The Quadratic Function Derived From Zeros of the

FunctionPresented by:

Christian Jett Morales

Dedication

I proudly dedicate this presentation first to God, for having a brilliant idea

for creating this slide show. Also I give thanks to my teacher for having

a opportunity to make this presentation. For Dianne Mejia, too, for inspiration and most of all… to

SNSD for inspiration also….

To God Be All the Glory

Lesson About the Topic

The Quadratic Function Derived From Zeros of the Function

• The zeros of the function f(x)= ax2+bx+c correspond to the roots of an equation ax2+bx+c=0.

• In solving for the zeros of f(x)= x2 -7x+10, the function is equated to 0. That is,

x2-7x+10=0(x-5)(x-2)=0

X-5=0 X-2=0X=5 X=2

The zeros of f(x)=x2-7x+10 are 2 and 5

The Quadratic Function Derived From Zeros of the Function

• The process in last slide can be reversed to find a quadratic function if the zeros are given. For instance, the quadratic function f(x) whose zeros are 5 and 6 are:

F(x)=(x-5)(x-6)F(x)=x2-11x+30

• In general, if the zeros are r1 and r2, you can express the quadratic function as:

F(x)= (x-r1)(x-r2).

The Quadratic Function Derived From Zeros of the Function

• Multiplying the two binomials results in:f(x)= x2-(r1)(x)-(r2)(x)-(r1)(r2) or

f(x)= x2-(r1-r2)(x)-(r1)(r2)• Now consider the quadratic function f(x)=

ax2+bx+c. Solving for the zeros of f(x) means equating to 0. That is, ax2+bx+c=0.

Or x2 + b/a(x)+c/a=0• Comparing the functions f(x)= x2-(r1-r2)(x)-(r1)

(r2) and f(x)= x2 + b/a(x)+c/a suggests that-(r1+r2)=b/a and (r1)(r2)=c/a

The Quadratic Function Derived From Zeros of the Function

• The comparison illustrates the relationships between zeros and the coefficients of a quadratic function.

Illustrative Examples

WELCOME TO…

The Quadratic Function Derived From Zeros of the Function

• EXAMPLE ONESunny challenged Taeyeon to solve for the

quadratic function derived from zeros of the function without a calculator.

Sunny gave her two numbers, 3/2 and -1 as zeros.

What will Taeyeon answer?

Sunny and Taeyeon

The Quadratic Function Derived From Zeros of the Function

• SOLUTION• Since it is only integers involved, then this

may come easy for you…!b/a= -(r1+r2)= -(3/2 – 1)

= -1/2

The Quadratic Function Derived From Zeros of the Function

c/a=(r1)(r2)=(3/2)(-1)

= -3/2

Thus, the quadratic equation is x2+1/2x-3/2

The Quadratic Function Derived From Zeros of the Function

• But if integer coefficients are required, multiply both sides by 2 (the least common denominator of the coefficients), resulting in:

F(x)=2x2-x-3

The Quadratic Function Derived From Zeros of the Function

• EXAMPLE TWO• Yoona wonders what quadratic function applies when the zeros

of the function g(x) are 2+√3 and 2-√3.

Yoona

The Quadratic Function Derived From Zeros of the Function

• SOLUTIONb/a = -(r1+r2)

= -{(2+√3)+(2 - √3)}= -4

The Quadratic Function Derived From Zeros of the Function

c/a=(r1)(r2)=(2+√3)(2-√3)

= 4+2 √3 - 2 √3 -3= 1

Thus, the quadratic equation is x2 -4x+1=0And the quadratic function is f(x)=x2 - 4x+1

The next examples are much difficult…

Behold and prepare!!!

The Quadratic Function Derived From Zeros of the Function

• EXAMPLE THREESooyoung plot three points on her Cartesian

coordinate plane. The points are, (1,2), (-2,23), and (3,8).

