Algebra II - Graphing Functions with Complex Numbers

Solve for

$\ln 1=x$

Explanation

Use the change of base formula for logarithmic functions and incorporate the fact that $\ln 1=0$ and $e^{\ln }=1$

$\log_ax=\frac{\log_bx}{\log_ba}$

$\log_ax=y$ can be solved using

$\ln 1=x$

$e^{x}=1$

Write the complex number $1-\sqrt{3}i$ 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, $\theta$ , measured in radians.

$ewline r=2 ewline ewline \theta = \frac{5 \pi}{6}$

$ewline r=1 ewline ewline \theta = \frac{\sqrt{3} \pi}{6}$

$ewline r=\sqrt{3} ewline ewline \theta =2 \pi$

$ewline r=2 ewline ewline \theta = \sqrt{3} \pi$

$ewline r=\frac{1}{\sqrt{3}} ewline ewline \theta = \frac{3 \pi}{5}$

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 $\theta$ , we can find its supplementary angle $\Psi$ and subtract it from $\pi$ radians, so $\theta = \pi - \Psi$ .

Using trigonometric ratios, $\tan \Psi = \frac{1}{\sqrt{3}}$ and $\arctan({\frac{1}{\sqrt{3}}})= \Psi = \frac{\pi}{6}$ .

Then $\theta = \frac{5\pi}{6}$ .

To find the distance , we need to find the distance from the origin to the point $(-\sqrt{3}, 1)$ . Using the Pythagorean Theorem to find the hypotenuse , $r = \sqrt{ \sqrt{3}^2 + 1^2 }$ or .

Where would fall on the number line? $(i^{2}= -1 )$

Cannot be determined

to the right of

to the left of

Explanation

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

** If the question asked where $4i^{2}$ falls on the number line, the answer would be to the left of 0, because $4i^{2}=-4$ .

Where does fall on the number line?

To the right of 0

To the left of 0

At 0

Cannot be determined

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.

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.

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 .

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.

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 .

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 .

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

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.

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 .

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.

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 .

Graphing Functions with Complex Numbers

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