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Graphing and Functions

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As we know that in a function each input value is instantly lined with one corresponding output value of that function. General concept of a graph is characterized by graph of relation. A function is always recognized by its graph but they are not same because it happens that two Functions that have different co- domain must have same graph. If we test graph of a function then we use vertical line test and if we test whether function is one – to – one then we use horizontal line test. If we have Inverse Function then graph of inverse function can be obtained by plotting graph of function along the line v = u. (here 'u' is along x - axis and 'v' is along y – axis. In a graph a curve is one – to – one function if and only if it is a function. Now we will try to understand Graphing and functions with help of an example.

Suppose we have a function: y = x – x2, we can plot graph of this function as shown below.
We need to follow some steps for Graphing Functions:
Step 1: Our function here is y = x – x2.

Step 2: We assume different values of x – coordinate to find values of y – coordinate.
If we value of x – coordinate as '0' then we get the value of y – coordinate as '0'. In same way if we put value of x – coordinate as -2, -1, 1, 2, 3 then we get value of y – coordinate as -6, -2, 0, -2, -6 respectively.

X

-2

-1

0

1

2

3

Y

-6

-2

0

0

-2

-6


Step 3: Now plot these coordinate on graph. So graph of given function is:


This is all about Graphing and Functions.

Graphing

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Graphing can be defined as process of plotting Functions on graphs. Abstract equations, functions, expressions and concepts become more clearly visible when projected on an x - y coordinate plane, also known as a Cartesian coordinate system. This method for determining 'x' and y- coordinates for an equation becomes more complicated, going into the depth of mathematics.

The first step we follow to graph any mathematical concept is to find x and y - axes on a regular coordinate plain. A coordinate or a rectangular plain looks like a big plus sign with Numbers marked on axes at particular intervals and arrowheads at ends of the lines. Horizontal line is x- axis, on which the x- coordinates are graphed, and vertical line is y- axis, with y- coordinates. Point where two axes i.e. the x and y- axis is called as origin (It is a fixed Point for reference).
In coordinate system each and every point is represented by two numbers written in parentheses separated by a comma: (x, y). Where, 'x' represents coordinate graphed on x- axis and y- represents coordinate graphed on y- axis. As these points are just measures, we need to count number of units on x- axis (right or left) and number of units on y- axis (bottom or up). These measures are taken from a reference point that is origin (0, 0). For example, to graph a point (5, 12), count five units to right of origin, then move 12 units up and mark the point.
Sometimes you may find a need to solve for two coordinates by using general algebraic operations. For instance, given formula y = 2x + 4, pick values for 'x', say, 0, 1, -2 and 3, and put them into formula to get 4, 6, 0 and 10 respectively, giving you coordinates (0, 4), (1, 6), (-2, 0), (3, 10). Graph these points and draw a line connecting them to get desired expression.

Graphing Lines

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An equation is said to represent a line if highest power of variable it consists is 1. Most general form of linear equation representing a line is given as: y = mx + c Remember to resolve the equation if it is present in different form. Graphing Lines is an easy task to be done by working following steps:

In above equation “ c ” represents y- intercept. So, to start Graphing by drawing a dot on y- axis at y = c distance from origin. This Point at which graph would be crossing y- axis after its completion. Next select any random value of 'x' other than 0 and mark it on horizontal x - axis you've drawn. Draw vertical line that crosses this point on your x-axis for further help in plotting the graph. Now substitute this value of 'x' just marked into your equation i.e. y = mx + c and solve it for 'y'. After you get the point on y- axis, mark it and draw a horizontal line that crosses this value y- axis.
Put a dot where two lines (drawn horizontally and vertically) intersect. From this point draw a Straight Line to y- intercept. When you extend this line in both directions towards the edges of the plot, it represents graph of the given equation.
There are certain points to be remembered like value of "m" in y = mx + c is the Slope of graph. Slope can be calculated as Ratio of vertical change to horizontal change between two points on graph.
For Example: Graph of equation y = 4x + 10 has Slope of 4 and crosses y- axis at an intercept y = 10.

Graphing Circles

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Circle is a conic section with its both axes (major and minor) equal in length. The key features of a Circle are its radius (half of the measure of any axis or Diameter), its circumference, chords (diameter being the longest), center, arcs, sectors etc. In other words a circle can be defined as Set of all those points which are equidistant from center Point. Circles can also be defined by Intersection of a plane and a cone. Graphing circles means Graphing Equations of circles. General form of a circle equation is given as:

(x – h)2 + (y – k)2 = r2 …..............equation 1.

