Quadratic function is a kind of function that is formed using the basic rules of arithmetic. The graph of a quadratic function is a curved line called parabola, which has properties that make it useful for representing certain types of complex mechanisms such as satellite dishes, car headlights, or radio telescopes.
The quadratic function can also be applied to solve such problems regarding the motion of projectiles, the path of a baseball, description of cables in suspension bridges and parabolic arches, as well as the maximization of revenues.
REMEMBER:
A quadratic function is defined by f(x) =ax2+bx+c, where a, b, and c are real numbers and a is not equal to 0.
Functions play a fundamental role in all areas of mathematics, as well as in other sciences and engineering. However, the intuition pertaining to functions, notation, and even the very meaning of the term "function" varies between the fields. More abstract areas of mathematics, such as set theory, consider very general types of functions, which may not be specified by a concrete rule and are not governed by any familiar principles. The characteristic property of a function in the most abstract sense is that it relates exactly one output to each of its admissible inputs. Such functions need not involve numbers and may, for example, associate to each nation the name of its capital, as discussed below.
Forms of a quadratic function:
A quadratic function can be expressed in three formats:
is called the general form or polynomial form,
Example: f(x)=x2+4x+3, g(x)=4x2-3x-5
is called the factored form, where r1 and r2 are the roots of the quadratic equation.
Examples: f(x)=-2(x-1)(x+1), g(x)=5(x-4)(x+5)
is called the standard form or vertex form.
Examples: f(x) =-2(x+6)2-3, where the opposite value of h is the x coordinate of the vertex and k is the y coordinate.
To convert the general form to factored form, one needs only the quadratic formula to determine the two roots r1 and r2. To convert the general form to standard form, one needs a process called completing the square. To convert the factored form (or standard form) to general form, one needs to multiply, expand and/or distribute the factors.
Finding the Roots or Zeroes of The Quadratic Function:
The simplest form of finding the roots of the quadratic function is simply by factoring the given equation. But if the equation can not be factored, then the quadratic formula should be used.
This formula is called the quadratic formula.
The triangle represents the discriminants that determine what kind of roots or zeros are contained in the function.
Let
If 0\,\!" type="#_x0000_t75">, then there are two distinct roots since is a positive real number.
If , then the two roots are equal, since is zero.
If , then the two roots are complex conjugates, since is imaginary.
By letting and or vice versa, one can factor as .
Saturday, August 9, 2008
Linear Functions
In elementary algebra and analytic geometry, the term linear function is sometimes used to mean a first degree polynomial function of one variable. These functions are called "linear" because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line.
Linear Functions is the most basic algebraic function. It is particularly useful in describing real-life situation. For instance, the Fahrenheit temperature scale is related to the Celsius temperature scale; the distance that a plane or a ship travels is related to the time it travels; the monthly salary of an employee is related to the number of hours the person has worked, etc. Each of these relationships can be described by a linear function.
Such a function can be written as: f(x) =mx+b where m and b are real constants and x is a real variable. The constant m is often called the slope or gradient, while b is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis. Changing m makes the line steeper or shallower, while changing b moves the line up or down.
These functions are in slope-intercept form:
Examples: f(x)=3x+2 or y=3x+2
g(x)=4x+1
Linear Functions is the most basic algebraic function. It is particularly useful in describing real-life situation. For instance, the Fahrenheit temperature scale is related to the Celsius temperature scale; the distance that a plane or a ship travels is related to the time it travels; the monthly salary of an employee is related to the number of hours the person has worked, etc. Each of these relationships can be described by a linear function.
Such a function can be written as: f(x) =mx+b where m and b are real constants and x is a real variable. The constant m is often called the slope or gradient, while b is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis. Changing m makes the line steeper or shallower, while changing b moves the line up or down.
These functions are in slope-intercept form:
Examples: f(x)=3x+2 or y=3x+2
g(x)=4x+1
Circular and Trigonometric Functions
Trigonometry can be studied using two different approaches. The first approach makes usse of circular functions which involve angles and angle rotations, extended to include th definitions of trigonometric functions based on the unit circle. With the real numbers as the domain, the periodic functions are defined in terms of the unit circle. The second aaproach makes use of the trigonometric functions to study triangles and their applications.
