Linear VS. Exponential - Algebra-1

1. Fundamental Concepts

Linear Relationships: They describe a constant rate of change, following the linear function model , where:
- m is the slope (constant rate of change, representing the fixed amount added or subtracted per unit of time/variable).
- b is the y-intercept (initial value when the independent variable ).
- The relationship shows a straight-line graph, with the dependent variable changing by the same absolute amount for each unit increase in the independent variable.
Exponential Relationships: They describe a proportional rate of change, following the exponential function model , where:
- a is the initial value (quantity when ).
- b is the growth/decay factor ( for growth, for decay; it represents the multiplier applied to the current quantity per unit of time/variable).
- The relationship shows a curved graph, with the dependent variable changing by a constant percentage or ratio for each unit increase in the independent variable.

Rate of Change:
- Linear: Has a constant absolute rate of change (e.g., increasing by 5 units every month).
- Exponential: Has a constant relative rate of change (e.g., increasing by 5% every month, or doubling every 3 years).
Graphical Differences:
- Linear: Produces a straight line on a coordinate plane.
- Exponential: Produces a curved line—rising steeply (for growth, ) or flattening out (for decay, ).
Long-Term Behavior:
- Linear: Grows or decreases at a steady, unchanging pace; the growth/decay remains predictable and moderate over time.
- Exponential: Grows extremely rapidly (for ) or shrinks toward zero (for ) over time, often outpacing or falling behind linear relationships significantly in the long run.
Key Identifiers in Context:
- Linear: Clues include "adds/subtracts a fixed number each time," "constant amount per period," or "steady rate of increase/decrease."
- Exponential: Clues include "multiplies by a factor each time," "doubles/halves every period," "grows by a percentage," or "scales by a ratio per unit time."

Easy Level
Identify whether the following relationships are linear or exponential:
1. A savings account starts with 100 dollars and earns 20 dollars each month.
2. A bacteria colony starts with 100 cells and doubles every hour.
Answers:
1. Linear (constant addition of $20 per month, following ).
2. Exponential (doubling every hour, following ).
Medium Level
Compare the two scenarios over 5 years:
- Scenario A: A plant is 2 feet tall and grows 0.5 feet each year.
- Scenario B: A plant is 2 feet tall and grows by 20% each year.
Analysis:
- Scenario A (linear): . After 5 years, height = feet.
- Scenario B (exponential): . After 5 years, height = feet. Conclusion: Scenario B is taller after 5 years.
Hard Level
Determine if each table is linear or exponential, and explain why:

Table A:

x -5 -4 -3 -2

y 16 4 1 0.25

Table B:

x 2 5 6 10

y 5 11 13 21

Answers:
- Table A: Exponential. The ratio between consecutive y-values is constant (, , etc.), so .
- Table B: Linear. The difference between consecutive y-values is constant (, [wait, corrected: for x increasing by 3 (5-2), y increases by 6 (11-5); x increasing by 1 (6-5), y increases by 2 (13-11); x increasing by 4 (10-6), y increases by 8 (21-13)], so slope , following .

x	-5	-4	-3	-2
y	16	4	1	0.25

x	2	5	6	10
y	5	11	13	21

Identify the Type of Relationship:
- For linear: Look for a constant absolute change in y for each unit change in x (e.g., "increases by 5 each time").
- For exponential: Look for a constant relative change in y for each unit change in x (e.g., "multiplies by 0.5 each time" or "grows by 10% each time").
Analyze Graphs or Data Tables:
- Linear: Graph is a straight line; in tables, the difference between consecutive y-values is constant (regardless of x increments, if normalized to unit x change).
- Exponential: Graph is curved; in tables, the ratio between consecutive y-values is constant.
Formulate Equations:
- Linear: Use , where (slope) and b is the initial value.
- Exponential: Use , where a is the initial value and b is the growth/decay factor (calculated as or ).
Compare and Predict:
- For short-term periods, linear relationships may be larger (if initial growth is higher).
- For long-term periods, exponential relationships (with ) will eventually surpass linear ones due to compounding growth.
- Verify predictions by plugging in specific values for x and comparing results.