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Learn to compare two functions even when one is a graph, another is an equation, and a third is a table — by extracting the same key properties from each.
For centuries, mathematicians struggled with a basic question: how do you describe a relationship between two quantities? A farmer might know from experience that adding more fertilizer increases crop yield, but turning that knowledge into something precise — something you could write down, draw, or compute with — took a long time. The concept of a function evolved over hundreds of years, and with it came multiple ways to represent the same relationship.
y = x² could be drawn as curves on a coordinate plane. This was the birth of graphing — one of the key representations you'll use in this lesson.f(x) that we still use today, making algebraic representation the standard way to write functions.The central challenge this lesson addresses is straightforward but powerful: if two functions are shown in different formats, how do you figure out which one is bigger, faster, or steeper? You'll learn to extract the same properties — slope, y-intercept, rate of change — from any representation, so you can make fair comparisons.
Before we compare functions, you need to know what "properties" to look for. A function's properties are the measurable traits that describe how it behaves. Think of it like comparing two cars: you wouldn't just look at color — you'd compare speed, fuel efficiency, and price. Similarly, we compare functions using specific, well-defined properties.
m. A steeper slope means faster change.b in slope-intercept form, or the point (0, b).f(x) = 0.The key idea in this lesson is that the representation doesn't change the function itself. Whether you show a function as an equation, a graph, a table, or a sentence, the slope is still the slope and the y-intercept is still the y-intercept. Your job is to extract these properties from whatever form you're given, then line them up side by side.
The diagram below shows the exact same linear function presented in all four representations. Notice how the same information — a slope of 2 and a y-intercept of 1 — appears differently depending on the format. This is the core visual concept you need to internalize.
In the diagram above, all four boxes describe the same function. The algebraic box shows the equation directly. The graph shows the line with its slope triangle and y-intercept point. The table lists input-output pairs where you can calculate the rate of change by finding the difference in outputs (Δf(x) = +2) for each +1 change in x. The verbal description uses everyday language to convey slope and starting value. When you compare two functions, they might come from different boxes — and your job is to pull the same numbers out of each.
Here's your toolkit — the formulas and strategies for reading key properties out of each representation type. Master these, and you can compare any two functions no matter how they're presented.
If a function is given algebraically in slope-intercept form, you can read the slope and y-intercept directly. If the equation is in a different form — like standard form Ax + By = C — you'll need to rearrange it first by solving for y.
When you have a table of values, choose any two rows and compute the change in y divided by the change in x. If the function is linear, every pair of rows will give you the same slope. To find the y-intercept, look for the row where x = 0. If x = 0 isn't in the table, use the slope and one known point to calculate it with b = y − mx.
Reading a graph is visual: spot where the line crosses the y-axis for the y-intercept, then pick two points with whole-number coordinates and count the vertical and horizontal distance between them. Rise over run gives the slope. Remember that a line going downward from left to right has a negative slope.
Verbal descriptions use everyday language. Phrases like "charges $5 per hour" tell you the slope is 5, and "has a $20 base fee" tells you the y-intercept is 20. Once you decode the words into numbers, you have the same slope and intercept you'd get from any other representation.
When a problem asks you to compare two functions, follow a systematic process. The diagram below shows how to extract properties from a graph and a table, then compare them in a unified format.
The comparison table below summarizes which properties to look for and where to find them in each representation. Use this as a reference when solving problems.
| Property | Equation | Graph | Table | Verbal |
|---|---|---|---|---|
| Slope / Rate of Change | The coefficient of x (the m in y = mx + b) | Rise ÷ run between two points | (y₂ − y₁) ÷ (x₂ − x₁) | Look for "per," "each," or "rate" |
| Y-Intercept | The constant term (b) | Where the line crosses the y-axis | The f(x) value when x = 0 | "Starts at," "initial," "base fee" |
| X-Intercept (Zero) | Set y = 0, solve for x | Where the line crosses the x-axis | Find the row where f(x) = 0 (or interpolate) | "Reaches zero when…" |
| Increasing or Decreasing | Positive m → increasing; Negative m → decreasing | Line goes up left-to-right → increasing | As x increases, does f(x) go up or down? | "Grows," "declines," "loses" |
Let's walk through a complete comparison. Suppose you're given two functions:
Function f is defined by the equation f(x) = −2x + 8.
