Curved Line - Types of Curved Lines & Examples
By the end of this lesson, you will be able to
- define curved line.
- describe types of curves.
- demonstrate examples of curved lines.
- distinguish between simple and non-simple curved lines.
- compare open and closed curve.
- relate line and curve.
- interpret algebraic curve and quadratic curve.
Curved line or curve
A curved line or curve is a smoothly-flowing line that line need not to be necessarily straight.
Generally speaking, a curve means a line that must bend. That is, a curve is a line that always changes its direction. Again, different type of mathematical curves change their direction in different fashion.
But in geometry, a line is strictly straight. A curve is a generalization of line.
As, in mathematics, the curved line is a path that need not to be necessarily bent. This implies that a curve may be straight path. Thus a curved line may be a straight path or bent.
So, a curve is a generalization of a straight line.
Another way, a line or straight line is a special form of a curve.
Thus a curve with curvature zero is a straight line.
Types of Curved Lines
A list of different kinds of curved lines are given below.
- Open Curve
- Closed Curve
- General Curve
- horizontal curve
- simple curve
- compound curve
- reverse curve
- spiral curve
- Algebraic Curve
- Quadratic Curve
- cubic curve
- transcendental curve
- parabolic curve
- polynomial curve
- hyperbolic curve
- square root curve
- log curve
Open Curve
A curve is said to be open if its endpoints don't meet.
A parabola is a perfect example of an open curved line.
The end points never meet to each other.
Closed Curve
A curve is said to be closed if the starting point of the curve is same as its endnig point.
Ellipse and circle are the suitable examples of a closed curve.
General Curve
A general curve means the curved line that represents all the curves of same shape and figure. That is, a general curve represents all curves of same shape and same look.
Examples of Curved Lines
A curve is shown in the figure. This is a curve of parabola. The equation of this parabola is y = ax2 where a>0.
Its general equation is y = ax2 where a≠0.
If one draws this parabola on a piece of graph paper, some graphs would be too narrow and some would be too wide. It completely depends on the different values of a.
If the values of a increases, the graph gets narrow. On the other hand, if the values of a decreases, the graph gets wide to wider. You can have a check by drawing graph of y = ax2 for different values of a.
If you want to dive deep into general graph, let us go and start the next section.
Consider the general equation of curved line, y = axn where a>0 and n is even. It represents several graphs depending on the different values of a and n.
General curve of parabola: case 1
Let the general equation of parabola be y = axn where a>0 and n is even. It represents all parabolic graphs of same look but some of them would be wider or narrow. For a>0, the parabolas are opening to the top.
More precisely, if a = 1 and n = 2, the equation takes the form:
y = x2.
Similarly,
if a = 2, 3, 4, 5, ... and n = 4, 6, 8, 10, ... respectively, the equation takes the form:
y = 2x4,
y = 3x6,
y = 4x8,
y = 5x10
respectively.
We can arrange several values of a and n; and the resulting equations above on a table.
| a; a>0 | n; n is even | y = axn |
|---|---|---|
| 1 | 2 | y = x2 |
| 2 | 4 | y = 2x4 |
| 3 | 6 | y = 3x6 |
| 4 | 8 | y = 4x8 |
| 5 | 10 | y = 5x10 |
| ... | ... | ... |
General curve of parabola: case 2
Consider the general equation, y = axn where a<0 and n is even. It produces several equations of graph depending upon for some values of a and n.
Let the general equation of parabola be y = axn where a<0 and n is even. For a<0, the parabolic graphs are opening to the bottom.
Set some values for a and n.
Set a = -1, -2, -3, -4, -5, ... and n = 2, 4, 6, 8, 10, ... respectively, the equation takes the form:
y = -x2,
y = -2x4,
y = -3x6,
y = -4x8,
y = -5x10
respectively.
Arrange all values of a and n; and the resulting equations above on a table.
| a; a<0 | n; n is even | y = axn |
|---|---|---|
| -1 | 2 | y = -x2 |
| -2 | 4 | y = -2x4 |
| -3 | 6 | y = -3x6 |
| -4 | 8 | y = -4x8 |
| -5 | 10 | y = -5x10 |
| ... | ... | ... |
General curve of parabola: case 3
Consider the general equation, x = ayn where a>0 and n is even. For some values of a and n, it produces a set of equations of graphs.
Let the general equation of parabola be x = ayn where a>0 and n is even. It gives all parabolic graphs of same look but for some values of a; some of them would be wider or narrow. For a>0, the parabolas are opening to the right.
More precisely,
if a = 1, 2, 3, 4, 5, ... and n = 2, 4, 6, 8, 10, ... respectively, the equation takes the form:
x = y2
x = 2y4
x = 3y6
x = 4y8
x = 5y10
respectively.
