function that relates price per}_{The function that relates price per unit and demanded quantity is}_{unit and demanded quantity is}

called a demand function. Meanwhile the function

called a demand function. Meanwhile the function that relatesthat relates price per unit and supplied quantity is

price per unit and supplied quantity is called supply function.called supply function.

•• _{For quadratic function :}_{For quadratic function :}

i.i. If If a a is is positivepositive (a>0),(a>0), ::

-- the function has athe function has a minimum point/vertex (U)minimum point/vertex (U)

-- supply functionsupply function

ii.

ii. If a is negative (a<0),If a is negative (a<0), ::

-- the function has athe function has a maximum point/vertex (maximum point/vertex (∩∩),),

-- demand function.demand function.

Price per unit (p) Price per unit (p)

Supply Function Supply Function

Quantity Supplied(q) Quantity Supplied(q)

Price per unit (p) Price per unit (p)

Demand Function Demand Function

Quantity Demanded(q) Quantity Demanded(q)

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Chapter

ATTENTION : In this course, we only consider the price per unit and the ATTENTION : In this course, we only consider the price per unit and the quantity in the first quarter of the plane!!

quantity in the first quarter of the plane!!

The Supply function y = f(q) for a product is in

The Supply function y = f(q) for a product is in a forma form quadratic functionquadratic function.. Three points that reside on the functions are (1,11), (0,6) and (2,18).

Three points that reside on the functions are (1,11), (0,6) and (2,18). Form the supply function of the product.

Form the supply function of the product.

A market research done by manufacturers of a product comes out with A market research done by manufacturers of a product comes out with a supply function in a form of

a supply function in a form of quadraticquadratic. The manufacturers were asked. The manufacturers were asked on the amount (quantity) of products they will produced at a certain on the amount (quantity) of products they will produced at a certain price per unit. The result of the research found that, at the price of

price per unit. The result of the research found that, at the price of RM6,RM6, RM30 and RM48, the manufacturers will produce 4, 8 and 10 units

RM30 and RM48, the manufacturers will produce 4, 8 and 10 units of theof the product.

product.

a)

a) Form a quadratic supply function based on the informations given.Form a quadratic supply function based on the informations given.

Example 9:

Example 10:

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Chapter

Chapter 4: 4: Quadratic Quadratic Function Function 101101 •• _{The equilibrium point is the point where the supply meets}_{The equilibrium point is the point where the supply meets}

demand. demand. b)

b) If 20 units of the product were produced, how much is If 20 units of the product were produced, how much is the price per the price per unit?

unit?

c)

c) If the price per unit of the product is If the price per unit of the product is set to RM70, how many of theset to RM70, how many of the product should be produce?

product should be produce?

b) Equilibrium point: b) Equilibrium point:

•• _{Here, the quantity demanded = quantity supplied at the}_{Here, the quantity demanded = quantity supplied at the same price}_{same price}

per unit. per unit.

•• _{The equilibrium point can be obtain by solving both function}_{The equilibrium point can be obtain by solving both function}

simultaneously: simultaneously:

-- Substitution MethodSubstitution Method

-- Elimination MethodElimination Method

-- Matrix (Inverse or Cramer’s Rule)Matrix (Inverse or Cramer’s Rule)

Attention !! For this course, the equilibrium point is only considered in the Attention !! For this course, the equilibrium point is only considered in the first quarter of the plane!

first quarter of the plane!

Price per unit

Price per unit (p) (p) SupplySupply

Equilibrium Equilibrium

b

b Point (a,b)Point (a,b)

Demand Demand

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Chapter

Total Revenue (TR) =

Total Revenue (TR) = Price

Price per

per unit

unit x

x Quantity

Quantity of

of product

product Sold

Sold

= p x q

Given two quadratic functions :

Given two quadratic functions : p = f(q) = -q

p = f(q) = -q22 _{-4q +12}_{-4q +12}

p= g(q) = q + 6 p= g(q) = q + 6

a) Determine which one is the demand/supply function. a) Determine which one is the demand/supply function.

b) Find the equilibrium point of the two functions. b) Find the equilibrium point of the two functions.

4.5 TOTAL REVENUE, TOTAL COST, PROFIT AND

BREAK-EVENT POINT (B.E.P)

a) Total Revenue : a) Total Revenue :

•• _{Total revenue is define by the}_{Total revenue is define by the product between price per unit and}_{product between price per unit and}

quantity of the product sold. quantity of the product sold.

•• _{Let say the}_{Let say the price per unit (p)}_{price per unit (p) is}_{is determine by the demand}_{determine by the demand (in linear}_{(in linear}

form). form).

p = -mq + c ---demand function p = -mq + c ---demand function

where

where pp is theis the price per unit price per unit (RM)when(RM)when qq unitsunits of the product wereof the product were demanded

demanded

Example 11:

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Chapter

Chapter 4: 4: Quadratic Quadratic Function Function 103103 •• _{Here, the Total Revenue is}_{Here, the Total Revenue is in quadratic form because:}_{in quadratic form because:}

0

Now, the total revenue is in a form of quadratic!!

