(chapter 3) quadratic function

QQM1023 Managerial Mathematics

4.1:: INTRODUCTION TO QUADRATIC FUNTION 4.1 • •

Quadratic function was described as polynomial function of degree 2. A function f is f is a quadratic function if and only if f ( x)  x) can be written in the form of:

2

ƒ(x) = ax + bx + c where a

• • •

≠

0 and a, b and c are constant

The graph of the quactratic function is called parabola. parabola. If the value of a for a quadratic function is positive is  positive,, therefore the graph (parabola) will open upward ( concave up  U) – minimum – minimum Meanwhile, if the value for  a is negative, negative, therefore the graph (parabola) will open downward (concave ( concave down - ∩) - maximum. maximum.

y

 y

vertex (max)

 y-intercept  c  x-intercept 

c  y-intercept   x 

 x  x-intercept 

vertex(min) 2

eg: y eg: y = x  + 7x + 10 - a is positive (a = 1) - concave up - minimum vertex point 

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eg: y eg: y =-3 x  + 6x + 9 - a is negative (a = 1) - concave down - maximum maximum vertex vertex point  point 

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Example 1: Determine whether each given function below is a quadratic function or not. If it is, then state s tate the value of a, b and c and the shape of the graph (parabola) – concave up/down? a) g(x) c) y

= 5x

2

3

 f(x) = 7x-2

b)

2

= 2x + 4x – 2x + 5

d) f(v)

2

= -10v – 6 

4.1.1: Vertex point

•

• •

The lowest (minimum) or the highest point (maximum) of a quadratic function is called the “vertex “ vertex”. ”.

If the value of a is greater than 0 (positive), then the quadratic function will have a minimum vertex point . Meanwhile, if the value of a is less than 0 (negative) then the quadratic function will have a maximum vertex. vertex . 2

 y = ax +bx +c

; a>0

y = -ax2 + bx + c ;

a 0 •

•

x

•

The parabola DO NOT  pass/touch the x-axis.

• b2-4ac < 0

The value/s of the x-intercept/s can be gain in 2 ways: quadratic formula factorization

a) Quadratic Formula : 2 Given the quadratic equation: ax + bx + c = 0, The value for x can be determine using the formula;

 x =

− b ± b 2 − 4ac 2a

Attention !!! : If b2 – 4ac < 0 ; therefore they do not intercept x-axis.

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Example 4: Solve the following quadratic equation (to find the value/s of x) : a) 0 = x2 + x -12 Solution: i. Determine the value of a, b and c: a = 1, b = 1 and c = -12 ii. Replace the value into the formula:

 x =

 x =  x =  x =

− b ± b 2 − 4ac 2a

− 1 ± 12 − 4(1)(−12) 2(1)

− 1 ± 49 2 −1 + 7 2

or

x=

−1− 7 2

  x=3 x = -4 therefore the function intercept the x-axis at (3,0) and (-4,0) b) 0 = x2 + x

c) 0 = -3x2 + 2x + 8

d) 0 = 2x2 + 5x – 3

e) 0 = x2 + 4x + 6

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b) Solving a quadratic equation using factorization :

Example 5: Solve the following quadratic equation (to find the value/s of x ): a) 0 = x2 + x -12

 

2

x

x

x

 x

(x+4)

(x-3) 4

 x

 x

-3

-12 x

+

4x

x2 + x -12 = 0 (x+4)(x-3) = 0   x + 4 = 0 or thus, x = -4 or

-3x

x

therefore

x–3=0 x=3

b) 0 = x2 + x

c) 0 = -3x2 + 2x + 8

d) 0 = 2x2 + 5x – 3

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4.2 : SKETCHING THE GRAPH OF QUADRATIC FUNCTION

• The graph of a quadratic function is in the form of parabola • Steps to sketch the quadratic function graph; y = ƒ(x) = ax2 + bx + c: 1. Determine shape of the graph (concavity) : Look at the value of a: a : positive  concave up (U) a : negative  concave down ( ∩)

2. Find the vertex point (x,y) using the formula : x = -b 2a

y = 4ac – b 4a

2

3. Find the y-intercept : replace x = 0 into the function  y = (a x 0 2) + (b x 0) + c  y=c 4. Find the x-intercept : replace y=0 into the function and find the value of x using the quadratic formula or the factorization method:

 x =

− b ± b 2 − 4ac 2a

5. Draw the axis and tick all of the points (vertex, yintercept, x-intercept/s) 6. Draw a parabola that connects all of the points and label the graph.

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Example 6: Sketch the graph for each of the following quadratic functions: a) f(x) = x2 + x -12

b) f(x) = x2 + x

c) f(x) = -3x 2 + 2x + 8

d) f(x) = 2x 2 + 5x – 3

e) f(x) = x2 + 4x + 6

4.3 FORMING A QUADRATIC EQUATION • •

To form a quadratic equation, we need to know at least 3 points that reside on the function/parabola. Steps: - Substitute all three coordinates of x and y into the general form of the quadratic equation; y = ax2 + bx + c. c. - Therefore, we will have 3 equations in the mean of a, b and c. s imultaneously (using either the - Solve this three equations simultaneously substitution, elimination, or inverse matrix, Cramer’s rule method) to find the value of a, b dan c that satisfy the three equations. equation by replacing the value of a, b and c. - Finally, rewrite the equation

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Example 7: Form a quadratic equation that passes through the points (1,8), (3,20) and (-2,5) i.

Substitude all three coordinates into the gerenal form of  quadratic equation y = ax 2 + bx + c.

ii.

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

iii.

Rewrite the equation y = ax 2 + bx + c by replacing the value of a, b and c into the equation.

Example 8: Form a quadratic equation that passes through the points (0,12) , (-6,0) and (2,0).

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4.4 APPLICATIONS - DEMAND AND SUPPLY FUNCTION, EQUILIBRIUM Many situations in economics can be described by using quadratic functions.

a) Demand and Supply Function : •

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

•

For quadratic function : i.

If a is positive (a>0), : - the function has a minimum point/vertex (U) supply function  Price per unit (p) Supply Function

Quantity Supplied(q)

ii.

If a is negative (a Total Cost Lost : Total Revenue < Total Cost

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d.Break-Event Point • Break-event point is thepoint where the Total Cost and Total Revenue intersect. • Here, the Total Cost = Total Revenue, there are no profit or loss.

Total Revenue = Total Cost OR 

Profit/Loss = 0 ATTENTION!! : For this course, we only considerBEP in the first quarter of the plane.

Example 14: The total revenue for a product is given given by the function R(q) = 2.5q, and the Total Cost function is C(q) = 100 + 2q – 0.01q 2 Determine

a) Profit Function

b)

Profit gain, if 100 unit of the products were sold

c)

Break Event Point (BEP)

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