TRACING OF CURVES
Given the equation of a curve explicitly as 𝑦 = 𝑓(𝑥) or implicitly as 𝑔(𝑥, 𝑦) = 𝑐 , a constant, many
properties of the curve can be determined easily by knowing its graph. Here we will study the method of
tracing a curve whose equation is given in Cartesian, polar or parametric equations.
Procedure for tracing curves of Cartesian equation.
1. Symmetry
The various kinds of symmetry arising from the form of the equation are as follows:
• i) If the equation of the curve has an even power of 𝑥, then the curve is symmetric about the
𝑦 -axis.
• ii) If the equation of the curve has an even power of 𝑦, then the curve is symmetric about xaxis.
• iii) If the equation of the curve has an even power of 𝑥 and 𝑦, then the curve is symmetric
about the origin.
2. Domain
Region of existence can be determined by finding out the set of permissible (real) values of 𝑥
and 𝑦. The curve does not lie in the region whenever 𝑥 or 𝑦 is imaginary.
3. Intercept
X-intercept Substitute 𝑥 = 0 in the equation of the curve and solve for y.
Y- intercept Substitute 𝑦 = 0 and solve for 𝑥.
4. Origin
If we get 𝑓(0,0) = 0 then the curve passes through the origin.
5. Tangents at the origin
A double point is a point through which two branches of the curve pass.
• A double point is called a cusp if two branches of the curve have the real and same tangents.
• A double point is called a node if two branches of the curve have real and distinct tangents.
• The equations of the tangents to the curve at the origin is obtained by equating the lowest
degree terms in x and y in the given equation to zero, provided the curve passes through the
origin.
6. Asymptotes
1. If 𝑓(𝑥) = 𝑎 then 𝑦 = 𝑎 is a horizontal asymptote
2. If 𝑓(𝑥) = ∞ then 𝑥 = 𝑎 is a vertical asymptote
7. Special points
If we need some more information about the curve then we need special points.
Example:
Trace the curve 𝑦 2 (𝑎 − 𝑥) = 𝑥 3 , 𝑎 > 0
Soln: The given equation is 𝑦 2 (𝑎 − 𝑥) = 𝑥 3
1. Symmetry: Since equation contains only even power of 𝑦 therefore the curve is symmetrical
about the x-axis.
2. Domain:
𝑥3
𝑎−𝑥
Since 𝑦 = ±√
𝑦 is defined if 𝑥 is in [0,a) otherwise not. So curves exist in [0,a).
3. Intercept:
x- intercept is 0
Y- intercept is 0
4. Origin: The curve is passes through (0,0) ie 𝑓(0,0) = 0
5. Tangents at the origin:
Since the curve passes through the origin. Now we will check tangents at origin.
The lowest degree term is 𝑎𝑦 2 . When we equate this to zero we get two coincident
tangents at (0,0). That is 𝑦 = 0 which is the equation of the x-axis and also the given curve
is symmetrical about x-axis. Therefore we get cusp at origin.
6. Asymptotes:
Here we get 𝑥 = 𝑎 is a vertical asymptote and we don’t have any horizontal asymptote.
Example:
Trace the curve 𝑦 2 (𝑎 − 𝑥) = 𝑥 2 (𝑎 + 𝑥), 𝑎 > 0
Soln: The given equation is 𝑦 2 (𝑎 − 𝑥) = 𝑥 3
1. Symmetry: Since equation contains only even power of 𝑦 therefore the curve is symmetrical
about the x-axis.
2. Domain:
𝑎+𝑥
Since 𝑦 = ±𝑥√𝑎−𝑥 𝑦 is defined if 𝑥 is in [-a,a) otherwise not. So curves exist in [-a,a).
3. Intercept:
x- intercepts are −𝑎, 0
Y- intercept is 0
4. Origin:
The curve is passes through (0,0) ie 𝑓(0,0) = 0
𝑥-intercept at (−𝑎, 0)
5. Tangents at the origin:
Since the curve passes through the origin. Now we will check tangents at origin.
The lowest degree term is 𝑎𝑦 2 − 𝑎𝑥 2 . When we equate this to zero we get two distinct
tangents at (0,0). That is 𝑦 = ±𝑥. Therefore we get node at origin.
6. Asymptotes:
Here we get 𝑥 = 𝑎 is a vertical asymptote and we don’t have any horizontal asymptote.
Procedure for tracing curves of parametric function 𝑥 = 𝑓(𝜃) & 𝑦 = 𝑔(𝜃)
1. Symmetry:
If x is even and y is an odd function then the curve is symmetrical about the x-axis.
If x is odd and y is even function then the curve is symmetrical about the y-axis.