Find the quadratic function f(x)

Sooyoung

The Quadratic Function Derived From Zeros of the Function

• SOLUTIONIn the quadratic function f(x)=ax

2+bx+c, substitute the first ordered pair (1,2) for (x, f(x))

2=a(1)2+b(1)+c2=a+b+c

The Quadratic Function Derived From Zeros of the Function

• Then substitute the other two ordered pairs (-2,23) and (3,8) for (x,

f(x)).23=a(-2)2+b(-2)+c

23=4a-2b+c8=a(3)2+b(3)+c

8=9a+3b+c

The Quadratic Function Derived From Zeros of the Function

• Transform equations 1, 2, and 3 into a system of linear equations which can be solved by linear combination.

a+b+c=24a-2b+c=23

9a+3b+c

3a-3b=215a+5b=-15

a-b=7a+b=-3

2a=4 a=2

The Quadratic Function Derived From Zeros of the Function

• Then substitute 2 for a in equation 4

a-b=72-b=7-b=5b=-5

The Quadratic Function Derived From Zeros of the Function

• Then substitute 2 for a and -5 for b in the first equation.

2+(-5) +c= 2-3+c=2c+=5

The Quadratic Function Derived From Zeros of the Function

•Therefore, the desired function is:

f(x)=2x2-5x+5

The Quadratic Function Derived From Zeros of the Function

• EXAMPLE FOURJessica wants to solve this quirky

question in this book. It says that find the quadratic function f(x) whose graph has its vertex at (-2,3) and

contains the point (4, 12).

Jessica

The Quadratic Function Derived From Zeros of the Function

• SOLUTION• Since the vertex of the parabola is

given, it would be easier to use the form f(x)=a(x-h)2+k

Substitute (-2,3) for the vertex (h, k)f(x)=a{x-(-2)}2+3

The Quadratic Function Derived From Zeros of the Function

• You only need ONE ordered pair to evaluate the constant a

Substitute (4,12) for(f, f(x)).12=a(4+2)2+3

9=36a1/4=a

The Quadratic Function Derived From Zeros of the Function

• Therefore,f(x)=1/4(x+2)2+3f(x)=1/4x2+x+4

The Quadratic Function Derived From Zeros of the Function

• EXAMPLE FIVEHyoyeon and Seohyun challenged Tiffany and Yuri to answer their question. Seohyun

gave her 7 as the sum of r1 and r2, and Hyoyeon gave her -8 as the product of r1

and r2.

What is Tiffany and Yuri’s answer?

Hyoyeon and Seohyun Vs Tiffany and Yuri

The Quadratic Function Derived From Zeros of the Function

• SOLUTION

• This problem is too easy to discuss…

• r1+r2= 7 so -(r1+r2)=-7

• (r1)(r2)=-8 so (r1)(r2)=-8

The Quadratic Function Derived From Zeros of the Function

•The answer is:x2-7x-8

Done yet?There’s an activity for you!!!

Activity

The Quadratic Function Derived From Zeros of the Function

• Answer the activity to decode the answer of this riddle…

• What adjective, which refers to charity or alms, comes from the

Latin term?• Note: Next slide, I have the answers

for this activity… the left side of them are the decoders… ENJOY

F(x)=x2-6x+5

F(x)=x2-7x-4/10

F(x)=x2 -12x+12

F(x)=x2

-6x+5

F(x)=x2 -6x+5

F(x)=x2 -16x-37

F(x)=7/15 x2-3/2 x-1

F(x)=x2 +14x+40

F(x)=5/9 x2+ 10/3 x-6

L-(3+√10, 4-√10)

S-r1+r2=6; (r1)(r2)=5

M-(8, 4)

R-vertex at (3, -1),containing (0, -6)

A-r1+r2=-14; (r1)(r2)=40

E-(1,5)

O-(-4, -3)

Y-(8-√3),(8+√3)

N-(1, -1),(-4, -4), (-1, 2)

There are the letters for the activity!!!

Decode Box

Do you decode the message properly?

Agreement• Make a problem out of this presentation and solve it with your friends…

Thank you for listening

Presented by:

Christian Jett Morales