# Algebra II : Graphing Functions with Complex Numbers

## Example Questions

### Example Question #1 : Graphing Functions With Complex Numbers

Solve for

Explanation:

Use the change of base formula for logarithmic functions and incorporate the fact that  and

Or

can be solved using

### Example Question #2 : Graphing Functions With Complex Numbers

Where would  fall on the number line?

to the left of

Cannot be determined

at

to the right of

Cannot be determined

Explanation:

Imaginary numbers do not fall on the number line-- they are by definition not real numbers.

** If the question asked where  falls on the number line, the answer would be to the left of 0, because .

### Example Question #3 : Graphing Functions With Complex Numbers

Write the complex number  in polar form, that is, in terms of a distance from the origin  on the complex plane and an angle from the positive -axis, , measured in radians.

Explanation:

To see what the polar form of the number is, it helps to draw it on a graph, where the horizontal axis is the imaginary part and the vertical axis the real part. This is called the complex plane.

To find the angle , we can find its supplementary angle  and subtract it from  radians, so .

Using trigonometric ratios,    and  .

Then .

To find the distance , we need to find the distance from the origin to the point . Using the Pythagorean Theorem to find the hypotenuse  or .