# 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 #1 : Graphing Functions With Complex Numbers

Where would  fall on the number line?

to the left of

to the right of

Cannot be determined

at

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, where polar form expresses the result 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 .

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

Where does  fall on the number line?

Cannot be determined

To the left of 0

To the right of 0

At 0

To the left of 0

Explanation:

Imaginary numbers do not fall on the number line by definition, since they are not real numbers. However, although i is an imaginary number equal to the square root of -1,  is a real number since . Therefore, . Negative numbers fall to the left of 0 on a number line.

### Example Question #5 : Graphing Functions With Complex Numbers

Which complex number does this graph represent?

Real numbers are represented by the x-axis, and imaginary numbers are represented by the y-axis.

Explanation:

In complex numbers of the form , a represents the real portion of the number and b represents the imaginary portion of the number. To graph  on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 2 units right and 3 units above the origin, so the complex number represented is .

### Example Question #6 : Graphing Functions With Complex Numbers

Which complex number does this graph represent?

Real numbers are represented by the x-axis, and imaginary numbers are represented by the y-axis.

Explanation:

In complex numbers of the form , a represents the real portion of the number and b represents the imaginary portion of the number. To graph  on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 5 units left and 2 units below the origin, so the complex number represented is .

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

Which of the following represents the real component of the complex number ?

Explanation:

In complex numbers of the form , a represents the real portion of the number and b represents the imaginary portion of the number. In the complex number and .

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

Which of the following represents the imaginary component of the complex number -3 + ki, in which k is a constant?

Explanation:

In complex numbers of the form represents the real portion of the number and  represents the imaginary portion of the number. In the complex number and

### Example Question #9 : Graphing Functions With Complex Numbers

Which complex number does this graph represent?

Real numbers are represented by the x-axis, and imaginary numbers are represented by the y-axis.

Explanation:

In complex numbers of the form , a represents the real portion of the number and b represents the imaginary portion of the number. To graph  on a plane in which real numbers are represented by the x-axis and imaginary numbers are represented by the y-axis, place a point a units right of the origin and b units above the origin. The graph shows a point 4 units right and 7 units below the origin, so the complex number represented is .

### Example Question #10 : Graphing Functions With Complex Numbers

Which complex number does this graph represent?

Real numbers are represented by the x-axis, and imaginary numbers are represented by the y-axis.