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Exponential graphs and circles - Higher

Exponential graphs and circles - Higher

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Summary

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 xx and yy.



Exponential graphs

Exponential equations have the form AxA^x or AxA^{-x} where AA is a positive constant. 


Note: Ax=1AxA^{-x}=\frac1{A^x} which is the same as (1A)x(\frac1A)^x. So really, AxA^x and AxA^{-x} do not actually represent different types of exponential graphs.


An example of an exponential graph is given below. This is y=2xy=2^x:​

Maths; Graphs; KS4 Year 10; Exponential graphs and circles - Higher

If AA is bigger than 11, then the exponential graph will have a very similar shape. They all pass the yy-axis at 11, and never touch the xx-axis. Another example is given below. This is y=(12)xy=(\frac12)^x (which is the same as y=2xy=2^{-x}):​

Maths; Graphs; KS4 Year 10; Exponential graphs and circles - Higher

If AA is between 00 and 11, it will have this shape. Notice that the curve still passes through (0,1)(0,1) and never touches the xx-axis. The difference is that it is reflected in the yy-axis.


Exponential graphs span the whole xx-axis, so any xx-coordinate inserted into an exponential equation will return a yy-coordinate. This is not true of yy-coordinates: notice that there are no points on the graph such that y0y\leq0.



Circles

Circles can be graphed using the general equation 

x2+y2=r2x^2+y^2=r^2


where rr 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:

Maths; Graphs; KS4 Year 10; Exponential graphs and circles - Higher

The equation of this circle is x2+y2=1x^2+y^2=1 since the radius is 11 and 12=11^2=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:

Maths; Graphs; KS4 Year 10; Exponential graphs and circles - Higher

You say that the tangent is at point AA​. 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 AA with coordinates (xA,yA)(x_A,y_A)​.
2.
Calculate the gradient of the radius to point AA​. This will be yAxA\frac{y_A}{x_A} (using m=change in ychange in xm=\frac{\text{change in }y}{\text{change in }x}) since the centre is at the origin.​
3.
The gradient of the tangent is the negative reciprocal of this. This is 1(yAxA)=xAyA-\frac{1}{(\frac{y_A}{x_A})}=-\frac{x_A}{y_A}​.


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)A(4,3)​ on the circle with equation

x2+y2=25x^2+y^2=25


You already have the point where the tangent touches the circle. Now the gradient of the radius. This is 34\frac34. The gradient of the tangent is the negative reciprocal of this. This is 1(34)=43-\frac1{(\frac34)}=\underline{-\frac43}​.​



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