••

_{Thus, the}_{Thus, the Total Revenue}_{Total Revenue ::}

-- The Total Revenue function always starts at theThe Total Revenue function always starts at the origin (0,0).origin (0,0).

-- As theAs the a is negativea is negative, therefore the Total Revenue function will have a, therefore the Total Revenue function will have a maximum

maximum point (concave down).point (concave down).

-- The maximum point (The maximum point (vertexvertex) here represent the) here represent the quantity thatquantity that maximize the total revenue

maximize the total revenue (on x-axis) and the(on x-axis) and the maximum revenuemaximum revenue (on y-axis) for the product.

(on y-axis) for the product.

Total

Total Revenue

Revenue =

= p x

p x q

q

Therefore,

Total

Total Revenue

Revenue =

= (-mq

(-mq +

+ c)

c) q

q =

= -mq

-mq

22

+ cq

Where; Where; p = -mq+c (demand) p = -mq+c (demand)

y = ax

22

+ bx + c

Total Revenue(RM) Total Revenue(RM)

Max. Total Revenue= Max. Total Revenue=

a a b b ac ac 4 4 4 4 −− 22 Quantity (units) Quantity (units) Quantity that will maximizeTR=

Quantity that will maximizeTR= a a b b 2 2 − −

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Chapter

Total Cost = Fixed Cost + Variable Cost

The demand function

The demand function for a product is given for a product is given by p = 1200 – 3q by p = 1200 – 3q where where p isp is the price per unit(RM) and q is

the price per unit(RM) and q is the quantity demanded.the quantity demanded.

a)

a) Determine the Total Revenue function for Determine the Total Revenue function for the product.the product.

b)

b) How many product should be produce in order to maximize theHow many product should be produce in order to maximize the total revenue?

total revenue?

c)

c) How much is the maximum total revenue?How much is the maximum total revenue?

b)

b) Total CostTotal Cost

•• _{Total Cost}_{Total Cost – is the sum}_{– is the sum of fixed cost and variable cost.}_{of fixed cost and variable cost.}

REMEBER!! What is Fixed Cost and what is Varible cost from the

REMEBER!! What is Fixed Cost and what is Varible cost from the previousprevious chapter??

chapter??

-- However, in this chapter, the total cost function is in the However, in this chapter, the total cost function is in the formform of of quadraticquadratic -- C(q) = aqC(q) = aq22_{+ bq + c}_{+ bq + c where;}_where; Fixed Cost Fixed Cost Variable cost Variable cost

Example 12:

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Chapter

Profit / Loss = Total Revenue – Total Cost

The Total Cost to produce 10 unit of pencils is

The Total Cost to produce 10 unit of pencils is RM380. Meanwhile the TotalRM380. Meanwhile the Total Cost to produce 20 unit of the

Cost to produce 20 unit of the pencils is RM1060. However, if no pencils arepencils is RM1060. However, if no pencils are produce, the b

produce, the business still needs to pay usiness still needs to pay RM100 for the Total RM100 for the Total Cost Cost ::

Find: Find:

a)Total Cost Function. a)Total Cost Function.

b)How much is the Fixed Cost? b)How much is the Fixed Cost?

c) The amount of product (quantity) to be produce when the total cost c) The amount of product (quantity) to be produce when the total cost isis

RM 740. RM 740.

c.

c. Profit / LossProfit / Loss

•• _{Obtain by substracting the Total Cost from the Total Revenue :}_{Obtain by substracting the Total Cost from the Total Revenue :}

-- Profit : Profit : Total Revenue Total Revenue > Total > Total CostCost

-- Lost Lost : : Total Total Revenue Revenue < < Total Total CostCost

Example 13:

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Chapter

Total Revenue = Total Cost

OR

Profit/Loss = 0

d.Break-Event Point d.Break-Event Point

•• _{Break-event point is thepoint where the Total Cost and Total Revenue}_{Break-event point is thepoint where the Total Cost and Total Revenue}

intersect. intersect.

•• _{Here, the Total Cost = Total Revenue, there are no profit or loss.}_{Here, the Total Cost = Total Revenue, there are no profit or loss.}

ATTENTION

!! : For this course, we only considerBEP in the !! : For this course, we only considerBEP in the firstfirst quarter of the plane.

quarter of the plane.

The total revenue

The total revenue for a product is gifor a product is given by the function ven by the function R(q) = 2.5q, R(q) = 2.5q, andand the Total Cost function is C(q) = 100

the Total Cost function is C(q) = 100 + 2q – 0.01q+ 2q – 0.01q22

Determine

Determine a) a) Profit Profit FunctionFunction

b)

b) Profit Profit gain, if 100 gain, if 100 unit of the unit of the products were products were soldsold

c)

c) Break Event Point (BEP)Break Event Point (BEP)