Cartesian Coordinate System

Plotting Points and Quadrants

The Cartesian coordinate system is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Points are plotted as ordered pairs (x, y).

• Quadrants: The plane is divided into four quadrants:

  • Quadrant I: (x > 0, y > 0)

  • Quadrant II: (x < 0, y > 0)

  • Quadrant III: (x < 0, y < 0)

  • Quadrant IV: (x > 0, y < 0)

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

• Example: Plot the points (-5, 3), (-2, -4), (1, 3), (-1, -4).

Graphing Functions by Plotting Points

To graph a function, calculate y-values for selected x-values and plot the resulting points.

• Example: Graph by plotting points:

Intercepts

• x-intercept: Where the graph intersects the x-axis ().

• y-intercept: Where the graph intersects the y-axis ().

• Example: For , the y-intercept is at (0, -2).

Linear and Rational Equations

Solving Linear Equations

A linear equation is an equation of the form .

• Example: Solve

• Step-by-step solution:

Solving Rational Equations

Rational equations contain fractions with polynomials in the numerator and denominator.

• Example: Solve

• Find the least common denominator (LCD): 12

• Multiply both sides by 12:

Types of Equations

• Identity: Always true for all real numbers (e.g., ).

• Conditional: True for one or more real numbers (e.g., ).

• Inconsistent: Never true (e.g., ).

Applications of Equations

Number Problems

Translate word problems into equations to solve for unknowns.

• Example: Find the number such that 80% of the number plus the number equals 252.

• Equation:

Investment Problems

Set up equations to solve for amounts invested at different interest rates.

• Example: Nick earned a $10,000 bonus and wants to invest in two stocks paying 6% and 11% annual interest to earn $800 in the first year.

• Let = amount invested at 6%, = amount invested at 11%

• Equation:

• Solve for :

  4. So, at 6%, at 11%

Geometry Problems

Use algebraic equations to solve for dimensions.

• Example: Find the dimensions of a rectangle with perimeter and given relationships between and .

Complex Numbers

Definition and Properties

A complex number is a number of the form , where is the real part and is the imaginary part. The imaginary unit is defined as .

• Examples: ,

• Imaginary unit:

Operations with Complex Numbers

• Addition/Subtraction: Combine like terms.

• Multiplication: Use distributive property and .

• Example:

• Complex Conjugate: For , the conjugate is . Used to divide complex numbers.

Division of Complex Numbers

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.

• Example:

• Multiply numerator and denominator by :

• Denominator:

• Numerator:

• Final answer:

Summary Table: Complex Number Operations

Additional info:

• Some steps and explanations have been expanded for clarity and completeness.

• All equations are provided in LaTeX format for mathematical accuracy.