Math: Algebra·Linear Functions

SAT Math — Linear Functions

Slope-Intercept Form

The most important form of a linear equation is slope-intercept form:

y = mx + b

  • m = slope (rate of change)
  • b = y-intercept (where the line crosses the y-axis, i.e., the value of y when x = 0)

    • Reading the equation: > y = 4x − 7 > Slope = 4 (y increases by 4 for every 1-unit increase in x) > Y-intercept = −7 (the line crosses the y-axis at (0, −7))

    Calculating Slope

    Slope = rise / run = (y₂ − y₁) / (x₂ − x₁)

    Given two points (1, 3) and (4, 9): > m = (9 − 3) / (4 − 1) = 6/3 = 2

    Slope tells you:

  • Positive slope: Line goes up left to right
  • Negative slope: Line goes down left to right
  • Zero slope: Horizontal line (y = constant)
  • Undefined slope: Vertical line (x = constant) — can't be expressed as a function

    • Writing the Equation of a Line

    Given slope and y-intercept: Plug directly into y = mx + b.

    Given slope and one point: 1. Use slope-intercept form: y = mx + b 2. Plug in the known point (x, y) and slope m 3. Solve for b

    Example: Slope = 3, passes through (2, 8) > 8 = 3(2) + b → 8 = 6 + b → b = 2 > Equation: y = 3x + 2

    Given two points: 1. Find the slope using the slope formula 2. Use slope and one point to find b (as above)

    Interpreting Linear Functions in Context

    The SAT loves to ask: "In the context of the problem, what does the slope (or y-intercept) represent?"

    Template:

  • Slope = the rate of change = how much [y] changes per unit of [x]
  • Y-intercept = the starting value = the value of [y] when [x] = 0

    • Example: A plumber charges $75 for the first hour plus $50 per additional hour. Total cost: C = 50h + 75.

  • Slope (50): The plumber charges $50 for each additional hour worked
  • Y-intercept (75): The base charge before any hours are counted (the minimum fee)

    • Real-world example: A car's fuel tank starts with 12 gallons and uses 0.04 gallons per mile. The equation is G = −0.04d + 12, where G is gallons remaining and d is miles driven.

  • Slope = −0.04: The car uses 0.04 gallons per mile (decreasing)
  • Y-intercept = 12: The starting amount of fuel (12 gallons when d = 0)

    • ---

    Key Terms

  • Slope (m): Rate of change; rise over run; (y₂ − y₁)/(x₂ − x₁)
  • Y-intercept (b): The value of y when x = 0; where the line crosses the y-axis
  • Slope-intercept form: y = mx + b
  • Positive slope: Line rises from left to right
  • Negative slope: Line falls from left to right
  • Zero slope: Horizontal line; y is constant
  • Linear function: A function whose graph is a straight line; every unit increase in x changes y by the same amount (the slope)
  • Rate of change: The amount y changes per 1-unit increase in x (equals the slope)

---

Quiz Questions:

Q1. A line has a slope of −2 and passes through the point (3, 4). What is the y-intercept?

A) 10 B) −2 C) 2 D) −10

Answer: A — Use y = mx + b: 4 = −2(3) + b → 4 = −6 + b → b = 10. Y-intercept = 10.

---

Q2. Two points on a line are (−1, 5) and (3, −3). What is the slope?

A) −2 B) 2 C) 1/2 D) −1/2

Answer: A — m = (−3 − 5)/(3 − (−1)) = −8/4 = −2.

---

Q3. A scientist tracks the temperature inside a freezer after it is turned on. The temperature starts at 65°F and drops 8°F every minute. Which equation represents this, where T is temperature (°F) and m is minutes?

A) T = 8m + 65 B) T = −8m + 65 C) T = 65m − 8 D) T = 8 − 65m

Answer: B — Temperature starts at 65 (y-intercept) and decreases by 8 per minute (slope = −8). T = −8m + 65.

---

Q4. The equation P = 0.75t + 20 models the number of pages P a student has read after t minutes. What does 0.75 represent in this context?

A) The total number of pages in the book B) The number of pages the student had read before the timer started C) The rate at which the student reads, in pages per minute D) The number of minutes needed to finish the book

Answer: C — The slope (0.75) represents the rate of change — in this context, the rate at which the student reads (0.75 pages per minute). The 20 is the y-intercept: the student had already read 20 pages before the timer started.

---

Q5. Which of the following lines is parallel to y = 3x − 5?

A) y = −3x + 5 B) y = 3x + 10 C) y = (1/3)x − 5 D) y = −(1/3)x + 2

Answer: B — Parallel lines have the same slope. The slope of y = 3x − 5 is 3. Choice B (y = 3x + 10) also has slope 3 and a different y-intercept, making it a parallel line. Choice A has slope −3 (negative reciprocal — that's perpendicular, not parallel). Choices C and D have slope 1/3 or −1/3.