{"id":5166,"date":"2021-08-31T11:24:27","date_gmt":"2021-08-31T05:54:27","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5166"},"modified":"2026-02-17T01:37:32","modified_gmt":"2026-02-16T20:07:32","slug":"what-is-curve-tracing","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-curve-tracing\/","title":{"rendered":"What is curve tracing?"},"content":{"rendered":"\n

Curve tracing – An introduction<\/strong><\/h2>\n\n\n\n

curve tracing<\/a>– a picture speaks more clearly than words and numbers. A curve which is the image of functional relationship gives us a lot information about the relation. <\/p>\n\n\n\n

We can get information analyzing the equation itself but the associated curve is often easy and understandable.<\/p>\n\n\n\n

Let\u2019s study about how to trace the curves of various equations in different forms like Cartesian, parametric and polar.<\/p>\n\n\n\n

Important definitions for curve tracing<\/strong><\/h2>\n\n\n\n

Here are some terms that we use in curve tracing:<\/p>\n\n\n\n

1. Double point- <\/strong>when a curve passes two times through this point is known as double point.<\/p>\n\n\n\n

2. Node- <\/strong>a double point at which two real tangents can be drawn.(tangents should not be coincide)<\/p>\n\n\n\n

3. Cusp – <\/strong> a double point is a cusp when two tangents are coincide on it.<\/p>\n\n\n\n

First we will understand the concept of asymptotes, which are used in curved tracing.<\/p>\n\n\n\n

Asymptotes<\/strong><\/h2>\n\n\n\n

Introduction-<\/strong> A line that a curve approaches is known as asymptote<\/a>. Any graph (curve) approaches it but never touches it.<\/p>\n\n\n\n

There are total three types of asymptotes<\/a>–<\/p>\n\n\n\n

1.     Vertical asymptote<\/p>\n\n\n\n

2.     Horizontal asymptote<\/p>\n\n\n\n

3.     Oblique\/slant asymptote<\/p>\n\n\n\n

Definition-<\/strong> An asymptote of a curve of function y = f(x) is a line which does not intersect on the graph.<\/p>\n\n\n\n

Types of asymptotes<\/strong><\/h2>\n\n\n\n
\"\"\/<\/figure>\n\n\n\n

Vertical asymptote-<\/strong><\/h3>\n\n\n\n

A line x = a which is straight is a vertical asymptote of the graph of the function y = f(x) if atleast one of the following condition does it follow-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Horizontal asymptote-<\/strong><\/h3>\n\n\n\n

A line y = L is called horizontal asymptote to the curve of the function f(x)-<\/p>\n\n\n\n

If f(x)\u2192 L as x\u2192\u221e or as x\u2192-\u221e<\/p>\n\n\n\n

Example: Find the horizontal asymptote of the function-<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Sol. In order to find the horizontal asymptote-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Hence the horizontal asymptote is y = 2.<\/p>\n\n\n\n

Oblique asymptote or slant asymptote-<\/strong><\/h3>\n\n\n\n

A straight line y = mx + c where m \u2260 0 will be an oblique\/slant asymptote to the graph of the function \u2018f\u2019 if-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Note-<\/strong> the value of m can be find as-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

And the value of c can be find as-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example: Find the slant asymptote of the function f(x) = x + <\/strong>x .<\/strong><\/p>\n\n\n\n

Sol. Find the value of m-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Hence the y = x + c,<\/p>\n\n\n\n

Now find c<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Here c must not be infinite.<\/p>\n\n\n\n

So we can say that f does not have a slant asymptote at \u221e.<\/p>\n\n\n\n

Procedure to find the asymptotes parallel to axes of a polynomial function<\/strong><\/h2>\n\n\n\n

Theorem-<\/strong><\/p>\n\n\n\n

Suppose f(x , y) is a polynomial in x and y.<\/p>\n\n\n\n

A straight line y = c is an asymptote of a curve f(x , y) = 0 if an only if y \u2013 c is a factor of the co-efficient of the highest power of x in f(x , y).<\/p>\n\n\n\n

