Chapter 1 (1.1): Equations and Inequalities - Graphs and Graphing Utilities

Definition: A system formed by drawing a horizontal line and a vertical line that intersect at right angles.
- x-axis: The horizontal line.
- y-axis: The vertical line.
- Origin: The point of intersection of the x-axis and y-axis. This represents their zero points.
Number Placement:
- Positive numbers are shown to the right of the origin along the x-axis and above the origin along the y-axis.
- Negative numbers are shown to the left of the origin along the x-axis and below the origin along the y-axis.

Every point in this system corresponds to an ordered pair of real numbers, represented as (x, y).
x-coordinate (first number): Denotes the distance and direction (left/right) from the origin along the x-axis.
y-coordinate (second number): Denotes the vertical distance and direction (up/down) from the origin along the y-axis.
Example 1a: Plotting the point (-2, 4).
- Move 2 units to the left of the origin (due to -2 for the x-coordinate).
- Move 4 units up from that position (due to 4 for the y-coordinate).
Example 1b: Plotting the point (4, -2).
- Move 4 units to the right of the origin (due to 4 for the x-coordinate).
- Move 2 units down from that position (due to -2 for the y-coordinate).

Equation in Two Variables: A relationship between two quantities typically expressed as an equation involving variables like x and y, e.g., y = 4 - 2 (which simplifies to y = 2).
Solution of an Equation in Two Variables: An ordered pair of real numbers (x, y) such that when the x-coordinate is substituted for x and the y-coordinate is substituted for y in the equation, it results in a true statement.

Method: To graph an equation, select various values for x, find the corresponding y values, plot these ordered pairs, and then connect the plotted points to form the graph.
Example 3: Graphing the equation y = |x+1|.
- Step 1: Select integer values for x. For this example, choose x values from -4 to 2: -4, -3, -2, -1, 0, 1, 2.
- Step 2: Find the corresponding y values for each x.
  - If x = -4, y = |-4+1| = |-3| = 3
  - If x = -3, y = |-3+1| = |-2| = 2
  - If x = -2, y = |-2+1| = |-1| = 1
  - If x = -1, y = |-1+1| = |0| = 0
  - If x = 0, y = |0+1| = |1| = 1
  - If x = 1, y = |1+1| = |2| = 2
  - If x = 2, y = |2+1| = |3| = 3
- Step 3: Create a table of ordered pairs.
  | x | y = |x+1| | (x, y) |
  |---|-------|--------|
  | -4 | 3 | (-4, 3) |
  | -3 | 2 | (-3, 2) |
  | -2 | 1 | (-2, 1) |
  | -1 | 0 | (-1, 0) |
  | 0 | 1 | (0, 1) |
  | 1 | 2 | (1, 2) |
  | 2 | 3 | (2, 3) |
- Step 4: Plot these points in the rectangular coordinate system and connect them. The graph will form a V-shape, characteristic of an absolute value function.

x-intercept: The x-coordinate of a point where the graph intersects the x-axis.
- The y-coordinate corresponding to an x-intercept is always 0.
- To find x-intercept(s), set y = 0 in the equation and solve for x.
y-intercept: The y-coordinate of a point where the graph intersects the y-axis.
- The x-coordinate corresponding to a y-intercept is always 0.
- To find y-intercept(s), set x = 0 in the equation and solve for y.
Example 5: Identifying Intercepts from a Graph.
- If a graph crosses the x-axis at (-3, 0), the x-intercept is -3.
- If a graph crosses the y-axis at (0, 5), the y-intercept is 5.

Graphs are often used to visually represent data and relationships, allowing for easy interpretation of information.
Example 6: Desirable Heart Rate during Exercise.
- A graph might show desirable heart rates based on age and gender.
- From the graph, the desirable heart rate of a 20-year-old woman during exercise is 130 (beats per minute).
- From the graph, the desirable heart rate of a 20-year-old man during exercise is 150 (beats per minute).