Understanding 8.F.1- Functions and Linear Models in 8th Grade
What Is 8.F.1 and Why It Matters
8.F.1 is a Common Core math standard for 8th graders. It asks students to understand what a function is and compare functions that are represented in different ways. That's it. No hidden complexity—just a clear focus on recognizing functions and analyzing how they behave.
Most 8th graders encounter functions for the first time here. The jump from basic algebra to understanding relationships between variables catches a lot of students off guard. But once you see how functions work, the whole thing clicks.
What Is a Function? The Core Idea
A function is a rule that assigns exactly one output to every input. You put something in, you get exactly one thing out. That's the whole definition.
Think of it like a machine:
- Drop in a number (the input)
- The function does something to it
- You get a number back (the output)
- Same input always gives the same output
If you can find even one input that produces two different outputs, it's not a function.
Function Notation
You'll see functions written as f(x). This isn't multiplication—it's notation. It means "the function f evaluated at x."
If f(x) = 2x + 3, then:
- f(2) = 2(2) + 3 = 7
- f(5) = 2(5) + 3 = 13
- f(-1) = 2(-1) + 3 = 1
The letter doesn't matter. You might see g(x), h(x), or k(x). They're just names.
Comparing Functions: Four Different Representations
8.F.1 requires students to compare functions presented in four common ways:
- Description with words
- A table of values
- A graph
- An equation
You need to be able to extract information from any of these formats and compare them.
Extracting Rate of Change from Each Format
The key comparison point is usually the rate of change—how fast the output changes when the input increases.
From an equation like y = 3x + 5, the rate of change is the coefficient of x. It's 3.
From a table, find the change in y divided by the change in x between any two points.
From a graph, pick two points and calculate the slope—rise over run.
From a description, look for language like "increases by 2 every time x goes up by 1." That's your rate.
Identifying Initial Values
After rate of change, the next thing to compare is the starting value or initial value.
In y = 3x + 5, the initial value is 5. That's where the line crosses the y-axis.
In a table, look for when x = 0 (or extrapolate backward).
In a description, look for "starts at" or "begins at."
Linear Functions: The Most Common Type
A linear function creates a straight line when graphed. Its equation takes the form y = mx + b, where m is the slope and b is the y-intercept.
Every linear function has a constant rate of change. The slope never changes. That's what makes it linear.
Non-linear functions curve. Their rates of change vary. Quadratic functions, exponential functions, and absolute value functions are all non-linear.
Linear vs. Non-Linear Quick Test
If the difference between consecutive outputs is always the same, it's linear. If the differences change, it's non-linear.
Example table—linear:
- x: 1, 2, 3, 4
- y: 5, 8, 11, 14
- Differences: +3, +3, +3 (constant → linear)
Example table—non-linear:
- x: 1, 2, 3, 4
- y: 2, 4, 8, 16
- Differences: +2, +4, +8 (changing → non-linear)
How to Identify If a Relation Is a Function
The vertical line test works for graphs. If a vertical line touches the graph in more than one place, it's not a function.
For tables, check if any input value appears more than once with different outputs. If input 3 shows up twice with y = 7 and y = 9, that's not a function.
For ordered pairs, same rule applies. (2, 5) and (2, 7) can't both be in the same function.
Comparing Functions: Worked Example
Problem: Compare two functions.
- Function A: described as "y increases by 4 for every increase of 1 in x, and starts at 2"
- Function B: y = 2x + 6
- Function C: table with points (0, 3), (1, 7), (2, 11), (3, 15)
Step 1: Put each into equation form.
- A: y = 4x + 2
- B: y = 2x + 6
- C: Rate is (7-3)/(1-0) = 4, starts at 3 → y = 4x + 3
Step 2: Compare rates of change.
- A: slope = 4
- B: slope = 2
- C: slope = 4
A and C have the same rate. B is slower.
Step 3: Compare starting values.
- A starts at 2
- B starts at 6
- C starts at 3
A starts lowest. B starts highest.
Common Mistakes Students Make
- Confusing the input and output. Remember: input goes in, output comes out.
- Forgetting that x = 0 gives you the y-intercept in most cases.
- Mixing up slope and y-intercept when reading equations.
- Assuming all graphs are functions without checking the vertical line test.
- Skipping the "exactly one output" rule. Two outputs for one input means it's not a function.
Quick Reference: Function Comparison Table
| Representation | How to Find Rate of Change | How to Find Initial Value |
|---|---|---|
| Equation (y = mx + b) | m (coefficient of x) | b (constant term) |
| Table of values | (y₂ - y₁) ÷ (x₂ - x₁) | y when x = 0, or extrapolate |
| Graph | Rise over run between two points | Where line crosses y-axis |
| Description | Look for "increases by ___ when x increases by 1" | Look for "starts at," "begins at," or "initial value" |
Getting Started: How to Approach These Problems
Step 1: Identify all the information you're given. What format is it in?
Step 2: Convert everything to equation form if needed. Get the slope and y-intercept for each function.
Step 3: Compare slopes first. Which function changes fastest? Which is flat?
Step 4: Compare starting values. Where does each function begin?
Step 5: Answer the specific question. Are they asking which has a greater rate? Which starts higher? Whether a relation is a function?
Bottom Line
8.F.1 comes down to two skills: recognizing functions and comparing them across formats. Once you can extract the slope and y-intercept from any representation, you're set. The standard doesn't ask for anything more complicated than that.