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Recognize
Constant Rate Changes

Learn to identify when one quantity changes at a constant rate per unit interval relative to another: the hallmark of linear functions.

Algebra 2

What Is a Constant Rate of Change?

A quantity has a constant rate of change when equal changes in the input always produce equal changes in the output. No matter where you look in the data, the ratio of output change to input change is the same.

Seeing It on a Graph

x y 0 2 4 6 8 1 2 3 4 +1 +2 +1 +2 \( y = 2x \)

Why Does Constant Rate Matter?

Constant rate of change is the defining feature of linear functions. Recognizing it allows you to predict outputs, write equations, and distinguish linear growth from exponential or other non-linear patterns.

Prediction
If the rate is constant, you can extrapolate any value from just two data points.
Modeling
Real-world scenarios like hourly wages, speed, and unit pricing are all constant-rate situations.
Classification
Identifying constant rate lets you separate linear from exponential or quadratic growth.

The Formal Definition

This ratio is the slope \( m \). Any function satisfying this condition can be written in slope-intercept form:

The Slope Triangle

x y run \( = 2 \) rise \( = 3 \) (1, 2.5) (3, 5.5) slope \( = \frac{3}{2} \)

Constant vs. Non-Constant Rate

Linear (Constant Rate) Exponential (Non-Constant)
Linear
Equal \(\Delta x\) always produces equal \(\Delta y\). Graph is a straight line.
Exponential
Equal \(\Delta x\) produces changing \(\Delta y\). The change in \(y\) is not constant. The graph is a curve.

Rate of Change vs. Common Ratio

Constant Rate (Linear) Constant Ratio (Exponential)
Pattern Add \( m \) each step Multiply by \( r \) each step
Formula \( f(x) = mx + b \) \( f(x) = a \cdot r^x \)
First differences Constant Not constant
Consecutive ratios Not constant Constant
Graph shape Straight line Curve

How to Identify Constant Rate

1
List the data. Organize inputs and outputs into a table with equally spaced \( x \)-values.
2
Compute first differences. For each pair, calculate \( \Delta y = f(x_{n+1}) - f(x_n) \).
3
Check consistency. If every \( \Delta y \) is the same, the rate is constant.
4
State the rate. The constant difference is \( m \), the slope: \( m = \frac{\Delta y}{\Delta x} \).

Example 1: Reading a Table

A plant grows over several weeks. Is the growth rate constant?

Week (\(x\)) 0 1 2 3 4
Height cm (\(y\)) 5 8 11 14 17
\(\Delta y\) +3 +3 +3 +3

Example 2: A Taxi Fare

A taxi charges a $3.00 base fare plus $2.50 per mile. Is this a constant rate situation?

1
Identify the quantities. Input: miles driven (\(x\)). Output: total fare (\(y\)).
2
Write the equation. \( y = 2.50x + 3.00 \). This is in \( y = mx + b \) form.
3
Check: Each additional mile adds $2.50. That never changes, so the rate is constant.

Example 3: Not a Constant Rate

A population of bacteria doubles every hour. Is the rate constant?

Hour (\(x\)) 0 1 2 3 4
Bacteria (\(y\)) 100 200 400 800 1600
\(\Delta y\) +100 +200 +400 +800

Example 4: Negative Constant Rate

hours gallons 0 50 1 2 3 4 +1 -10

Language Signals for Constant Rate

Certain keywords in word problems tell you the rate is constant. Look for these clues:

"per"
$5 per hour, 3 cm per week, 60 miles per hour
"each"
$2 for each item, 4 points each round
"every"
Grows 6 in. every month, loses 5 lbs every week

Common Mistakes

Confusing rate with ratio
"Doubles every hour" sounds constant, but that is a constant ratio (exponential), not a constant rate of change (linear).
Checking only two points
Two points always produce a straight line. You need at least three data points to confirm that the rate truly stays constant.
Ignoring unequal spacing
If the \( x \)-values are not equally spaced, the differences in \( y \) will naturally vary even for a linear function. Always use \( \frac{\Delta y}{\Delta x} \), not just \( \Delta y \) alone.

Quick Reference Summary

Question Constant Rate (Linear) Non-Constant Rate
First differences? All equal Vary
Graph shape? Straight line Curve
Equation form? \( y = mx + b \) \( y = a \cdot r^x \), \( y = ax^2 + \ldots \), etc.
Real-world clue? "per", "each", "every" "doubles", "triples", "percent increase"
Slope formula \( m = \frac{\Delta y}{\Delta x} \) (constant) Varies by interval

Question 1 of 5

A gym membership costs $25 per month plus a $50 sign-up fee. Which equation models the total cost \( C \) after \( m \) months?

Question 2 of 5

Given the table below, does \( f(x) \) have a constant rate of change?

\(x\): 0, 2, 4, 6
\(f(x)\): 3, 9, 15, 21

Question 3 of 5

Which situation describes a constant rate of change?

Question 4 of 5

The table shows values for a function \(g(x)\):

\(x\): 1, 2, 3, 4
\(g(x)\): 4, 8, 16, 32

What is the best description of this function?

Question 5 of 5

A phone plan charges a flat $15 monthly fee plus $0.10 per text message. If you send 200 texts, what is the total monthly cost, and what is the constant rate?