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Recognizing
Linear Functions

Learn to identify linear functions from tables and graphs by spotting a constant rate of change.

Pre-Algebra

What Is a Function?

A function is a rule that assigns exactly one output to each input. We often write it as \( y = f(x) \), where \( x \) is the input and \( y \) is the output.

Input \( x \)
The value you plug in: hours worked, items bought, etc.
Output \( y \)
The result the rule produces: total pay, total cost, etc.

What Makes a Function Linear?

A linear function is one whose graph forms a straight line. Its equation can always be written in the form:

The key feature: no matter which two points you pick on the line, the rate of change between them is always the same.

The Constant Rate of Change

\(x\)
\(y\)
0
1
2
3
4
0
1
2
3
4
+1
+1
+1
+1
run
rise

The Core Rule

In a Table
Equal steps in \( x \) always produce equal steps in \( y \).
On a Graph
The points form a perfectly straight line.

Recognizing from a Table

\(x\)
\(y\)
0
3
1
5
2
7
3
9
4
11
Differences:
\( \Delta x = +1 \)
\( \Delta x = +1 \)
\( \Delta x = +1 \)
\( \Delta x = +1 \)
\( \Delta y = +2 \)
\( \Delta y = +2 \)
\( \Delta y = +2 \)
\( \Delta y = +2 \)

Linear vs Non-Linear

Linear
Straight line
Non-Linear
Curved line
Linear
Constant rate of change. Graph is a straight line. Example: \( y = 2x + 1 \)
Non-Linear
Changing rate of change. Graph curves. Example: \( y = x^2 \)

A Non-Linear Table

\(x\)
\(y\)
0
0
1
1
2
4
3
9
4
16
Differences:
\( \Delta x = +1 \)
\( \Delta x = +1 \)
\( \Delta x = +1 \)
\( \Delta x = +1 \)
\( \Delta y = +1 \)
\( \Delta y = +3 \)
\( \Delta y = +5 \)
\( \Delta y = +7 \)

Calculating the Slope

The slope (rate of change) between any two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

1
Pick any two rows from the table (or two points on the graph).
2
Subtract the \( y \)-values (rise): \( y_2 - y_1 \).
3
Subtract the \( x \)-values (run): \( x_2 - x_1 \).
4
Divide: if the result is the same for every pair, the function is linear.

How to Check for Linearity

1
Look at the \( x \)-values. Are they evenly spaced? If yes, proceed to step 2. If not, you will need to compute the slope between each pair.
2
Find the differences in \( y \). Compute \( \Delta y \) for each consecutive pair of rows.
3
Compare all \( \Delta y \) values. If they are all equal, the rate of change is constant.
4
Conclude: constant rate = linear. Changing rate = non-linear.

Example 1: Table Check

\( x \) 0 2 4 6 8
\( y \) 5 11 17 23 29
1
\( x \) steps: \( 2 - 0 = 2 \), \( 4 - 2 = 2 \), etc. All \( \Delta x = 2 \).
2
\( y \) steps: \( 11 - 5 = 6 \), \( 17 - 11 = 6 \), \( 23 - 17 = 6 \), \( 29 - 23 = 6 \). All \( \Delta y = 6 \).
3
Slope: \( m = \frac{6}{2} = 3 \). Constant: this is a linear function.

Example 2: Graph Check

0
1
2
3
4
0
2
4
6
8
(0, 8)
(1, 6)
(2, 4)
(3, 2)
(4, 0)

Example 3: Tricky Non-Linear Table

\( x \) 1 2 3 4 5
\( y \) 2 4 8 16 32
1
\( \Delta y \) values: \( 4-2=2 \), \( 8-4=4 \), \( 16-8=8 \), \( 32-16=16 \).
2
The differences are 2, 4, 8, 16: they keep doubling! The rate of change is not constant.
3
Conclusion: This is non-linear (it is actually \( y = 2^x \), an exponential function).

Common Mistakes to Avoid

✕ Checking Only Two Points
Two points always form a straight line. You must check at least three pairs to confirm the rate of change is constant.
✕ Ignoring the \( x \)-spacing
If \( x \)-values are not evenly spaced (e.g., 1, 2, 5, 10), you cannot just compare \( \Delta y \). You must compute \( \frac{\Delta y}{\Delta x} \) for each pair.
✕ Thinking "Goes Up = Linear"
A function can increase and still be non-linear (e.g., \( y = x^2 \) increases but curves). The rate of increase must be constant.

Summary Comparison

Linear Non-Linear
Rate of change Constant Changes
Graph shape Straight line Curve
Equation form \( y = mx + b \) \( y = x^2 \), \( y = 2^x \), etc.
Table test Equal \( \Delta y \) for equal \( \Delta x \) Unequal \( \Delta y \) for equal \( \Delta x \)
Real-world example Earning \(\$10/\text{hour}\) Population doubling

Question 1 of 5

A table shows: when \( x = 1, y = 4 \); when \( x = 2, y = 7 \); when \( x = 3, y = 10 \); when \( x = 4, y = 13 \). Is this function linear?

Question 2 of 5

Which of the following equations represents a linear function?

Question 3 of 5

A table shows: \( x \): 0, 1, 2, 3 and \( y \): 1, 3, 9, 27. What is the rate of change between consecutive rows?

Question 4 of 5

A graph passes through \( (1, 5) \), \( (3, 11) \), and \( (5, 17) \). What is the slope, and is the function linear?

Question 5 of 5

A student says: "The table below is linear because \( y \) increases every time." Is the student correct?

\( x \): 1, 2, 3, 4   \( y \): 2, 6, 12, 20