An angle is formed by two rays with a common endpoint. One side of the angle rotates about the common endpoint and the other side remains stationary. The stationary ray is the initial side of the angle and the rotating ray is the terminal side. The angle formed by the rotating he terminal side exactly once in the counterclockwise direction until it coincides with the initial side has a degree measure of 360(1 revolution). Revolution is the motion of the bodyabout a center or about its axis. One degree is equal to 1/360 revolution.
If the terminal side of a central angle coincides with the coordinate axis such that its initial side coincides with the positive side of the x-axis the angle is called quadrantal angle.
Radian is the measure of a central angle of a circle whose rays subtend an arc on the circle whose length is equal to the radius of the circle.
The length of an arc of a circle of radius r subtended by a central angle theta is given by the product of radius and the measure of the central angle in radians. That is, s=r(theta)
The reference angle for a nonquadrantal angle greater than 90 degrees is the smallest nonnegative angle between the terminal side and the x-axis when the angle is in standard position.
Unit Circle
Trigonometric Functions and Identities
The major functions where the trigonometric functions are based or formed by using the abscissa, ordinate, and the radius vector of a point on the terminal side of an angle in standard position.
sin theta= ordinate/radius vector= y/r
cos theta= abscissa/radius vector= x/r
tan theta= ordinate/abscissa= y/x, where x is not equal to 0
csc theta= radius vector/ordinate= r/y
sec theta= radius vector/abscissa = r/x
cot theta= abscissa/ordinate= x/y, where y is not equal to 0
When an equality is true for all values of the uknown quantity or quantities, the equality is called an ientity. By means of proveen idntities, it is possible to prove other identities.
sin2theta+cos2theta=1
tantheta= sintheta/costheta
cot theta= costheta/sintheta
sectheta= 1/costheta
csctheta= 1/sintheta
An angle is formed by two rays with a common endpoint. One side of the angle rotates about the common endpoint and the other side remains stationary. The stationary ray is the initial side of the angle and the rotating ray is the terminal side. The angle formed by the rotating he terminal side exactly once in the counterclockwise direction until it coincides with the initial side has a degree measure of 360(1 revolution). Revolution is the motion of the bodyabout a center or about its axis. One degree is equal to 1/360 revolution.
If the terminal side of a central angle coincides with the coordinate axis such that its initial side coincides with the positive side of the x-axis the angle is called quadrantal angle.
Radian is the measure of a central angle of a circle whose rays subtend an arc on the circle whose length is equal to the radius of the circle.
The length of an arc of a circle of radius r subtended by a central angle theta is given by the product of radius and the measure of the central angle in radians. That is, s=r(theta)
The reference angle for a nonquadrantal angle greater than 90 degrees is the smallest nonnegative angle between the terminal side and the x-axis when the angle is in standard position.
Unit Circle
Trigonometric Functions and Identities
The major functions where the trigonometric functions are based or formed by using the abscissa, ordinate, and the radius vector of a point on the terminal side of an angle in standard position.
sin theta= ordinate/radius vector= y/r
cos theta= abscissa/radius vector= x/r
tan theta= ordinate/abscissa= y/x, where x is not equal to 0
csc theta= radius vector/ordinate= r/y
sec theta= radius vector/abscissa = r/x
cot theta= abscissa/ordinate= x/y, where y is not equal to 0
When an equality is true for all values of the uknown quantity or quantities, the equality is called an ientity. By means of proveen idntities, it is possible to prove other identities.
sin2theta+cos2theta=1
tantheta= sintheta/costheta
cot theta= costheta/sintheta
sectheta= 1/costheta
csctheta= 1/sintheta
Operations on Functions
operation on functions
let f and h be two functions:
A. their sum, f and h,is defined as:
(f + h)(x) = f(x) + h(x)
the domain of f + h consists of the number x that are in the domain of f and in the domain of h.
B. their difference, f - h,is defined as:
(f - h)(x) = f(x) - h(x)
the domain of f - h consists of the numbers x that are in the domain f and in the domain of h.
let f and h be two functions:
A. their sum, f and h,is defined as:
(f + h)(x) = f(x) + h(x)
the domain of f + h consists of the number x that are in the domain of f and in the domain of h.
