The Cartesian coordinate system is the foundation for graphing in algebra. It lets us plot points and equations on a two-dimensional plane, using x and y coordinates to pinpoint locations. This system is crucial for visualizing mathematical relationships and solving problems graphically.

Interpreting graphs involves analyzing key features like intercepts, distances, and midpoints. By examining these elements, we can extract valuable information about equations and functions, helping us understand their behavior and make predictions about real-world situations they represent.

The Cartesian Coordinate System

Points in Cartesian coordinates

• Two-dimensional plane formed by intersecting horizontal (x-axis) and vertical (y-axis) number lines at the origin (0, 0)
• x-coordinate represents horizontal distance from origin
  • Positive values to the right, negative values to the left
• y-coordinate represents vertical distance from origin
  • Positive values above, negative values below
• To plot a point, start at origin, move horizontally by x-coordinate, then vertically by y-coordinate
• Coordinates written as ordered pair (x, y)
  • First value is x-coordinate, second value is y-coordinate
• Examples: (3, 4) is 3 units right and 4 units up, (-2, -5) is 2 units left and 5 units down

Graphing equations with technology

• To graph by plotting points:
  1. Create table of x and y values satisfying equation
  2. Plot points from table on coordinate plane (coordinate plane)
  3. Connect points with smooth curve or straight line
• Graphing calculators and software graph equations efficiently
  • Enter equation into technology
  • Adjust window settings to view desired portion of graph
• Examples: $y = 2x + 1$ (linear), $y = x^2 - 4$ (quadratic)

Interpreting Graphs

Intercepts of graphs

• x-intercept: point where graph crosses x-axis
  • y-coordinate always 0
  • To find, set y = 0 and solve for x
• y-intercept: point where graph crosses y-axis
  • x-coordinate always 0
  • To find, set x = 0 and solve for y
• Examples: x-intercept of $y = 2x - 6$ is (3, 0), y-intercept of $y = -x + 4$ is (0, 4)

Distance formula for lengths

• Calculates length of line segment between points $(x_1, y_1)$ and $(x_2, y_2)$
  • $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• To use:
  1. Identify coordinates of two points
  2. Substitute coordinates into formula
  3. Simplify and calculate result
• Example: Distance between (1, 2) and (4, 6) is $\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{25} = 5$

Midpoint formula for segments

• Finds coordinates of point dividing line segment into two equal parts
  • For segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$, midpoint is $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$
• To use:
  1. Identify coordinates of endpoints
  2. Substitute coordinates into formula
  3. Simplify and calculate result
• Example: Midpoint of segment with endpoints (-3, 1) and (5, 7) is $(\frac{-3+5}{2}, \frac{1+7}{2}) = (1, 4)$

Information from graph analysis

• Shape and behavior determine equation or relationship type
  • Linear equations: straight lines
  • Quadratic equations: parabolas
  • Exponential equations: curves increasing or decreasing rapidly
• Identify key features
  • Intercepts, symmetry, increasing or decreasing behavior, turning points (maximums or minimums)
• Use information to make predictions or draw conclusions about variable relationships
• Examples: Parabola opens upward (positive leading coefficient), line with positive slope (increasing function)

Functions and their properties

• A function (function) is a rule that assigns each input value to exactly one output value
• The graph of a function is a visual representation of its behavior
• Domain: set of all possible input values for a function
• Range: set of all possible output values for a function
• Understanding these properties helps analyze and interpret functions in various contexts