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 $A^x$ or $A^{-x}$ where $A$ is a positive constant.

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

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

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

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 $y=(\frac12)^x$ (which is the same as $y=2^{-x}$ ):

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

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 $y\leq0$ .

Circles

Circles can be graphed using the general equation

$x^2+y^2=r^2$

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:

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

The equation of this circle is $x^2+y^2=1$ since the radius is $1$ and $1^2=1$ .

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 $A$ . 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 $(x_A,y_A)$ .
2.	Calculate the gradient of the radius to point $A$ . This will be $\frac{y_A}{x_A}$ (using $m=\frac{\text{change in }y}{\text{change in }x}$ ) since the centre is at the origin.
3.	The gradient of the tangent is the negative reciprocal of this. This is $-\frac{1}{(\frac{y_A}{x_A})}=-\frac{x_A}{y_A}$ .

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 $A(4,3)$ on the circle with equation

$x^2+y^2=25$

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