6.2 Linear Functions and Their Features
Key Takeaways
- MA.912.F.1.5 compares two linear functions represented in different forms (equation, graph, table, verbal description), so you must extract each feature from whichever representation you are given before comparing numerically.
- The five tested key features are the y-intercept (set x = 0), the x-intercept (set y = 0), the slope or rate of change (delta y over delta x), the domain, and the range.
- Mathematically every non-vertical linear function has domain and range of all real numbers, but a context-driven item asks for the restricted domain (often time >= 0 or quantity >= 0).
- Verbal descriptions encode slope as a signed rate: "drops 5 feet per second" means slope -5, not 5, and "starts at 6" identifies the y-intercept as 6.
- When reading slope from a graph, use the axis scale rather than counting grid squares, because a graph labeled in 5-unit increments does not have a slope of 1 when the line rises one square.
Quick Answer: MA.912.F.1.5 asks you to compare the key features of two linear functions when each is presented in a different representation — equation, table, graph, or verbal description. The features tested are the x-intercept, y-intercept, slope, domain, and range. You must read each feature out of whichever representation you're given, then compare numerically.
A linear function has a small, fixed set of key features. Memorize what each means and how to read it from each representation:
| Feature | Equation y = mx + b | Graph | Table | Verbal description |
|---|---|---|---|---|
| y-intercept | b (set x = 0) | point where line crosses y-axis | y-value when x = 0 | "starts at..." / "initial value" |
| x-intercept | set y = 0, solve for x | point where line crosses x-axis | x-value when y = 0 | "when output is zero" / "break-even" |
| Slope | m = (y2 - y1)/(x2 - x1) | rise over run between two lattice points | delta y / delta x between rows | "per" language (dollars per hour) |
| Domain / range | all reals unless restricted | projection on each axis | x-values / y-values listed | context bounds (often x >= 0) |
| Increasing / decreasing | m > 0 / m < 0 | line rises / falls left to right | y-values grow / shrink as x grows | "increases" / "decreases" |
Reading features from each representation
From an equation y = 3x - 6: slope is 3, y-intercept is (0, -6), x-intercept is found by 0 = 3x - 6, so x = 2, giving (2, 0). Domain and range are both all real numbers. The function is increasing because m > 0.
From a graph: locate where the line crosses each axis for the intercepts; pick two lattice points and compute rise over run. A line through (0, 4) and (2, 0) has slope (0 - 4)/(2 - 0) = -2, y-intercept 4, x-intercept 2.
From a table: slope is the constant difference in y divided by the constant difference in x. If x goes {1, 2, 3, 4} and y goes {5, 8, 11, 14}, delta y / delta x = 3/1 = 3. The y-intercept isn't in the table — extend backward one row (x = 0 gives y = 2) or solve y = 3x + b: 5 = 3(1) + b, so b = 2.
From a verbal description: "A water tank starts with 50 liters and drains 4 liters per minute" gives y-intercept 50 and slope -4 (negative because draining). The x-intercept — when empty — is 50/4 = 12.5 minutes. The domain is restricted to 0 <= x <= 12.5 because the tank cannot hold negative water.
Worked comparison item
Function A is given by y = 2x + 1. Function B is given by the table:
| x | -1 | 0 | 1 | 2 | | y | 5 | 3 | 1 | -1 |
Compare features. Function A: slope 2, y-intercept 1, x-intercept at x = -1/2. Function B: delta y / delta x = (3 - 5)/(0 - (-1)) = -2, y-intercept 3 (from the x = 0 row), x-intercept from 0 = -2x + 3, so x = 1.5. Function A is increasing with a smaller y-intercept; Function B is decreasing with a larger y-intercept. The y-intercept of B (3) is greater than that of A (1), and the slope of A (2) is greater than that of B (-2). A follow-up might ask which reaches 10 first — solve 2x + 1 = 10 for x = 4.5; only A reaches 10 for positive x.
Common traps
Trap 1: confusing the intercepts. The x-intercept is found by setting y = 0. The y-intercept is found by setting x = 0. If you flip these, every downstream answer is wrong. Memory hook: "x-intercept means x is the answer, so set y to zero."
Trap 2: ignoring units when comparing rates. A table in "meters per second" cannot be directly compared to an equation in "feet per second" without conversion. Convert both to the same unit first.
Trap 3: domain and range in context. Mathematically, every non-vertical linear function has domain and range of all real numbers. In context, the domain is usually restricted (time >= 0, quantity >= 0). An item asking for "the domain in the context" wants the context window, not all real numbers.
Trap 4: reading slope from grid squares instead of the axis scale. A graph whose x-axis is labeled in 5-unit increments does not have a slope of 1 when the line goes up one square — it has a slope of 5.
Trap 5: treating a table as the only points. A table gives a sample, not the whole function. To find a y-intercept not listed, extend the pattern backward or use algebra. To find an x-intercept not listed, set y = 0 and solve.
Trap 6: verbal "rate" without sign. "Drops 5 feet per second" encodes slope -5, not 5. The EOC pairs a verbal decreasing function with an equation that has a positive slope, then asks which is decreasing — only the verbal one.
Test-day workflow
- Read each representation and extract all five features for both functions before comparing.
- Reduce slope fractions; convert verbal rates to signed numbers.
- For domain and range, decide whether the item asks the mathematical domain (all reals) or the contextual domain (a bounded interval).
- When two features tie, look for the follow-up that breaks the tie with a third feature.
Function A is given by y = 3x - 2. Function B is given by the table with x = {0, 1, 2, 3} and y = {-1, 1, 3, 5}. Which statement correctly compares the two functions?
Function P is given by the equation y = 4x + 2. Function Q is described verbally as "starts at 6 and increases by 1 for every increase of 1 in x." Which function has the greater y-intercept?
A linear function is given by y = 2x - 6. What is the x-intercept of this function?