Curve Sketching

 

To sketch the graph of the curve, the following points should be considered:

·         Origin

 If the point (0, 0) satisfies the equation of the curve then it passes through the origin (For example: y² = 4ax, x³ + y³ = 3axy pass through the origin but x² + y² = a² does not pass through the origin.)

·         Symmetry

(i) If the equation of the curve remains unchanged when x is replaced by - x, then it is symmetrical about y-axis. For example, the curves x² = 4ay, x² + y² = 4 etc. are symmetrical about y-axis.

(ii)   If the equation of the curve contains only the even power of y, then the curve is symmetrical about x-axis. For example, the curves y² = 4ax, x² + y² = 4 etc are symmetrical about x-axis.

(iii) If the equation of curve remains unchanged when x and y are respectively changed to y and x, then the curve is symmetrical about the line y = x, for example x³ + y³ = 3axy.

(iv) If the equation of curve remains unchanged when x is replaced by - x, y is replaced by - y, then it is symmetrical about the line y = - x. For example, the curve xy = 4 is symmetrical the line y = -  x.

·         Point on axes

(i)    Putting x = 0, the value of y gives the point where the curve cuts the y-axis. For example, the curve y = 2x + 3 meet y-axis at (0, 3).

 (ii)  Putting y = 0, the value of x gives the point where the curve cuts the x-axis. For example, the curve y² = 8(x - 2) meets x-axis at (2, 0).

·         Even and odd function

(i)    For the curve y = f(x), if f(- x) = f(x), then f(x) is said to be even function. So, if (x, y) lies on the curve (- x, y) also lies on the same curve. For example, the function f(x) = x⁴ + x² + 1 is even.

(ii)  For the curve y = f(x), if f(- x) = - f(x), then f(x) is said to be odd function. So, if (x, y) lies on the curve, (- x, y) also lies on the same curve. For example, the function f(x) = x³ is odd.

·         Increasing and decreasing function

A function f(x) is increasing if f(x₁) < f(x₂) for x₁ < x₂. Similarly, f(x) is said to be decreasing if f(x₁) > f(x₂) for x₁ < x₂.

·         Periodic function

A function f(x) is said to be periodic with the period k if f(x + k) = f(x). For example sin (2p + q) = sinq, cos (2p + q) = cosq, tan (p + q) = tanq etc. are periodic function.

·         Asymptote

A line x = a parallel to y-axis is said to be a asymptote to the curve y = f(x) if y = ¥ when x = a.

Similarly, y = b parallel to x-axis is said to be an asymptote to the curve y = f(x) if x = ¥ when y = b.