Curve sketching.

Graph is one of the various ways of representing the function. For the graphical representation of a function values of y = f(x). Then points are plotted taking the value of x coordinates and the corresponding value of y as the co-ordinates.

Curve sketching (or curve tracing) includes techniques that can be used to produce a rough idea of overall shape of a plane curve given its equation without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features.

Characteristics of curve

Origin: If the value of the co-ordinate plotted is (0, 0) the curve passes through the origin.

Point on axes: if the curve lays on the axis then its x coordinate or y co-ordinates is zero.

Even function: f: A → B is said to be even function if f(x) = f(-x) ∀ x ϵ A , eg F(x) = x²

Odd function: f: A → B is said to be odd function if f( - x) = - f(x) ∀ x ϵ A eg F(x) = x + x³

Symmetric curve: 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.

Eg A curve is symmetric about the y-axis if no changes occur in function when y is replaced by –y

Increasing function: A function is "increasing" when the y-value increases as the x-value increases

.i.e If x₁ > x₂ ⇒ f(x₁ ) > f(x₂) ∀ x_{1 ,} x₂ ϵ (a,b)

Decreasing function: A function is "decreasing" when the y-value decreases as the x-value increases

.i.e If x₁ > x₂ ⇒ f(x₁ ) < f(x₂) ∀ x_{1 ,} x₂ ϵ (a,b)

Periodic function: A function is said to be periodic if it satisfies f(x+ k) = f(x) such that K> 0.

Eg sin (x +2π) = sinx

Asymptote

An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.

Transformation of graphs

If the given function shifted to the right or left by a unit it is called shifting or translation of the graph.

Reflection

The graph of y = f(-x) is the reflection of the graph of y – f(x) about y-axis. Again the graph of y = -f(x) is the reflection of the graph of y = f(x) about x-axis .

Examples

1. Determine the weather given function is even or odd.

F(x) = x^-3 + x^-1

Soln

we have

F(x) = x^-3 + x^-1 = $\frac{1}{{{{\rm{x}}^3}}} + \frac{1}{{\rm{x}}}$

now, f(-x) = $\frac{1}{{ - {{\rm{x}}^3}}}$ ⁺$\frac{1}{{ - {\rm{x}}}}$ = - $\left( {\frac{1}{{{{\rm{x}}^3}}} + \frac{1}{{\rm{x}}}} \right) = {\rm{\: }} - {\rm{f}}\left( {\rm{x}} \right)$

Since, f(-x)= -f(x) so,

f(x) is odd functions

2. Sketch the following functions

a. y= $\frac{{2{\rm{x}} - 3}}{{{\rm{x}} - 2}}$ = 2 + $\frac{1}{{{\rm{x}} - 2}}$.

soln

The given curve is y = $\frac{{2{\rm{x}} - 3}}{{{\rm{x}} - 2}}$ = 2 + $\frac{1}{{{\rm{x}} - 2}}$.
The characteristics of the curve are:
(i) The curve does not pass through the origin but cuts the axes at $\left( {\frac{3}{2},0} \right)$ and $\left( {0,\frac{3}{2}} \right)$.
(ii) The curve is not symmetrical about any axes.
(iii) The line x = 2 and y = 2 are the asymptotes.
(iv) X > 2 →y > 2and when 0 ≤ x ≤ $\frac{3}{2}$. Then y decreases from $\frac{3}{2}$ to 0 and when x < 0 then y < 2. Moreover when $\frac{3}{2}$ ≤ x < 2 then y decrease from 0 to - ∞..

 b. Y= x² = x- 2

Soln

The curve is y = x² = x- 2

The characteristics of the curve are

i.The curve doesn’t passes through the origin , cuts the y-axis at (0,2)and doesn’t intersects the x-axis at real points

ii. Since a = 1 and b= -1 , c = 0

so, vertex = (${\rm{\: }}\frac{1}{2}{\rm{\: }},{\rm{\: }}\frac{7}{4}{\rm{\: }})$

iii . Since a = 1> 0 so, the parabola turns upwards

iv. y= x² $ - {\rm{\: x\: }} + {\rm{\: }}2{\rm{\: }}$⁼ ${\left( {{\rm{x\: }} - \frac{1}{2}} \right)^2}{\rm{\: }} = {\rm{\: y\: }}--\frac{7}{4}$

so, the curve is symmetrical about the line x =$\frac{1}{2}$