Example: Determine the asymptotes parallel to axes of the curve:<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Sol. The given function can be written as-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Asymptote parallel to x-axis-<\/strong><\/p>\n\n\n\n

Equating the coefficient of the highest power of x to zero, we get y\u00b2 = 0 which means y = 0. This is an asymptote.<\/p>\n\n\n\n

Asymptote parallel to y-axis-<\/strong><\/p>\n\n\n\n

Equating the coefficient of the highest power of x to zero, we get x\u00b2 – a\u00b2 = 0<\/p>\n\n\n\n

Which gives x = \u00b1a , that means x = +a , x = -a     <\/p>\n\n\n\n

So these are the asymptotes.<\/p>\n\n\n\n

Tracing a curve- Cartesian form<\/strong><\/h2>\n\n\n\n

Let us the equation of a curve is f(x,y)=0 , now we will learn a few steps to simplify tracing of this curve.<\/p>\n\n\n\n

1. The first step is to find out the region of the plane. For example no point in the curve x= y\u00b2 in the second and third quadrant as we will always get a positive value on the x-axis. <\/p>\n\n\n\n

That means our curve will lie on the first and fourth quadrant only.<\/p>\n\n\n\n

2. The second step is to find out If the curve is symmetrical about any line or origin.<\/p>\n\n\n\n

Some examples of symmetrical curves are as below-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Steps to determine the symmetry of a curve<\/strong><\/h2>\n\n\n\n

1. If all the powers of x in function f(x,y)=0 are even, then f(x,y)= f(-x,y) and the curve is always symmetrical about the y-axis. Similarly we can draw the conclusion about x-axis<\/p>\n\n\n\n

2.If f(x,y)= 0 & f(-x,-y)= 0 , then the curve is symmetrical about the origin.<\/p>\n\n\n\n

3. If the equation of the curve does not change when we interchange x and y then it is symmetrical about the line y= x<\/p>\n\n\n\n

Let\u2019s understand this with the help of following table-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

4. The next step to determine the points where curve intersects the axes. <\/p>\n\n\n\n

If we put y = 0 in f(x, y)=0 and solve the equation, we get the points intersecting on x-axis. Similarly we get point on y- axis.<\/p>\n\n\n\n

5. Now we try to locate the points for discontinuity of the function.<\/p>\n\n\n\n

6. Calculate dy\/dx to locate the portion where the curve is rising(dy\/dx>0) or falling(dy\/dx<0)<\/p>\n\n\n\n

7. Calculate d\u00b2y\/dx\u00b2 to locate maxima and minima and the point of inflection<\/p>\n\n\n\n

For maxima = dy\/dx = 0,  d\u00b2y\/dx\u00b2<0  &     for minima = dy\/dx = 0,  d\u00b2y\/dx\u00b2>0<\/p>\n\n\n\n

For point of inflection –  d\u00b2y\/dx\u00b2 = 0<\/p>\n\n\n\n

8. The next step is to find the asymptotes,<\/p>\n\n\n\n

9. Another point to determine the singular point. The shape of the curve at these points generally more complex.<\/p>\n\n\n\n

10. Finally plot the points as many as we can. Also try to draw tangents to the curve at some points(calculate the derivative). Now join the plotted point by a smooth curve.<\/p>\n\n\n\n

Now we will understand curve tracing with some easy examples:<\/p>\n\n\n\n

Solved examples<\/h2>\n\n\n\n

Example-:  Trace the curve y = 1\/x\u00b2.<\/strong><\/p>\n\n\n\n

Sol.<\/strong>     <\/p>\n\n\n\n

As we can see y- coordinates of the curve can not be negative. So the curve must be above x-axis. The curve is also symmetric about y-axis so we can draw the graph only in single side.<\/p>\n\n\n\n

Here, we will find the first and second derivatives-<\/p>\n\n\n\n

 So, dy\/dx= -2\/x\u00b3 and d\u00b2y\/dx\u00b2 = 6\/ x\u2074 ,  here  dy\/dx <0 for all x>0 so we can say that the function is non- increasing so the graph falls as we increase x.<\/p>\n\n\n\n