B. their difference, f - h,is defined as:
(f - h)(x) = f(x) - h(x)
the domain of f - h consists of the numbers x that are in the domain f and in the domain of h.
C. Their product,(f) * (h), is defined as:
(f * h)(x) = f(x) * h(x)
the domain of f * h consists of numbers x that are in the domain of f and in the domain of h.
D. their quotient, f / h, is defined as:
(f / h)(x) = f(x)/h(x), h(x) is not = O
thee domain of f / h consisits of the number x for which h(x) is not = 0 that are in the domain of f and the domain of h.
Functions
Definiton of Functions
Functions is said to be the central idea in the study of mathmatics. In ievery situation, there is always a mathmtical function in which one quantity correspondence to another quantity accordin to some definite role.
There are many ways to represent a function: by a formula, by a plot or graph, by an algorithm that computes it, by a description of its properties. Sometimes, a function is described through its relationship to other functions. In applied disciplines functions are frequently specified by their tables of values, or by a formula. Not all ways apply to every possible kind of function, and one has to make a firm distinction between the function itself and multiple ways of presenting or visualizing it.
In definiton, set X is clled the domain of the function. For every element x in set X, the corresponding elementy in set Y is called the value of the function at x, or the image of x. The set of values or images of thee elements of the ddomain is caled th rang of the function.
Friday, August 8, 2008
Polynomial Funtions
Functions such as linear, quadratic, square and cube functions all belong under one type function, the POYNOMIAL FUNCTION. This type of function is one of the foundations of other certain kinds of functions.
The study of graphs of polyommial functions is of considerable importance to scientists, astronomers, chemists, and physicists because the properties of these functions affect the behavior and shape of collected data in scientific research. For instance, technology has its foundation in the mathematical concepts of parabola and the quadratic formula. They have properties that are useful in making complex mechanisms that are useful in the advancement of the human race, such as satellite dishes, car headlights, radio telescopes, and reflecting telescopes.
Polynomial Functions of degree n is a function of the of the form:
f(x)=anxn+an-1xn-1+an-2xn-2+…a0, where n is a non-negative integer, and
an,an-1,…a0 are real numbers, and an not equal to 0.
The degree of polynomial functions is determined by the highest power of its terms.
The function h(x) =6x4+2x3+x2-x+4+7x5 has six terms.
The degree of terms are 5, 4, 3, 2, 1,and 0. Therefore the degree of this function is 5.
Methods in Finding Values of Polynomial Functions:
· The Remainder Theorem:
Example:
Find the remainder and value of P(x)=(x3-3x2+x+4)÷(x-2)
Solution:
Since x-c is represented by x-2 then, you must equate it to 0. Whereas:
x-c=0 x-2=0
x=c x=2 Therefore, substitute the x in the function as 2.
P(2)=(2)3-3(2)2+(2)+4
=8-12+2+4
=2 So, P(2)=2
· Synthetic Division:
Synthetic Division, hand-in-hand with the remainder theorem can be used as a convenient way to find values of polynomial functions or the remainder. In this kind of division the, coefficients are copied and the divisor is equated to zero.
Example:
P(x)=2x3-8x2+19x-12 divided by 2=x-3.
Solution:
First equate the divisor to 0 then find x.
x-3=0
x=3
Then copy the coefficients of the given function. With the value of x located at the upper left-hand corner of the solution.
3 2 -8 19 -12
Then, bring down the first numerical coefficient then multiply it to the value of x. Then, add the product to the next term. Repeat this process until to last term.
3 2 -8 19 -12
6 -6 39
2 -2 13 27
Therefore, the remainder of the function is 27 or can also written as P(3)=27.
· Factor Theorem:
It states that you can use either the Remainder Theorem or Synthetic Division to determine of the divisor is a factor of the equation or not. If it is a factor, then the function would be equal to 0. But if not, it would result to the remainder of the function.
Example:
When P(x)=x3-x2-4x+4 is divided by x-2.