We can accumulate all values of a and n; and the resulting equations above on a table.
| a; a>0 | n; n is even | x = ayn |
|---|---|---|
| 1 | 2 | x = y2 |
| 2 | 4 | x = 2y4 |
| 3 | 6 | x = 3y6 |
| 4 | 8 | y = 4x8 |
| 5 | 10 | x = 5y10 |
| ... | ... | ... |
General curve of parabola: case 4
Consider the general equation, x = ayn where a<0 and n is even. It produces several equations of graph depending on for some values of a and n.
Let the general equation of parabola be x = ayn where a<0 and n is even. For a<0, the parabolic graphs are opening to the left.
Consider some values for a and n.
Set a = -1, -2, -3, -4, -5, ... and n = 2, 4, 6, 8, 10, ... respectively, the equation takes the form:
x = -y2,
x = -2y4,
x = -3y6,
x = -4y8,
x = -5y10
respectively.
Set the values of a and n; and the resulting equations above on a table.
| a; a<0 | n; n is even | x = ayn |
|---|---|---|
| -1 | 2 | x = -y2 |
| -2 | 4 | x = -2y4 |
| -3 | 6 | x = -3y6 |
| -4 | 8 | x = -4y8 |
| -5 | 10 | x = -5y10 |
| ... | ... | ... |
General curve: case 5
Consider the general equation, y = axn where a>0 and n is odd. It represents several equations of graphs depending on the different values of a and n.
Suppose the equation, y = axn where a>0 and n is odd. It gives all graphs of same look. For a = 1 and n = 1, it takes the form: y = x.
If you draw y = x, it will be drawn a straight line that is also a special curved line. That is, a straight line is always a curve with curvature zero. This line passes through the origin (0,0). And it also goes through ..., (-2,-2), (-1,-1), (1,1), (2,2), ... .
Again, for a = 1 and n = 3, 5, 7, 9, ..., the equation y = axn takes the form:
y = x3,
y = x5,
y = x7,
y = x9
respectively.
All the graphs of the equation above also passes through (-1,-1), (0,0), (1,1). This is a wonderful characteristics of the graph of the equations for a = 1 and for odd n.
Now, find out the equations for different values of a.
Set,
a = 1, 2, 3, 4, 5, ... and n = 1, 3, 5, 7, 9, ... respectively.
Then the equation takes the form:
y = x,
y = 2x3,
y = 3x5,
y = 4x7,
y = 5x9
respectively.
Now arrange a, n and resulting equations on a table.
| a; a>0 | n; n is odd | y = axn |
|---|---|---|
| 1 | 1 | y = x |
| 2 | 3 | y = 2x3 |
| 3 | 5 | y = 3x5 |
| 4 | 7 | y = 4x7 |
| 5 | 9 | y = 5x9 |
| ... | ... | ... |
If you draw graph of the any equation on the table, all of them look same shapes. Moreover, all the graphs always start from negative x to positive x and from negative y to positive y.
General curve: case 6
Let the general equation be y = axn where a<0 and n is odd. It produces number of equations of graphs depending on the different values of a and n.
Considering the equation, y = axn where a<0 and n is odd, it represents all graphs of same shape style. For a = -1 and n = 1 odd, it takes the form: y = -x.
If someone draws y = -x, its graph will be a straight line that is also a special cuved line. That is, a straight line, we have previously discussed already, is always a curved line with curvature zero. This straight line passes through the origin (0,0). And it also passes through ...,(2,-2), (1,-1), (-1,1), (-2,2),... .
It goes from positive x to negative x and negative y to positive y.
Again, for a = -1 and n = 3, 5, 7, 9, ..., the equation y = axn takes the form:
y = -x3
y = -x5
y = -x7
y = -x9
respectively.
All the graphs above also passes through (-1,1), (0,0), (1,-1). This is a fantastic property for the graph of the equations for a = -1 and for odd n.
Now, one can have a check the equations for different values of a.
Set,
a = -1, -2, -3, -4, -5, ... and n = 1, 3, 5, 7, 9, ... respectively.
Then the equation produces several equations:
y = -x
y = -2x3,
y = -3x5,
y = -4x7,
y = -5x9
respectively.
Let arrange a, n and resulting equations on a table.
| a; a<0 | n; n is odd | y = axn |
|---|---|---|
| -1 | 1 | y = -x |
| -2 | 3 | y = -2x3 |
| -3 | 5 | y = -3x5 |
| -4 | 7 | y = -4x7 |
| -5 | 9 | y = -5x9 |
| ... | ... | ... |
If you draw graph of the any equation on the table, all of them look same shapes. Again, all the graphs always start from positive x to negative x and from negative y to positive y.
General curve: case 7
Consider the general equation, x = ayn where a>0 and n is odd. It generates some equations of graphs for different values of a and n.
Suppose the equation, x = ayn where a>0 and n is odd. It represents all graphs of same look. For a = 1 and n = 1, it takes the form: x = y.
