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Common Graphs

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Graphs can be defined as plot of any expression or we can say representation of an expression on a graph. Here we will see some Common Graphs. Some common graphs are line graph, Parabola graph, Absolute value graph, Cubic graph, Square Root graph, Circle graph, Ellipse graph, Hyperbola graph, Semi – circle, Exponential, Logarithmic, Reciprocal, Linear function, Even degree polynomial, odd degree polynomial, Even root, Odd root, one over an odd power, sine and cosine function and Greatest Integer graph. These all are included in common graphs. Let's have small introduction of some of above mentioned graphs.
Slope of line graph: - As we know that the equation of Slope is given by: y = mx + b. In this case line has a y- intercept of (0, b) and 'm' is denotes Slope of line. Here Slope of a line is given by: m (slope) = rise / run
Graph to depict slope of a line is given below:


Figure of slope of a line
Now we will see the graph of Absolute Value Function. Absolute value function is given as:
| x | = x, if value of 'x' is greater than 0,
-x, if value of 'x' is less than 0,
Using this function we get graph of absolute value as:


Now we will see Parabola graph. General form of parabola is given by: f (x) = ax2 + bx + c, in this form, value of x – coordinate is given as: x = -b / 2a and value of y – coordinate as: y = f (-b / 2a). Suppose we have a parabola equation f (x) = -x2 + 2x + 3. If we solve parabola equation using above form then we get value of x – coordinate as -1 and 3. Now we can easily plot graph of parabola. Graph is shown below.


This is all about common graphs.

Lines, Circle and Piecewise Functions

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In mathematics, we study different types of shapes and Lines, Circle and Piecewise Functions are among them.

Circle can be defined as a line which has round or curved shape and every Point on line is at fixed distance from center. Some properties are also defined for Circle which is shown below:

Center –
All points lie on boundary of circle which is equidistant from center.
Radius – Radius of a circle can be defined as distance measured from center to any point on a circumference of circle. Note: - Radius of circle is always half of the Diameter.

Diameter – A line used to cut circle in two halves is known as diameter of a circle. Note: - Diameter of a circle is always double of radius.
Circumference – Distance around circle is known as circumference of circle. Circumference formula is given by: circumference of a circle = 2⊼R;
Area – Number of units or Square units which can be filled inside a circle is known as area of circle. Formula to find area is given by: area of a circle = ⊼r2,
If diameter of circle is given then area of circle is:
Area = ⊼D2 / 4, and if circumference of circle is given then area of circle is:
Area = C2 / 4⊼,
Now we will discuss lines. Line can be defined as a geometrical object which is straight, infinitely long and infinitely thin. Line has only one dimension. Generally equation of a Straight Line is given as:
Y = mx + b; or
Y = mx + c;
Where, ‘m’ represents the Slope (gradient) of a line. Now talk about piecewise function. A function which is defined on a sequence of intervals is known as piecewise function. We can write the piecewise function as:
| P | = -P for P < 0,
= 0 for P = 0.
= P for P > 0.
This is all about Lines, Circle and Piecewise Functions.

Transformations

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Translation, rotation, reflection or resizing geometrical forms is known as transformation in mathematics. Congruent shapes are suitable for translation as new shape that we attain will have same measurements (angle, area, lengths of lines). Resizing of shapes doesn’t Mean that their orientation will change; only measurements like line lengths and areas will differ. So transformations can be of four types, described as follows:

Rotation is a transformation in which you can turn your shape around a center Point. The point can be present anywhere, inside or outside the shape, or it can also be present on edge of shape. Distances between points lying on your shape and center will not change during entire rotation. Path for any point will end up with a Circle, if you trace it during rotation.
Next is reflection where you take reflection of your shape with respect to some line instead of a point. Here also you will find that distance of any point on shape is equidistant from the line. The line can even pass through structure of shape, along edge of shape.
When a shape is transformed from one direction to another it is said to be translated. For instance, sliding a ball from one place to another at an angle can be thought as an example of translation as long as it continues to face same direction. To represent it graphically you just need to add desired amounts to your 'x' and 'y' coordinate for each point to move the shape.
Resizing of a shape can be done by selecting a center point and drawing lines from each corner of your shape to that center point. If you want a shape of larger size, let the lines pass out through corners as well.

Symmetry

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Symmetry means two parts of figures which are exactly same in measure of their sides and angles. We say that figure is symmetrical about a line, if we fold figure at that particular line and find that two parts of figure exactly overlap each other.
This line which divides figure in two equal halves is called line of symmetry. Now we take different figures and find the lines of symmetry for each of them. Let us start with Square. A square has 4 lines of symmetry, which means figure can be folded from 4 different places, such that it is divided into two equal halves. These lines of symmetry are two diagonals and two lines which can be formed by joining mid points of opposite and Parallel Lines.
Next figure we take is a Rectangle. In a rectangle, we have opposite pair of sides as equal and parallel. We observe that on joining mid points of opposite sides of rectangle we get lines of symmetry of rectangle. Although diagonals of rectangles are not lines of symmetry. Thus we conclude that rectangle has two lines of symmetry.
Let us take a triangle now. We will start with Equilateral Triangle and observe that equilateral triangle has 3 lines of symmetry; on other hand we say that Isosceles Triangle has only one line of symmetry and Scalene Triangle has no lines of symmetry.
But if we take Circle, we conclude that the Diameter of circle is the line of symmetry of circle as it divides the figure in two equal parts. As there can be infinite number of diameters, so there can be infinite lines of symmetry for circle.