11.2 Comparing Proportional Relationships and Functions Represented Differently
Key Takeaways
- A proportional relationship's constant rate — its unit rate, slope, or 'k' value in y = kx — can be found from a table (y divided by x), a graph (rise over run), an equation (the coefficient of x), or a verbal description (compute directly).
- Assessment target A.7.a (comparing proportional relationships) and A.7.d (comparing linear/quadratic function properties) are both rated DOK level 2 — the highest cognitive-complexity function items on the GED.
- The classic GED comparison scenario is two moving objects — one described by a distance-time graph, the other by a distance-time equation — where you must extract and compare their speeds (slopes).
- When comparing two functions, first identify exactly which property the question asks about — rate of change, starting value (y-intercept), or maximum/minimum — before comparing, since each representation surfaces that property differently.
- For quadratic comparisons, the vertex reveals the maximum or minimum value; comparing two quadratics often means finding and comparing their vertex y-values, not just a rate of change.
Why This Topic Matters
Assessment targets A.7.a ("Compare two different proportional relationships represented in different ways") and A.7.d ("Compare properties of two linear or quadratic functions each represented in a different way") are both flagged by GED Testing Service's Assessment Guide at Depth of Knowledge level 2 — the highest complexity rating given to any function indicator on the test. These items are deliberately mixed-representation: one function might appear as a graph, the other as an equation, a table, or a written description, and you must translate each into a comparable form before answering. Because the skill combines slope-finding, table-reading, and graph-reading into a single question, it is one of the topics where GED candidates lose points not from lack of math knowledge but from rushing the translation step.
The Core Skill: Extracting the Rate from Any Representation
A proportional relationship is a relationship of the form y = kx, where k is the constant of proportionality — also called the unit rate when the context involves rates like speed, price per item, or pay per hour. For straight-line functions in general, this same number is the slope or rate of change. Whatever representation you're given, the goal is always the same: pull out that one number so you can compare it apples-to-apples against the other function's rate.
| Representation | How to Find the Rate (k / Slope) |
|---|---|
| Table of values | Divide any y-value by its paired x-value (for a proportional relationship, this ratio is constant); for a general linear table, compute (change in y) / (change in x) between two rows |
| Graph | Pick two clear points and compute rise over run: (y2 - y1) / (x2 - x1); for a line through the origin, read the y-value at x = 1 |
| Equation | If written y = kx or y = mx + b, the rate is the coefficient k or m sitting in front of x |
| Verbal description | Compute directly from the given numbers, e.g., "a car travels 150 miles in 3 hours" gives a rate of 150 / 3 = 50 miles per hour |
Worked Example: Comparing Two Proportional Relationships (A.7.a)
Scenario: Car A travels according to the equation d = 55t, where d is distance in miles and t is time in hours. Car B's distance is shown on a graph passing through the points (0, 0) and (2, 120). Which car is traveling faster?
- Car A's rate is read directly from its equation: the coefficient of t is 55, so Car A travels at 55 miles per hour.
- Car B's rate is the slope of its graph: (120 - 0) / (2 - 0) = 120 / 2 = 60 miles per hour.
- Compare: 60 mph > 55 mph, so Car B is traveling faster.
Common trap: comparing the wrong numbers — for instance, comparing Car A's coefficient (55) to Car B's single y-coordinate (120) instead of finishing the slope calculation for Car B first. Always reduce both functions to the same kind of number (a single rate) before comparing.
Worked Example: Comparing Two Linear Functions from Different Representations (A.7.d)
Scenario: Function P is given by the table below. Function Q is given by the equation Q(x) = 4x + 3. Which function has the greater rate of change?
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(x) | 5 | 8 | 11 | 14 |
- Find P's rate of change from the table: from x = 0 to x = 1, P(x) goes from 5 to 8, a change of 8 - 5 = 3 per unit of x. Checking the next pair confirms it: 11 - 8 = 3, and 14 - 11 = 3. So P's rate of change is 3.
- Find Q's rate of change from its equation: in Q(x) = 4x + 3, the coefficient of x is 4, so Q's rate of change is 4.
- Compare: 4 > 3, so Function Q has the greater rate of change.
Common trap: confusing the rate of change (3 or 4) with the starting value or y-intercept (5 for P, 3 for Q). If the question instead asked which function has the greater y-intercept, the answer would flip to Function P (starting value 5, versus Q's 3). Always re-read exactly which property — rate of change, y-intercept, or another feature — the question is asking you to compare.
Comparing Quadratic Functions
A.7.d also covers quadratic comparisons, which usually center on the vertex — the maximum or minimum point of the parabola — rather than a constant rate of change (which does not exist for a curved function).
Worked example: Function R(x) = x^2 - 6x + 5 is described algebraically. Function S is shown on a graph with a maximum point at (2, 9) that opens downward. Which function has the greater maximum value?
Since R(x) = x^2 - 6x + 5 opens upward (positive leading coefficient), it has a minimum, not a maximum — so before comparing maximums, recognize that R doesn't have one to compare. A GED item built this way is testing whether you notice the leading coefficient's sign, not just whether you can calculate a vertex. When both functions do open the same direction, compute each vertex (for R(x) = ax^2 + bx + c, the vertex x-coordinate is -b/(2a); substitute back in to get the vertex y-value) and compare those y-values directly.
Test-Day Strategy
When a GED item presents two functions in mixed formats, work one function completely to a single comparable number before touching the second. Label your work ("Function A's rate = ...", "Function B's rate = ...") so you don't accidentally compare an intercept to a slope, or a single point to a full rate.
Plan A charges according to the equation c = 25 + 2n, where n is the number of items ordered. Plan B's cost is shown in a table: n = 2 costs $34, and n = 5 costs $46. Which plan has the lower cost per additional item?
A cyclist's distance is modeled by d = 12t (t in hours). A second cyclist's distance is graphed passing through (0, 0) and (3, 42). Which cyclist is faster, and by how much?
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