Sol’n:
By Synthetic Division:
2 1 -1 -4 4
2 2 -4
1 1 -2 0
Or By Remainder Theorem: where x=2
P(x)=x3-x2-4x+4
P(2)=(2)3-(2)2-4(2)+4
=8-4-8+4
P(2)=0 Therefore, x-2 is a factor of the function P(x)=x3-x2-4x+4.
NOTE: The factors of the Function are also the zeros of that particular Function.
· Finding the Zeros of the Function:
As mentioned earlier, the factors of the function is also its zeros. But if it has a remainder, then the quadratic formula should be used to find its zeros.
Example:
Q(x)=x4-x3-11x2+9x+18 divided by (x+3), find its zeros.
By synthetic division:
-3 1 -1 -11 9 18
-3 12 -3 -18
1 -4 1 6 0
So, the quotient is x3-4x2+x+6 and x+3 is a factor of the function.
To find the other zeroes, factor the quotient, if not use the quadratic formula.
The study of graphs of polyommial functions is of considerable importance to scientists, astronomers, chemists, and physicists because the properties of these functions affect the behavior and shape of collected data in scientific research. For instance, technology has its foundation in the mathematical concepts of parabola and the quadratic formula. They have properties that are useful in making complex mechanisms that are useful in the advancement of the human race, such as satellite dishes, car headlights, radio telescopes, and reflecting telescopes.
Polynomial Functions of degree n is a function of the of the form:
f(x)=anxn+an-1xn-1+an-2xn-2+…a0, where n is a non-negative integer, and
an,an-1,…a0 are real numbers, and an not equal to 0.
The degree of polynomial functions is determined by the highest power of its terms.
The function h(x) =6x4+2x3+x2-x+4+7x5 has six terms.
The degree of terms are 5, 4, 3, 2, 1,and 0. Therefore the degree of this function is 5.
Methods in Finding Values of Polynomial Functions:
· The Remainder Theorem:
Example:
Find the remainder and value of P(x)=(x3-3x2+x+4)÷(x-2)
Solution:
Since x-c is represented by x-2 then, you must equate it to 0. Whereas:
x-c=0 x-2=0
x=c x=2 Therefore, substitute the x in the function as 2.
P(2)=(2)3-3(2)2+(2)+4
=8-12+2+4
=2 So, P(2)=2
· Synthetic Division:
Synthetic Division, hand-in-hand with the remainder theorem can be used as a convenient way to find values of polynomial functions or the remainder. In this kind of division the, coefficients are copied and the divisor is equated to zero.
Example:
P(x)=2x3-8x2+19x-12 divided by 2=x-3.
Solution:
First equate the divisor to 0 then find x.
x-3=0
x=3
Then copy the coefficients of the given function. With the value of x located at the upper left-hand corner of the solution.
3 2 -8 19 -12
Then, bring down the first numerical coefficient then multiply it to the value of x. Then, add the product to the next term. Repeat this process until to last term.
3 2 -8 19 -12
6 -6 39
2 -2 13 27
Therefore, the remainder of the function is 27 or can also written as P(3)=27.
· Factor Theorem:
It states that you can use either the Remainder Theorem or Synthetic Division to determine of the divisor is a factor of the equation or not. If it is a factor, then the function would be equal to 0. But if not, it would result to the remainder of the function.
Example:
When P(x)=x3-x2-4x+4 is divided by x-2.
Sol’n:
By Synthetic Division:
2 1 -1 -4 4
2 2 -4
1 1 -2 0
Or By Remainder Theorem: where x=2
P(x)=x3-x2-4x+4
P(2)=(2)3-(2)2-4(2)+4
=8-4-8+4
P(2)=0 Therefore, x-2 is a factor of the function P(x)=x3-x2-4x+4.
NOTE: The factors of the Function are also the zeros of that particular Function.
· Finding the Zeros of the Function:
As mentioned earlier, the factors of the function is also its zeros. But if it has a remainder, then the quadratic formula should be used to find its zeros.
Example:
Q(x)=x4-x3-11x2+9x+18 divided by (x+3), find its zeros.
By synthetic division:
-3 1 -1 -11 9 18
-3 12 -3 -18
1 -4 1 6 0
So, the quotient is x3-4x2+x+6 and x+3 is a factor of the function.
To find the other zeroes, factor the quotient, if not use the quadratic formula.
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