Math worksheet for practicing complex number operations, including absolute value calculations and graphing on the complex plane.

Problem Analysis:

The task involves working with complex numbers. Specifically, it requires:
1. Finding the absolute value of each given complex number.
2. Graphing complex numbers on the complex plane.
3. Identifying complex numbers from their graphs.

Let's solve each part step by step.

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Part 1: Find the Absolute Value of Each Complex Number

The absolute value (or modulus) of a complex number \( z = a + bi \) is given by:
\[
|z| = \sqrt{a^2 + b^2}
\]
where \( a \) is the real part and \( b \) is the imaginary part.

#### 1) \( |3 - 5i| \)
Here, \( a = 3 \) and \( b = -5 \).
\[
|3 - 5i| = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}
\]

#### 2) \( |-1 + 5i| \)
Here, \( a = -1 \) and \( b = 5 \).
\[
|-1 + 5i| = \sqrt{(-1)^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26}
\]

#### 3) \( |5 + 3i| \)
Here, \( a = 5 \) and \( b = 3 \).
\[
|5 + 3i| = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}
\]

#### 4) \( |3 + 2i| \)
Here, \( a = 3 \) and \( b = 2 \).
\[
|3 + 2i| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}
\]

Final Answers for Part 1:

\[
\boxed{\sqrt{34}, \sqrt{26}, \sqrt{34}, \sqrt{13}}
\]

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Part 2: Graph Each Complex Number

To graph a complex number \( z = a + bi \):
- The real part \( a \) corresponds to the horizontal (real) axis.
- The imaginary part \( b \) corresponds to the vertical (imaginary) axis.

#### 5) \( -1 - i \)
- Real part: \( -1 \)
- Imaginary part: \( -1 \)
- Plot the point at \( (-1, -1) \).

#### 6) \( 2 + 4i \)
- Real part: \( 2 \)
- Imaginary part: \( 4 \)
- Plot the point at \( (2, 4) \).

Graphs for Part 2:

- For \( -1 - i \): The point is in the third quadrant at \( (-1, -1) \).
- For \( 2 + 4i \): The point is in the first quadrant at \( (2, 4) \).

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Part 3: Identify Each Graphed Complex Number

#### 7)
- The point is located at \( (3, -2) \).
- This corresponds to the complex number \( 3 - 2i \).

#### 8)
- The point is located at \( (-2, 3) \).
- This corresponds to the complex number \( -2 + 3i \).

Final Answers for Part 3:

\[
\boxed{3 - 2i, -2 + 3i}
\]

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Summary of All Answers:

1. \( \sqrt{34}, \sqrt{26}, \sqrt{34}, \sqrt{13} \)
2. Graphs for \( -1 - i \) and \( 2 + 4i \) are plotted as described.
3. Identified complex numbers from graphs: \( 3 - 2i \) and \( -2 + 3i \).

\[
\boxed{\sqrt{34}, \sqrt{26}, \sqrt{34}, \sqrt{13}, 3 - 2i, -2 + 3i}
\]