Key Vocabulary
- Linear function
- A function whose graph is a non-vertical straight line.
- Linear equation
- Any equation that can be written as Ax + By = C.
- Solution
- An ordered pair (x, y) that makes the equation true. The graph = all solutions.
- Discrete function
- Graph is unconnected points (whole-number inputs).
- Continuous function
- Graph is an unbroken line with no gaps.
- x-intercept
- x-coordinate where the graph crosses the x-axis. Point (a, 0).
- y-intercept
- y-coordinate where the graph crosses the y-axis. Point (0, b).
Forms of a Linear Equation
Standard form:
Ax + By = C
(A and B not both 0)
Slope-intercept form:
f(x) = mx + b
m = slope, b = y-intercept
- A = 0 → y = C/B → horizontal line
- B = 0 → x = C/A → vertical line (not a function)
Is It Linear?
- Linear ✓ x and y both to the 1st power, no xy, no variable in a denominator
- Not linear ✗ x2, y2, x3, xy, 5/x
Table test:
1. x changes by the same amount?
2. Subtract consecutive y-values
3. All y-differences equal → LINEAR
y = x²: 1, 4, 9, 16, 25 → changes 3, 5, 7, 9 → not linear
Check x first! x: −1, 0, 2, 3, 4 with y up by 5 each time is NOT linear (x jumped by 2).
Equal Differences Proof
If x₂ − x₁ = x₄ − x₃
then f(x₂) − f(x₁) = f(x₄) − f(x₃)
Why: f(x₂) − f(x₁)
= (mx₂ + b) − (mx₁ + b)
= m(x₂ − x₁)
Change in y = m × (change in x). Equal x-steps always give equal y-steps.
Graphing from Standard Form
- Solve for y (example: 5x + y = 10 → y = 10 − 5x)
- Make a table with a few x-values
- Plot the points
- Connect them if continuous; leave as dots if discrete
Discrete vs. Continuous
| Discrete (dots) | Continuous (line) |
|---|
| DVDs, tickets, rooms | Time, distance |
| Basketballs made | Gallons of water |
| Pieces of jewelry | Pounds of apples |
f(x) = 2x per DVD → Domain {0, 1, 2, …}, Range {0, 2, 4, …}
Finding Intercepts
x-intercept: set y = 0, solve for x
y-intercept: set x = 0, solve for y
For Ax + By = C:
x-intercept = C / A
y-intercept = C / B
3x − 2y = 6
x-int: 3x = 6 → 2
y-int: −2y = 6 → −3
- Set the other variable to 0. x-intercept → y = 0.
- (5, 0) is an x-intercept, not a y-intercept.
- (0, 0) is the only point that is both.
Interpreting Intercepts
- y-intercept = starting value (when time = 0)
- x-intercept = when the amount reaches 0
- Negative y-intercept = starts below 0 (underground, below 0°, owing money)
Tram: f(x) = 4500 − 300x
x-int: 0 = 4500 − 300x → 15 min
y-int: f(0) = 4500 m
Graphing with Intercepts
- Rewrite in standard form Ax + By = C
- Find x-intercept (y = 0)
- Find y-intercept (x = 0)
- Plot both points and draw the line
3y = −5x − 30 → 5x + 3y = −30
Plot (−6, 0) and (0, −10)
Triangle Area from Intercepts
Line Ax + By = C with the axes
makes a right triangle:
base = C / A
height = C / B
Area = ½ · base · height
= C² / (2AB)
10x + 4y = 240 → base 24, height 60
Area = ½ · 24 · 60 = 720 ft²
Rate of Change & Slope
rate of change = change in y / change in x
slope = rise / run
Slope formula:
m = (y₂ − y₁) / (x₂ − x₁)
(2, 1) and (3, 4): m = (4 − 1)/(3 − 2) = 3
- y on top, x on bottom
- Subtract in the same order top and bottom
- Watch double negatives: 3 − (−1) = 4
Four Kinds of Slope
| Slope | Graph | Clue |
|---|
| Positive | Rises ↗ | y goes up as x goes up |
| Negative | Falls ↘ | y goes down as x goes up |
| Zero | Horizontal → | same y-values (rise = 0) |
| Undefined | Vertical ↑ | same x-values (run = 0) |
Steeper line = bigger rate of change (bigger |m|).
A linear function has the same slope between any two points.
Interpreting Slope
“For every 1 [x-unit],
y goes up/down by |m| [y-units].”
Units of slope = y-units per x-unit
- Heart rate (20, 180), (50, 150): m = −1 → drops 1 beat/min per year
- Earnings (4, 110), (8, 170): m = 15 → $15 per hour
- Balance (3, 860), (8, 585): m = −55 → loses $55 per month
- Ladder 19 up, 11 out: m ≈ 1.73
- Given slope, find run: 0.37 = 1/run → run ≈ 2.70