Algebra II : Graphing Functions with Complex Numbers

Study concepts, example questions & explanations for Algebra II

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Graphing Functions With Complex Numbers

Solve for 

Possible Answers:

Correct answer:

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? 

Possible Answers:

to the left of 

to the right of 

at 

Cannot be determined

Correct answer:

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.

Possible Answers:

Correct answer:

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.

Vecii

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 #4 : Graphing Functions With Complex Numbers

Where does  fall on the number line?

Possible Answers:

At 0

To the right of 0

To the left of 0

Cannot be determined

Correct answer:

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?

Screen shot 2020 08 26 at 9.25.41 am

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

Possible Answers:

Correct answer:

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?

Screen shot 2020 08 26 at 9.27.34 am

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

Possible Answers:

Correct answer:

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 #7 : Graphing Functions With Complex Numbers

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

Possible Answers:

Correct answer:

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 #8 : 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? 

Possible Answers:

Correct answer:

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?

Screen shot 2020 08 26 at 9.29.38 am

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

Possible Answers:

Correct answer:

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?

Screen shot 2020 08 26 at 9.31.52 am

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

Possible Answers:

Correct answer:

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 8 units left and 1 unit above the origin, so the complex number represented is .

Learning Tools by Varsity Tutors