Also second derivative is also non zero so there are no point of inflection.<\/p>\n\n\n\n

Here the curve is x\u00b2y=1 (rewritten), here both the axes are asymptotes of the curve.<\/p>\n\n\n\n

Here is the figure of the curve-:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Tracing a curve- parametric form<\/p>\n\n\n\n

Before we start tracing curves of the equations in parametric form, here first we understand the definition of parametric equations:<\/p>\n\n\n\n

Parametric equations:<\/p>\n\n\n\n

If x and y are the continuous functions of \u201ct\u201d on an interval I , then the equations:<\/p>\n\n\n\n

x = x(t)<\/p>\n\n\n\n

and<\/p>\n\n\n\n

y= y(t)<\/p>\n\n\n\n

are called parametric equations and t is called the parameter.<\/p>\n\n\n\n

Now let\u2019s understand how to trace the curves in parametric by examples:<\/p>\n\n\n\n

Example1: trace the curve of the following parametric equations:<\/p>\n\n\n\n

                  ,     x(t) = t-1 ,     y(t) = 2t+4         -3<\u2264t\u22642<\/p>\n\n\n\n

Sol. <\/strong> Here we will create the table for t, x(t) and y(t)  , t is independent variable in both case,<\/p>\n\n\n\n

      t<\/td>x(t)<\/td>y(t)<\/td><\/tr>
    -3<\/td>  -4<\/td>  -2<\/td><\/tr>
    -2<\/td>  -3 <\/td>  0<\/td><\/tr>
    -1<\/td>  -2<\/td>  2<\/td><\/tr>
      0<\/td>  -1<\/td>  4<\/td><\/tr>
      1<\/td>  0<\/td>  6<\/td><\/tr>
      2<\/td>  1<\/td>  8<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Here value of t lies between -3 to 2.<\/p>\n\n\n\n

By plotting these set of point we get a curve as below.<\/p>\n\n\n\n

Arrows in the graph indicates the orientation of the graph.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Tracing a curve- polar form<\/strong><\/h2>\n\n\n\n

To trace the polar curve  we follow the following steps:<\/p>\n\n\n\n

1. Symmetry \u2013 (a) <\/strong>if the equation is an even function of , then the curve is symmetrical about the initial line.<\/p>\n\n\n\n

(b) <\/strong>if the equation is an even function of r, then the curve is symmetrical about the origin\/<\/p>\n\n\n\n

(C ). <\/strong>If the equation remains same if we change   by  and r by \u2013r then the curve is symmetric about the line through the pole and perpendicular to the initial line.<\/p>\n\n\n\n

2. Region- <\/strong>Find the reason for for which r is defined and real<\/p>\n\n\n\n

3. Table- <\/strong>make table for the values of r determined by .<\/p>\n\n\n\n

4. Angle <\/strong><\/strong> –  <\/strong><\/strong> is the angle between the radius vector and tangent to the curve;<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Then angle tangents to the curve can be determined by angle <\/p>\n\n\n\n

5. Asymptotes-  <\/strong> find the asymptotes of the curve.<\/p>\n\n\n\n

Now we will understand of tracing of polar equations by an example:<\/p>\n\n\n\n

Example: Trace the curve <\/strong><\/strong><\/p>\n\n\n\n

Sol:<\/strong><\/p>\n\n\n\n

Here we can see clearly <\/strong><\/strong> hence the curve is symmetric about initial line.<\/strong><\/p>\n\n\n\n

Since <\/strong><\/strong>, the curve lies inside the circle r = 2a<\/strong><\/p>\n\n\n\n

 <\/strong><\/strong>, when <\/strong><\/strong>, thus r decreases as increases in the interval 0 to <\/strong><\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

this shows the angle between r and and the tangent is 0 or according to . Hence the line joining a point on the curve to the origin is orthogonal to the tangent when = 0 and coincides with <\/p>\n\n\n\n

From the above results, we can easily draw the graph above the initial line<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n