MathematicsExponential Change: Multiplying by a Constant Factor
Atlas stands beside a lab bench covered with folded paper sheets of increasing thickness, a compound-interest table, and a glowing screen showing a steeply curving line labeled y = 1.06^x alongside a straight line labeled y = 1 + 0.06x. He lifts one thick stack of folded paper and grins, pointing to how each fold doubles the layers.
- Explain why multiplying by a constant factor each step produces exponential, not linear, change.
- Identify the growth or decay factor in an exponential expression of the form f(x) = a · bˣ.
- Use a table of values to determine whether a sequence is exponential or linear by checking for a constant ratio or constant difference.
- Predict the value of an exponential quantity after a given number of steps by applying repeated multiplication.
- Calculate the constant ratio between successive terms to determine whether a sequence is exponential.
Key terms
- Growth factor
- The constant multiplier b greater than 1 applied each step in exponential growth.
- Decay factor
- The constant multiplier b between 0 and 1 that shrinks a quantity each step.
- Constant ratio
- The fixed quotient between consecutive outputs that signals an exponential relationship.
- Base of the exponential
- The value b raised to the variable power in f(x) = a · bˣ.
- Half-life
- The time required for a decaying quantity to fall to half its current amount.
Reading the Parameters a and b
In f(x) = a · bˣ the initial value a is the output when x equals zero, because b⁰ equals 1. The base b controls the multiplicative behavior: each unit increase in x multiplies the previous output by exactly b. When b exceeds 1 the curve rises ever more steeply; when b lies strictly between 0 and 1 the curve falls toward, but never reaches, zero. Converting a percent rate to a base is routine — a 6% annual increase gives b = 1.06, while a 6% annual loss gives b = 0.94 = 1 − 0.06.
Two Tests From a Table
Given a table of equally spaced inputs, two quick tests classify the pattern. Compute first differences by subtracting consecutive outputs; if they are constant, the relationship is linear with that common difference as the slope. Compute ratios by dividing each output by the previous one; if those are constant, the relationship is exponential with that ratio as the base. A sequence cannot have both a constant difference and a constant ratio unless it is trivially constant, so the two tests rarely conflict and together cover the cases you will meet.
Why Exponentials Overtake Lines
A linear function adds a fixed amount each step, so its total grows in proportion to the number of steps. An exponential function multiplies, so its increment itself grows with the running total — growth feeds on growth. Even a tiny base above 1, paired with a small starting value, will eventually surpass any linear function no matter how steep that line is. This compounding explains why interest, populations, and viral spread are modeled exponentially and why long-run projections must respect the multiplicative structure rather than averaging it away.
Worked examples
Find the exponential model for a population of 800 growing 5% per year, then its size after 3 years.
- Identify the initial value a = 800 and convert the 5% growth rate to a base b = 1 + 0.05 = 1.05.
- Write the model f(x) = 800 · (1.05)ˣ where x is the number of years.
- Evaluate at x = 3: (1.05)³ = 1.157625, so f(3) = 800 · 1.157625 = 926.1.
Answer: f(x) = 800 · (1.05)ˣ, giving about 926 individuals after 3 years.
A 240 mg drug dose decays 25% each hour; how much remains after 2 hours?
- Losing 25% each hour leaves 75%, so the decay factor is b = 1 − 0.25 = 0.75.
- Write f(x) = 240 · (0.75)ˣ with x in hours.
- Evaluate at x = 2: (0.75)² = 0.5625, so f(2) = 240 · 0.5625 = 135.
Answer: 135 mg remain after 2 hours.
Activity
Drag each sequence card into the correct category — Linear or Exponential — then identify the constant difference or constant ratio for each sequence.
Practice
A car worth $24,000 loses 12% of its value each year; write a model and find its value after 4 years.
Given the table 5, 15, 45, 135, decide whether it is linear or exponential and state the constant difference or ratio.
Common mistakes to avoid
- Exponential and linear are the sameLinear functions add a constant difference each step while exponential functions multiply by a constant ratio, producing fundamentally different long-run behavior.
- The percent rate is the baseThe base is one plus the growth rate or one minus the decay rate, not the raw percent, so 6% growth gives a base of 1.06 rather than 0.06.
Check your understanding
A savings account starts with $500 and earns 6% interest each year, so the balance is multiplied by 1.06 every year. Which expression gives the balance after x years?
The table below shows outputs for a function. Which statement is correct? x: 0, 1, 2, 3 f(x): 4, 12, 36, 108
A radioactive substance loses half its mass every 10 years. If it starts at 200 grams, how many grams remain after 30 years?
Recap
Exponential change multiplies by a constant factor b each step in f(x) = a · bˣ, growing when b exceeds one and decaying when b lies between zero and one; a constant ratio in a table identifies it, and compounding lets it eventually outpace any linear model.
Reflect
What everyday quantities in your life grow or shrink by a constant percentage?