Rectangular Coordinate System

Definition and Structure

The rectangular coordinate system (also known as the Cartesian plane) is a fundamental concept in algebra, used to visually represent points, lines, and other geometric objects. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0, 0).

• X-axis: The horizontal axis.

• Y-axis: The vertical axis.

• Origin: The point (0, 0) where the axes intersect.

Plotting Points (x, y)

Each point in the plane is represented by an ordered pair (x, y), where x is the horizontal position and y is the vertical position.

• To plot a point: Move horizontally to the x-value, then vertically to the y-value.

• Example: Plot the points (4, 2), (5, -3), (-3, 4), (-2, -5).

Midpoint Formula

Finding the Midpoint Between Two Points

The midpoint of a segment connecting two points (x_1, y_1) and (x_2, y_2) is the point exactly halfway between them. It is calculated by averaging the x-values and y-values:

• Midpoint Formula:

• Example: Find the midpoint between (1, 2) & (3, 4):

• Example: Find the midpoint between (-4, 3) & (1, -4):

Pythagorean Theorem

Relationship in Right Triangles

The Pythagorean Theorem relates the lengths of the sides of a right triangle. If a and b are the lengths of the legs, and c is the length of the hypotenuse:

• Formula:

• Example: For a triangle with sides 3, 4, and 5:

Distance Formula

Finding the Distance Between Two Points

The distance formula is derived from the Pythagorean Theorem and calculates the straight-line distance between two points (x_1, y_1) and (x_2, y_2):

• Formula:

• Example: Find the distance between (1, 2) & (9, 8):

• Example: Find the distance between (-3, 7) & (6, -10):

Slope of a Line

Definition and Interpretation

The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

• Slope Formula:

• Example: Find the slope between (1, 2) & (3, 5):

• Example: Find the slope between (-4, 3) & (1, -4):

Properties of Slope

The value of the slope determines the behavior of the line:

• Increasing Line: If , the line rises from left to right.

• Decreasing Line: If , the line falls from left to right.

• Horizontal Line: If , the line is flat (no rise).

• Vertical Line: If is undefined (division by zero), the line is vertical.

Applications of Slope

The slope formula can be used to calculate rates of change in various contexts, such as economics, science, and everyday life.

• Example: Annual rate of increase in milk price: In 2010, price = m = \frac{3.49 - 1.99}{2020 - 2010} = \frac{1.50}{10} = 0.15$ per year.

• Example: Annual rate of decline in college enrollment: 2015: 12,500 students; 2020: 9,850 students. students per year.

• Example: Cost rate per gigabyte for iPhone: 64GB: m = \frac{999 - 799}{256 - 64} = \frac{200}{192} \approx 1.04$ dollars per GB.

Recap Table

Summary of Key Concepts

Homework Practice Exercises

Suggested Textbook Problems

• Section 1.1: 3, 5, 63, 65, 67, 73, 83, 87

• Section 1.3: 5, 7, 9, 11, 15, 41, 45, 47

Additional info: These exercises are likely to reinforce concepts such as plotting points, using the midpoint and distance formulas, and calculating slope.