2. Origin:
If we get 𝑥 = 0 and 𝑦 = 0 for some value of 𝜃 = 0 then the curve passes through the origin and
finds points on the coordinate axis.
3. Tangents:
𝑑𝑦
𝑑𝑦/𝑑𝜃
If 𝑑𝑥 = 𝑑𝑥/𝑑𝜃 = 0 at(ℎ, 𝑘) then tangent is parallel to x-axis
If
𝑑𝑦
𝑑𝑥
=
𝑑𝑦/𝑑𝜃
𝑑𝑥/𝑑𝜃
= ∞ at(ℎ, 𝑘) then tangent is perpendicular to x-axis
4. Region of existence: Find the lowest and highest value of y.
5. Points: obtain 𝑥, 𝑦, 𝑑𝑦/𝑑𝑥 value at some successive point of t.
Example: Trace the curve 𝑥 = 𝑎(𝑡 + 𝑠𝑖𝑛𝑡), 𝑦 = 𝑎(1 − 𝑐𝑜𝑠𝑡)
Soln: The given equation is 𝑥 = 𝑎(𝑡 + 𝑠𝑖𝑛𝑡), 𝑦 = 𝑎(1 − 𝑐𝑜𝑠𝑡)
1. Symmetry:
The curve is symmetrical about y axis.
2. Tangents:
𝑑𝑥
𝑑𝑡
= 𝑎(1 + 𝑐𝑜𝑠𝑡),
𝑑𝑦
𝑑𝑡
= 𝑎𝑠𝑖𝑛𝑡 →
𝑑𝑦
𝑑𝑥
=
𝑎𝑠𝑖𝑛𝑡
2
𝑡
2
𝑡
2
= 𝑡𝑎𝑛( )
3. Origin:
The curve is passing through the origin.
4. Largest value of y is 2𝑎.
𝑡
0
𝑥
0
𝑦
𝑑𝑦
𝑑𝑥
0
0
𝜋
2
𝜋
𝑎( 2 + 1)
𝑎
1
𝜋
𝑎𝜋
2𝑎
∞
3𝜋
2
3𝜋
𝑎( + 1)
2
𝑎
−1
2𝜋
2𝑎𝜋
0
0
Example: Trace the curve 𝑥 2/3 + 𝑦 2/3 = 𝑎2/3 or 𝑥 = 𝑡 , 𝑦 = 𝑎𝑠𝑖𝑛3 𝑡
Soln: The given equation is 𝑥 = 𝑡 , 𝑦 = 𝑎𝑠𝑖𝑛3 𝑡
1. Symmetry
The curve is symmetrical about x axis
2. Tangents:
𝑑𝑥
𝑑𝑡
= −3𝑡 𝑠𝑖𝑛𝑡,
𝑑𝑦
𝑑𝑡
= 3𝑡𝑐𝑜𝑠𝑡
→
𝑑𝑦
𝑑𝑥
= 𝑡𝑎𝑛𝑡
3. Origin:
The curve does not pass through the origin.
4. Largest value of r is 𝑎.
5. Table
𝑡
0
𝑥
𝑦
𝑑𝑦
𝑑𝑥
𝑎
0
0
𝜋
2
0
𝑎
−∞
𝜋
−𝑎
0
0
3𝜋
2
0
−𝑎
∞
2𝜋
𝑎
0
0
Procedure for tracing polar curves
1. Symmetry:
A curve is symmetrical about the initial line OX, if it remains unchanged when 𝜃 is changed to – 𝜃
A curve is symmetrical about the initial line OY, if it remains unchanged when 𝜃 is changed to 𝜋– 𝜃
2. Pole:
If r=0 for some value of theta then we say that the curve is passing through the pole.
3. Largest value of r
4. points:
Obtain value of 𝑟 at some successive point of 𝜃.
Example: Trace the cardioid 𝑟 = 𝑎(1 − 𝑐𝑜𝑠𝜃)
0 ≤ 𝜃 ≤ 2𝜋
Soln:
1. Symmetry
When 𝜃 is changed to (−𝜃) the equation remains unaltered. Therefore the curve is symmetrical about
the initial line .
2. Pole
We have 𝑟 = 0 for 𝜃 = 0 𝑜𝑟 2𝜋. Therefore curve passes through the pole at which we get tangent line
at 𝜃 = 0 𝑜𝑟 2𝜋.
3. Largest value of 𝑟 is 2𝑎
4. Table
From the given equation we get the value of 𝑟 for some values of 𝜃 are tabulated below
𝜃
𝑟
0
0
𝜋/4
𝑎(1 − 1/√2)
𝜋/2
𝑎
2𝜋/3
3𝑎/2
𝜋
2𝑎
3𝜋/2
𝑎
2𝜋
0