<< Chapter < Page Chapter >> Page >

We are already acquainted with quadratic equation and its roots. In this module, we shall study quadratic expression from the point of view of a function. It is a polynomial function of degree 2. The general form of quadratic expression/ function is :

f x = a x 2 + b x + c ; a , b , c R , a > 0

Elements of quadratic equation

Quadratic equation

Quadratic equation is obtained by equating quadratic function to zero. General form of quadratic equation corresponding to quadratic function is :

a x 2 + b x + c = 0 ; a , b , c R , a > 0

Discriminant of quadratic equation

Nature of a given quadratic function is best understood in terms of discriminant, D, of corresponding quadratic equation. This is given as :

D = b 2 4 a c

Roots of quadratic equation

Quadratic equation is obtained by equating quadratic function to zero. Quadratic equation has at most two roots. The roots are given by :

α = - b D 2 a = - b b 2 4 a c 2 a

β = - b + D 2 a = - b + b 2 4 a c 2 a

Properties of roots of quadratic equation

1 : If D>0, then roots are real and distinct.

2 : If D=0, then roots are real and equal.

3 : If D<0, then roots are complex conjugates with non-zero imaginary part.

4 : If D>0; a,b,c∈T (rational numbers) and D is a perfect square, then roots are rational.

5 : If D>0; a,b,c∈T (rational numbers) and D is not a perfect square, then roots are radical conjugates.

6 : If D>0; a=1;b,c∈Z (integer numbers) and roots are rational, then roots are integers.

7 : If a quadratic equation has more than two roots, then the function is an identity in x and a=b=c=0.

8 : If a quadratic equation has one real root and a,b,c∈R, then other root is also real.

Elements of quadratic function

Zeroes of quadratic function

The real roots of the quadratic equation are zeroes of quadratic function. The zeroes of quadratic function are real values of x for which value of quadratic function becomes zero. On graph, zeros are the points at which graph intersects y=0 i.e. x-axis.

Graph of quadratic function

Graph reveals important characteristics of quadratic function. The graph of quadratic function is a parabola. Working with the quadratic function, we have :

y = a x 2 + b x + c = a x 2 + b a x + c a

In order to complete square, we add and subtract b 2 / 4 a 2 as :

y = a x 2 + b a x + b 2 4 a 2 + c a b 2 4 a 2

y = a { x + b 2 a 2 - b 2 4 a c 4 a }

y + b 2 4 a c 4 a = a x + b 2 a 2

y + D 4 a = a x + b 2 a 2

Y = a X 2

Where,

X = x + b 2 a and Y = y + D 4 a

Graph of quadratic function

The graph is parabola.

Clearly, Y = a X 2 is an equation of parabola having its vertex given by (-b/2a, -D/4a). When a>0, parabola opens up and when a<0, parabola opens down. Further, parabola is symmetric about x=-b/2a.

Maximum and minimum values of quadratic function

The graph of quadratic function extends on either sides of x-axis. Its domain, therefore, is R. On the other hand, value of function extends from vertex to either positive or negative infinity, depending on whether “a” is positive or negative.

When a>0, the graph of quadratic function is parabola opening up. The minimum and maximum values of the function are given by :

y min = - D 4 a at x = - b 2 a

y max

Clearly, range of the function is [-D/4a, ∞).

When a<0, the graph of quadratic function is parabola opening down. The maximum and minimum values of the function are given by :

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

Ask