Where,
h, k and c are constants (h and k being the coefficients).
'r' is Radius of Circle and point (h, k) represents the Center of Circle. For origin i.e. (0, 0) to be the center, values of 'h' and 'k' must be equals to zero giving simple equation for a circle.
(x – 0)2 + (y – 0)2 = r2,
(x)2 + (y)2 = r2.
To graph equation 1, first plot the center i.e. (h, k). Next draw all points “r” units away from center by substituting values of 'x' for getting the corresponding values of 'y'. You can plot a few of them and then connect dots in a circular shape using a protractor.
Taking examples of two circles whose equations can be given as:
(x)2 + (y)2 = 25 and
(x - 4)2 + (y - 4)2 = 25
First equation has its origin at (0, 0) while second equation has its origin at (4, 4). Radii of both circles are equal. These circle equations can be graphed as follows as shown in figure:

definition of a function

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The Definition of a Function in mathematics can be given as representation of relationship between a Set of variables and constants. These are used to solve equations for unknown variables. A function possesses unique output or value for each and every input given to it. A function in its simplest form can be written as: y = x. For any value of 'x' we insert in the function, same will be value of 'y'. So, here 'y' can be said to be dependent on 'x'. Mostly complex Functions involve mathematical operations being applied to 'x' to determine final value of 'y'. For example, if we take y = x2 + 5x + 6. Here, first we need to simplify the function in 'x' using Factorization method. Then for two possible values of 'x' we get two values of 'y'. In functions, values will change, but relationship between variables remains constant.

A much known form to represent a function is: f(x).

Mostly functions are written with f(x) in place of 'y'. For example, f (x) = 4x. In this notation, function of 'x' is equal to four times value of 'x'. So, for any value of 'x' say 2, function of 'x', or f(x) is equal to 8.
Evaluating a function means solving a mathematical problem or equation involving a function. For this we need to provide an input. For each input given for variable 'x', there can be only one output for function.
For example, in function f(x) = 20x, inputs may be given as:
x = 2,
x = 3,
x = 5,
and corresponding values or outputs of functions that we get will be:
x=2, f(x) = 40,
x=3, f(x) = 60,
x=5, f(x) = 100,
Functions are of importance in various fields of maths, physics, science etc.

Combining Functions

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In Algebraic calculations, you would most deal with the systems of Linear Equations that can also be correlated with finding the Combination of two Functions. Combining functions can Mean several possible operations like adding up two functions together, subtracting one function from the other, multiplying one function by the other, dividing one function by the other, or substituting one function into the other.
To perform combining of functions arrange them such that similar terms are present in the same order. For instance, let us consider two functions given as:
F (x) = x2 - 4x + 5 and
G (x) = 4x + 4 – y.
Here y represents the function G (x). So, we need to rearrange the second equation by adding “y” both sides to obtain 2G (x) = 4x + 4, then divide both sides of the equation by 2 to obtain G (x) = 2x + 2.
Now let us perform the combining operations on the functions F (x) and G (x). First we go for addition where the like terms of the function G(x) are added to those of the function F (x). We will add 2x to - 4x to obtain -2x, and you will add 2 to +5 to obtain 7. Thus,
F (x) + G (x) = x2 - 2x + 5
Similarly,
F (x) - G (x) = x2 - 4x + 5 – (2x + 2) = x2 - 6x + 3
F (x) / G (x) = x2 - 4x + 5 * (2x + 2) = 2x3 – 8x2 + 10x + 2x2 – 8x + 10 = 2x3 – 6x2 + 2x + 10
F (G (x)) = (2x + 2) 2 – 4 (2x + 2) + 5 = 4x2 + 4 + 8x – 8x – 8 + 5 = 4x2 +1

Graphing Functions

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In a function each input value is directly lined with one corresponding output value of that function. For example: If we have a relation that is defined by function rule f (s) = s2, then it compares its input’s’ to its Square, both values are real. If we put input value as -8 then we get its output as 64. In function form we can also write it as: f (-8) = 64. Real number graph is indistinguishable for representation of a function. Representation of a graph is not applied for general function. General concept of a graph is characterized by graph of relation. A function is always recognized by its graph but they are not same because it happens that two Functions that have different value of co- domain must have same graph. If we test graph of a function then we use vertical line test and if we test whether function is one – to – one then we use horizontal line test. If we have Inverse Function then graph of inverse function can be obtained by plotting graph of function along the line q = p. (here 'p' is along x - axis and 'q' is along y – axis. In a graph a curve is one – to – one function if and only if it is a function. For example: Suppose we have a function:

F (x) = p, if p = 1,
q, if p = 2,
r, if p = 3. Then Graphing Functions of given values.
Here we get function value (1, p), (2, q), (3, r). We can also write function in cubical form as:
F (p) = p2 – 9p, on plotting graph of this function, we get curve shown below:

This is how we plot functions on a graph.