Exponential graphs and circles
In a nutshell
Equations of lines and curves show relationships. One such relationship is an exponential relationship, which tracks a very small positive number to infinity, or vice versa. Some curves look like shapes, for example, circles can be drawn using equations. These involve quadratic terms in both x and y.
Exponential graphs
Exponential equations have the form Ax or A−x where A is a positive constant.
Note: A−x=Ax1 which is the same as (A1)x. So really, Ax and A−x do not actually represent different types of exponential graphs.
An example of an exponential graph is given below. This is y=2x:
If A is bigger than 1, then the exponential graph will have a very similar shape. They all pass the y-axis at 1, and never touch the x-axis. Another example is given below. This is y=(21)x (which is the same as y=2−x):
If A is between 0 and 1, it will have this shape. Notice that the curve still passes through (0,1) and never touches the x-axis. The difference is that it is reflected in the y-axis.
Exponential graphs span the whole x-axis, so any x-coordinate inserted into an exponential equation will return a y-coordinate. This is not true of y-coordinates: notice that there are no points on the graph such that y≤0.
Circles
Circles can be graphed using the general equation
x2+y2=r2
where r is a constant and is the radius of the circle.
Note: Circles using this equation will have their centre at the origin.
An example is given below:
The equation of this circle is x2+y2=1 since the radius is 1 and 12=1.
Finding the gradient of a tangent to a circle
A tangent is a line that touches a curve at one place, then continues on. For example, a tangent to a curve is given below:
You say that the tangent is at point A. Since tangents are straight lines they have the same gradient everywhere. At the point where a tangent touches the circle the tangent is perpendicular to the radius. To calculate the gradient of the tangent follow the procedure below:
Procedure
1.
| Identify the point on the circle where the tangent touches. Call this point A with coordinates (xA,yA). |
2.
| Calculate the gradient of the radius to point A. This will be xAyA (using m=change in xchange in y) since the centre is at the origin. |
3.
| The gradient of the tangent is the negative reciprocal of this. This is −(xAyA)1=−yAxA. |
Note: The reciprocal of a fraction is the same fraction but with the numerator and the denominator swapped.
Example
Find the gradient of the tangent at point A(4,3) on the circle with equation
x2+y2=25
You already have the point where the tangent touches the circle. Now the gradient of the radius. This is 43. The gradient of the tangent is the negative reciprocal of this. This is −(43)1=−34.