If you draw x = y, it produces a straight line. But we know a straight line is also a curved line. That is, a straight line is always a curve with curvature zero. This line passes through the origin (0,0). And it also goes through ...,(-2,-2), (-1,-1), (1,1), (2,2),... .
Again, for a = 1 and n = 3, 5, 7, 9, ..., the equation x = ayn takes the form:
x = y3,
x = y5,
x = y7,
x = y9
respectively.
All the graphs of the equation above also passes through (-1,-1), (0,0), (1,1). This is a wonderful characteristics of x = ayn for a = 1 and for odd n.
Now, find out the equations for different values of a.
Set,
a = 1,2,3,4,5,... and n = 1, 3, 5, 7, 9, ... respectively.
Then the equation takes the form:
x = y,
x = 2y3,
x = 3y5,
x = 4y7,
x = 5y9
respectively.
Now arrange a, n and resulting equations on a table.
| a; a>0 | n; n is odd | x = ayn |
|---|---|---|
| 1 | 1 | x = y |
| 2 | 3 | x = 2y3 |
| 3 | 5 | x = 3y5 |
| 4 | 7 | x = 4y7 |
| 5 | 9 | x = 5y9 |
| ... | ... | ... |
One can draw graph of the any equation on the table, all of them look same shapes. Moreover, all the graphs always start from negative x to positive x and from negative y to positive y.
General curve: case 8
Consider the general equation, x = ayn where a<0 and n is odd. It renders unique equation of graph for unique value of a and n.
Assume the general equation, x = ayn where a<0 and n is odd. It represents all graphs of same look. For a = -1 and n = 1 odd, it takes the form: x = -y.
If someone draws x = -y, it renders a straight line that is also a curve. That is, a line is always a curved line. This line passes through the origin (0,0). And it also goes through ...,(-2,2), (-1,1), (1,-1), (2,-2),... .
Again, for a = -1 and n = 3, 5, 7, 9, ..., the equation x = ayn takes the form:
x = -y3,
x = -y5,
x = -y7,
x = -y9
respectively.
All the graphs rendered by the equations above also passes through (-1,1), (0,0), (1,-1). This is an exceptional properties of x = ayn for a = -1 and for odd n.
Now, let us produce some equations for several values of a.
Set,
a = -1, -2, -3, -4, -5, ... and n = 1, 3, 5, 7, 9, ... respectively.
Then the equation takes the form:
x = -y,
x = -2y3,
x = -3y5,
x = -4y7,
x = -5y9
respectively.
Now arrange a, n and resulting equations on a table.
| a; a<0 | n; n is odd | x = ayn |
|---|---|---|
| -1 | 1 | x = -y |
| -2 | 3 | x = -2y3 |
| -3 | 5 | x = -3y5 |
| -4 | 7 | x = -4y7 |
| -5 | 9 | x = -5y9 |
| ... | ... | ... |
One can draw graph of the any equation on the table, all of them look same shapes. Moreover, all the graphs always start from positive x to negative x and from negative y to positive y.
Simple Curve
The curve that doesn't intersect itself is called simple curve. Different curves go on different fashion. Some of them are self-intersecting and some are not. A simple curve never intersects itself.
On the other hand, the curve that intersects itself is called a non-simple curve.
Algebraic Curve
A plane algebraic curve is a curve formed by the locus of points whose coordinates are the roots of the polynomial in two variables.
Another way...
A plane algebraic curved line is a path formed by the set of points whose coordinates x and y satisfy P(x,y)=0 where P(x,y) is a polynomial in two variables.
A circle with center (0,0) of radius 4; that is x2 + y2 = 16, for example, is perfectly an algebraic curve because it contains the set of roots of the polynomial x2 + y2 - 16.
Here the coordinates (-4,0) and (4,0) satisfies the polynomial P(x,y) = x2 + y2 - 16 independently.
That is, P(-4,0) = 0 and P(-4,0) = 0.
Quadratic Curve
A one variable quadratic function in the variable x is the form:
f(x) = ax2 + bx + c; a ≠ 0.
This is the standard quadratic function of single variable.
When f(x) is set to 0, it takes take the form:
f(x) = 0. That is,
ax2 + bx + c = 0
This equation is called the quadratic equation and well known as standard quadratic equation. The solution of the quadratic equation is called the roots of the quadratic function. The quadratic curved line shows two roots of the quadratic equation. The one variable or single variable function is also well known as univariate function. And the single variable equation is familiar to univariate equation.
If the quadratic equation is drawn on a piece of graph paper, it is a curve of parabola.
The axis of symmetry of this parabolic curve is always parallel to y-axis.
A quadratic function of two variables x and y is the form:
f(x,y) = ax2 + by2+ hxy + gx + fy + c
at least one of a,b,h is not equal to 0.
If f(x,y) is set to zero, it is the form:
f(x,y) = 0. That is
ax2 + by2+ hxy + gx + fy + c = 0
This equation represents the conic section consists of circle, ellipse, parabola and hyperbola.
