CHAPTER 2 —

Graphing Basics

  • Axis Definitions:
    • x-axis: Horizontal axis in the coordinate plane.
    • y-axis: Vertical axis in the coordinate plane.
  • Coordinate Points: Represented as ((x,y)), where:
    • A) ((2,3))
    • B) ((-2,3))
    • C) ((-2,-3))
    • D) ((2,-3))
    • E) ((-2,0)) on x-axis
    • F) ((0,3))

Graphing a Line by Completing a Table (T-Chart)

  • Example: To graph the equation (2x - y = 9).
  • Method: Solve for (y) first.
    • Rearrange equation: (y = 2x - 9)
  • T-Chart construction may help with plotting points.
  • x- and y-values can be derived from chosen values for (x).

Finding Intercepts of a Linear Equation

  • Definition of Intercepts:
    • x-intercept: Point where the graph crosses the x-axis; set (y = 0).
    • y-intercept: Point where the graph crosses the y-axis; set (x = 0).
  • Example: Find Intercepts for the line (2x - y = 9):
    • X-intercept Calculation:
    • Set (y = 0): (2x - 0 = 9 \Rightarrow 2x = 9 \Rightarrow x = \frac{9}{2} = 4.5)
    • X-int: ((4.5, 0))
    • Y-intercept Calculation:
    • Set (x = 0): (2(0) - y = 9 \Rightarrow -y = 9 \Rightarrow y = -9)
    • Y-int: ((0, -9))

Distance Formula

  • Formula: (d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2})
  • Example: Find the distance between the points ((-3, -1)) and ((2, 3)) using the formula:
    • Calculate the differences: (x2 - x1 = 2 - (-3) = 5);
    • (y2 - y1 = 3 - (-1) = 4);
    • Substitute: (d = \sqrt{(5)^2 + (4)^2} = \sqrt{25 + 16} = \sqrt{41}).

Midpoint Formula

  • Formula: (M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right))
  • Example: Find the midpoint of the line segment with endpoints ((7,-2)) and ((9,5)).
    • Calculate: (M = \left( \frac{7 + 9}{2}, \frac{-2 + 5}{2} \right) = \left(8, \frac{3}{2}\right)).

Linear Equations in One Variable

  • Standard Form: Equations like (Ax + B = C).
  • Example Problems: Solve the following: a. (2x + 3 = 17):
    • Rearranging gives (2x = 14 \Rightarrow x = 7 ).
      b. (3(x-6) = 6x-x):
    • Expand: (3x - 18 = 6x - x)
    • Rearranging gives (-3x = 18 \Rightarrow x = -6).

Slope and Linear Equations

  • Slope Definition: (m = \frac{y2 - y1}{x2 - x1})
    • Positive slope: Line rises left to right.
    • Negative slope: Line falls left to right.
    • Zero slope: Horizontal line; no rise or fall.
    • Undefined slope: Vertical line.
  • Point-Slope Form: (y - y1 = m(x - x1))

Examples of Finding and Using Slope

  1. Find the slope of the line through points ((-5, 4)) and ((3, -6)):
    • (m = \frac{-6 - 4}{3 - (-5)} = \frac{-10}{8} = -\frac{5}{4} ).
  2. Given a slope, find equation using Point-Slope Form:
    • If slope = 4, through point ((1,3)):
    • Substitute to get (y - 3 = 4(x - 1))`.
    • Rearranging gives: (y = 4x - 1).

Parallel and Perpendicular Lines

  • Parallel Lines: Lines that have the same slope and never intersect.
  • Perpendicular Lines: Lines that intersect at right angles; their slopes are negative reciprocals of each other.

Writing Equations for Lines

  • To write equations for lines, find the slope and y-intercept for standard and slope-intercept forms:
    1. For a line through ((-4,0)) with slope (-2):
    • Standard Form: (y = -2x - 8)
    1. For a line through ((-4,2)) perpendicular with slope (3/2):
    • Use Point-Slope Form to derive and simplify to find standard form: (y = \frac{3}{2}x + 4)
    • Rearranging gives (5y - 3x - 10 = 0).

Conclusion

  • Mastery of these concepts and their applications is critical for success in calculus and beyond. Familiarity with graphing techniques, linear equations, distances, and midpoints will aid in understanding more complex topics.