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College Algebra Study Guide: Coordinate Geometry and Set Theory

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Q1. State the coordinates of each of the points shown in the Cartesian plane.

Background

Topic: Rectangular Coordinate System (Cartesian Plane)

This question tests your understanding of how to read and write the coordinates of points on the Cartesian plane, which is foundational for graphing and analyzing equations in College Algebra.

Key Terms:

  • Cartesian Plane: A two-dimensional plane defined by a horizontal x-axis and a vertical y-axis.

  • Coordinates: An ordered pair representing the horizontal and vertical positions of a point relative to the origin (0, 0).

Step-by-Step Guidance

  1. For each labeled point, identify its position on the x-axis (horizontal) and y-axis (vertical).

  2. Write the coordinates as an ordered pair , where is the horizontal value and is the vertical value.

  3. Remember: If a point is to the left of the origin, is negative; if below the origin, is negative.

  4. Repeat this process for each point (A, B, C, D, E, F, G, P, Q).

Try solving on your own before revealing the answer!

Final Answer:

A (3, 2); B (5, 5); C (7, 8); D (−6, 4); E (−3, 7); F (−4, −6); G (−8, −8); P (9, −9); Q (6, −5)

Each coordinate pair correctly identifies the position of the point on the Cartesian plane.

Q2. Find the distance between the points.

Background

Topic: Distance Formula in the Coordinate Plane

This question tests your ability to use the distance formula to calculate the straight-line distance between two points in the plane.

Key Formula:

  • and are the coordinates of the two points.

Step-by-Step Guidance

  1. Label the coordinates of the two points as and .

  2. Subtract from to find the difference in the x-coordinates, and from for the y-coordinates.

  3. Square both differences: and .

  4. Add the squared differences together.

  5. Take the square root of the sum to find the distance .

Try solving on your own before revealing the answer!

Final Answers:

  • a) 5

  • b)

  • c)

  • d)

Each answer is found by applying the distance formula to the given pairs of points.

Q3. Find the midpoint between the points.

Background

Topic: Midpoint Formula in the Coordinate Plane

This question tests your ability to find the point exactly halfway between two given points in the plane.

Key Formula:

  • and are the coordinates of the two points.

Step-by-Step Guidance

  1. Label the coordinates of the two points as and .

  2. Add the -coordinates together: .

  3. Add the -coordinates together: .

  4. Divide each sum by 2 to find the average for and .

  5. Write the midpoint as an ordered pair using these averages.

Try solving on your own before revealing the answer!

Final Answers:

  • a)

  • b)

  • c)

  • d)

Each answer is found by averaging the and coordinates of the given points.

Q4. Find the indicated intersection or union. Express your answer in interval notation.

Background

Topic: Set Operations with Intervals

This question tests your understanding of how to find the union and intersection of sets described by intervals on the real number line, and how to express the result in interval notation.

Key Terms:

  • Union (): All elements in either set or both.

  • Intersection (): Only elements common to both sets.

  • Interval Notation: A way to describe subsets of the real numbers using parentheses and brackets.

Step-by-Step Guidance

  1. For each pair of intervals, sketch or visualize them on the number line.

  2. For intersections (), identify the overlap between the two intervals.

  3. For unions (), combine all values covered by either interval.

  4. Express the result using correct interval notation, using parentheses for open endpoints and brackets for closed endpoints.

  5. For cases with no overlap (empty set), indicate this appropriately.

Try solving on your own before revealing the answer!

Final Answers:

  • a)

  • b)

  • c)

  • d) (Empty Set)

  • e)

  • f)

Each answer is the result of applying the union or intersection operation to the given intervals.

Q5. Use the following sets to find the indicated intersection or union. Express your answer in set notation.

Background

Topic: Set Operations (Union and Intersection)

This question tests your ability to perform union and intersection operations on sets, and to express the results using set notation.

Key Terms:

  • Union (): The set of all elements that are in either set or both.

  • Intersection (): The set of all elements that are in both sets.

  • Set Notation: Curly braces are used to list elements of a set.

Step-by-Step Guidance

  1. Identify the elements in each set (A, B, C, D) as described.

  2. For intersections (), list only the elements that appear in both sets.

  3. For unions (), list all elements that appear in either set, without duplicates.

  4. Express your answer using set notation, with curly braces and ellipses for infinite sets.

Try solving on your own before revealing the answer!

Final Answers:

  • a)

  • b)

  • c)

  • d)

  • e)

  • f)

Each answer is the result of the indicated set operation, written in set notation.

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