Function g is defined by this table:
| x | g(x) |
|---|---|
0 | 2 |
2 | 5 |
4 | 8 |
6 | 11 |
Question: Which function has the greater rate of change? Which has the greater y-intercept? At what input value will g(x) catch up to f(x)?
f(x) = −2x + 8 is already in slope-intercept form. We can read the properties directly:slope = (5 − 2) / (2 − 0) = 3 / 2 = 1.5. Let's verify with another pair, (2, 5) and (4, 8): slope = (8 − 5) / (4 − 2) = 3 / 2 = 1.5 ✓. The rate of change is constant, confirming g is linear.g(x) = 1.5x + 2. Then: −2x + 8 = 1.5x + 2 → 6 = 3.5x → x = 6 / 3.5No single representation is "best." Each has advantages and drawbacks, which is exactly why it's so valuable to move between them. The table below highlights what each format does well and where it falls short.
| Representation | Strengths | Limitations |
|---|---|---|
| Equation | Precise, compact, easy to manipulate algebraically. You can find any output for any input instantly. | Hard to "see" the overall shape. Students sometimes mis-read coefficients or forget to rearrange non-standard forms. |
| Graph | Shows the big picture at a glance — direction, steepness, intercepts, and where two functions cross. Great for visual learners. | Reading exact values can be imprecise (especially between grid lines). Requires careful axis scaling. |
| Table | Gives exact input-output pairs. Easy to spot patterns and compute rate of change numerically. | Shows only a few points. Might miss behavior between listed values. The y-intercept might not be included. |
| Verbal | Connects math to real-world context. Helps you understand what the function models and why it matters. | Can be ambiguous. Different people might interpret the same description differently. Requires careful keyword extraction. |
In this lesson, most of our examples have been linear functions — functions with a constant rate of change that produce straight-line graphs. But the skill of comparing properties across representations applies to every type of function you'll encounter in your math career.
| Feature | Linear (This Lesson) | Quadratic & Beyond (Coming Up) |
|---|---|---|
| Rate of change | Constant slope m | Changes at every point (you'll learn about "average rate of change" over intervals) |
| Key features to compare | Slope, y-intercept, x-intercept | Vertex, axis of symmetry, direction of opening, maximum/minimum values |
| Graph shape | Straight line | Parabolas, curves, exponential growth/decay |
| Same strategy? | Yes — extract, then compare | Yes! The "extract properties, then compare" approach works the same way |
When you study quadratic functions, you'll compare things like the vertex (highest or lowest point) and whether a parabola opens up or down. When you study exponential functions, you'll compare growth factors and initial values. The representations will still be equations, graphs, tables, and verbal descriptions — and the strategy you're learning right now will carry you through all of them.
In more advanced courses like Algebra 2 and Precalculus, you may even compare a linear function against an exponential one to determine which function "wins" in the long run. The same extraction-and-comparison framework applies, just with more properties to track.
Try these five problems on your own. Click "Show Answer" to check your work. Each problem increases in difficulty.
y = 4x + 1. Function B is described verbally: "B starts at 5 and increases by 3 for each unit increase in x." Without doing any calculations, which function has the greater rate of change, and which has the greater y-intercept?g(x) = 5x − 3. Find the slope and y-intercept of each function. Table for f: x = 0 → f(x) = 7; x = 1 → f(x) = 10; x = 2 → f(x) = 13; x = 3 → f(x) = 16.C = 0.10m + 30, where C is the monthly cost in dollars and m is the number of text messages sent. Plan B is given by a table: m = 0 → Cost = $50; m = 100 → Cost = $55; m = 200 → Cost = $60; m = 300 → Cost = $65. Which plan has the lower base cost? Which plan charges less per message? At how many messages do the two plans cost the same?Comparing two functions represented in different ways is a foundational skill in Algebra 1 that you'll use throughout your math education. The core strategy is always the same: extract key properties — slope (rate of change), y-intercept (starting value), x-intercept, and direction of change — from whatever representation you're given, whether that's an equation, a graph, a table of values, or a verbal description. Then you line those properties up side by side and make direct comparisons.
From an equation in slope-intercept form, you read the slope and intercept directly. From a graph, you use rise-over-run and find where the line crosses the y-axis. From a table, you compute the change in output divided by the change in input. From a verbal description, you identify keywords like "per," "starts at," and "rate." The representation changes — but the function's properties never do. Mastering this translation between forms is what separates surface-level math understanding from deep, flexible mathematical thinking.