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Curve Sketching-Overview

Curve sketching

According to Wikipedia, Curve Sketching are techniques for producing a rough idea of overall shape of a plane curve given its equation, without computing the large number of points required for a detailed plot.  It is an application of the theory of curves to find their main features. 


Characteristics of Curve Sketching

1.Origin: We must see if the curve passes through the origin(0, 0) or not. If the value of y=0 when we keep the value of x=0 or vice versa in the given equation then the curve passes through the origin.

2.Point on Axes: When we put, x=0 and y=0 successively in the equation and we get respective values of y and x. These are the point the curve meet at the axes(if we get imaginary values of y or x then the curve doesn't intersect at the given axis.)

3.Even Function: A function f: A→B is said to be even function if f(-x)=f(x) for all x ∈ A. In other words, if (x, y) and (-x, y) lies on the same curve then, the function is even function.
                Example;        cos(-x) = cos(x), so it f(x)=cos(x) is a even function.

4.Odd Function: A function f: A→B is said to be odd function if f(-x)=-f(x) for all x ∈ A. In other words, if (x, y) and (-x, -y) lies on the same curve then, the function is odd function.
                Example;        sin(-x) = - sin(x), so it f(x)=sin(-x) is a odd function.

5. Symmetry: A curve is symmetric about the axis if no changes occur in function when the co-ordinate of its axis is replaces by its respective negative co-ordinates.
            →A curve represented by the function y=f(x) is symmetric about y-axis if no change occurs in f(x) when x is replaced by -x i.e., if f(-x)=f(x).
            →A curve represented by the equation f(x, y) is symmetric about x-axis if no change occurs in the equation when y is replaced by -y.
            →A curve represented by the function y=f(x) is symmetric about the origin if f(-x) = -f(x) i.e., if f(x) is an odd function.

6.Increasing Function: A function y=f(x) is said to be increasing function in the interval (a, b) if for every x1, x2 ∈ (a, b) 
   x2 > x1 => f(x2 ) > f(x1)
Slope of tangent at any point of such curve is always positive.
i.e., f'(x)>0                [f'(x) is the 1st derivative of the given function of the curve]

7.Decreasing Function: A function y=f(x) is said to be decreasing function in the interval (a, b) if for every x1, x2 ∈ (a,b)
    x2 > x1 => f(x2 ) < f(x1)
Slope of tangent at any point of such curve is always negative.
i.e., f'(x)<0

8.Periodicity: A function f which satisfies f(x+k)=f(x) for all x belonging to its domain and k>0 is said tp be periodic function. The smallest value of k is known as the period of the function.
        Example; sin(x+2π)=sin(x), so, sin(x) is the periodic function with period 2π.

9.Asymptote: A line x=a is said to be an asymptote to the curve y=f(x) if y=∞ when x=a(i.e. x close to a makes y close to ∞)
        A line y=b is said to be an asymptote to the curve y=f(x) if x=∞ when y=b(i.e. y close to b makes x close to ∞)


Transformation of Graphs 
1.Shifting of Graph: The shifting of the given function to the right or left by some units is called shifting of graph.

    Here, the function y=x2 is shifted by a units to the right.

2. Reflection: The graph of y=f(-x) is the reflection of graph of y=f(x) about y-axis. Also, the graph of y=-f(x) is the reflection of graph of y=f(x) about x-axis. Example;
                              

The graphs of y=x2 for x ∈ [0, ∞) is the reflection for the curve y=x2 for x ∈ (-∞, 0]