Math: Linear Relations & Graphs

WHAT IS A LINEAR RELATION?

A relationship between two variables where the RATE OF CHANGE IS

CONSTANT. When you plot the points, they form a straight line.

  Linear: every time x goes up by 1, y goes up by the same amount
  Non-linear: the amount y changes is different each time

RATE OF CHANGE (SLOPE)

The rate of change tells you how fast y is changing relative to x.

  Rate of change = (change in y) / (change in x)

  From a table:
    x: 0, 3, 6     y: 60, 228, 396
    Rate = (228 - 60) / (3 - 0) = 168 / 3 = $56 per hour

  From a word problem:
    Temperature goes from -8°C at 2am to -2°C at 4am.
    Rate = (-2 - (-8)) / (4 - 2) = 6 / 2 = +3°C per hour

  IMPORTANT: The rate of change is the SLOPE (m in y = mx + b).
  It is NOT the y-intercept. A common trick: K = 2n + 7 has a rate
  of change of 2, not 7. The 7 is the starting value.

IDENTIFYING NON-LINEAR RELATIONS

Check if the FIRST DIFFERENCES are constant.

  First differences = the change in y between consecutive rows.

  LINEAR example: (1,5), (2,7), (3,9), (4,11)
    Differences: +2, +2, +2 → constant → LINEAR

  NON-LINEAR example: (1,-1), (2,2), (3,6), (4,11)
    Differences: +3, +4, +5 → NOT constant → NON-LINEAR

  If the differences keep changing, the relation is NOT linear.

READING GRAPHS — INTERSECTIONS

When two lines cross on a graph, that's the BREAK-EVEN POINT.

  Example: Two entrance fee options cross at 7 visits.
  - Before 7 visits: one option is cheaper
  - After 7 visits: the other option is cheaper
  - At exactly 7 visits: they cost the same

  TEST TIP: Read carefully which option is cheaper BEFORE vs AFTER
  the intersection point.

READING GRAPHS — Y-INTERCEPT

The y-intercept is where the line crosses the vertical axis (when x = 0).

  Example: Job A starts at $200 on the y-axis even at 0 sales.
  This means Job A has a BASE PAY of $200 regardless of sales.

  Job B starts at the origin (0,0).
  This means Job B has NO base pay — no sales means no money.

MATCHING EQUATIONS TO GRAPHS

  y = mx + b

  Step 1: Look at the y-intercept (where does the line cross the y-axis?)
    That tells you b.

  Step 2: Look at the slope (is the line going up or down? steep or gentle?)
    Going up = positive slope
    Going down = negative slope
    Steeper = larger absolute value of slope

  Example: y = -3x + 4
    y-intercept = 4 (crosses y-axis at 4)
    slope = -3 (goes DOWN steeply from left to right)
    The correct graph starts at (0, 4) and drops steeply.

EXTENDING PATTERNS FROM TABLES

  Step 1: Find the rate of change
  Step 2: Use the equation y = mx + b to predict new values

  Example: Arya earns: 0 trees=$40, 50=$48, 100=$56, 150=$64
    Rate = $8 per 50 trees = $0.16 per tree
    Starting value = $40
    Equation: E = 40 + 0.16t
    For 1,000 trees: E = 40 + 0.16(1000) = 40 + 160 = $200

CHECKING IF A POINT IS ON A LINE

Plug the x-value into the equation and see if you get the y-value.

  Example: Tickets cost $14.25 each. Is (13, 185) on the graph?
    14.25 × 13 = 185.25 ≠ 185 → NOT on the graph

  Example: Is (44, 627) on the graph?
    14.25 × 44 = 627 → YES, on the graph

INTERPRETING REAL-WORLD GRAPHS

  y-intercept = fixed/base cost (service fee, visit fee, setup fee)
  slope = variable rate (per hour, per item, per visit)

  Example: A plumber charges $100 visit fee + $200/hour.
    y-intercept = 100 (the graph starts at $100)
    slope = 200 (steep upward line — $200 for each hour)

KEY THINGS TO REMEMBER

  Rate of change = (change in y) / (change in x) = the slope
  Constant first differences = linear
  Changing first differences = non-linear
  y-intercept = the value when x = 0 = the starting/fixed amount
  Intersection of two lines = the break-even point
  Negative slope = line goes downward left to right
  To check if a point is on a line: plug in and verify