Exponential Function

Q: What is the difference between an exponential function and a linear function?

A linear function changes by a constant amount each step (e.g., adding 5 every time), while an exponential function changes by a constant ratio or multiplier each step (e.g., multiplying by 2 every time). This is why exponential functions grow much faster than linear functions for large values of x. Graphically, a linear function is a straight line, whereas an exponential function is a curve that gets steeper (growth) or flatter (decay) over time.

Q: Why can't the base b equal 1 or be negative in an exponential function?

If b = 1, then bˣ = 1 for every x, making y = a · 1 = a — just a constant, not an exponential function. If b were negative, bˣ would be undefined for many values of x (for example, (−2)^(1/2) is not a real number), so we restrict b to positive values other than 1.

Q: How do you find the equation of an exponential function from two points?

Set up two equations using y = a · bˣ with the coordinates of each point. Divide one equation by the other to eliminate a and solve for b. Then substitute b back into either equation to find a. For instance, given (0, 5) and (2, 45): from the first point, a = 5; from the second, 45 = 5 · b², so b² = 9 and b = 3.

Exponential Function
Exponential Model

A function of the form y = a·b^x where a > 0 and either 0 < b < 1 or b > 1. The variables do not have to be x and y. For example, A = 3.2·(1.02)^t is an exponential function.

Note: Exponential functions are used to model exponential growth, exponential decay, compound interest, and continuously compounded interest.

See also

Key Formula

y = a \cdot b^{x}

Where:

$y$ = The output (dependent variable)
$a$ = The initial value (the y-value when x = 0); must be greater than 0
$b$ = The base, or constant multiplier; b > 0 and b ≠ 1. If b > 1, the function models growth. If 0 < b < 1, it models decay.
$x$ = The exponent (independent variable)

Worked Example

Problem: A bacteria colony starts with 500 bacteria and triples every hour. Write an exponential function for the population P after t hours, then find the population after 4 hours.

Step 1: Identify the initial value a. At t = 0 the population is 500, so a = 500.

a = 500

Step 2: Identify the base b. The population triples each hour, so the constant multiplier is 3.

b = 3

Step 3: Write the exponential function using the form P = a · bᵗ.

P = 500 \cdot 3^{t}

Step 4: Substitute t = 4 to find the population after 4 hours.

P = 500 \cdot 3^{4} = 500 \cdot 81 = 40{,}500

Answer: The exponential function is P = 500 · 3ᵗ, and after 4 hours the population is 40,500 bacteria.

Another Example

This example shows exponential decay (0 < b < 1) and demonstrates how to convert a percentage decrease into the base b, contrasting with the growth example above.

Problem: A car worth $20,000 depreciates by 15% each year. Write an exponential function for the car's value V after t years, and find its value after 3 years.

Step 1: Identify the initial value. The car starts at $20,000, so a = 20,000.

a = 20{,}000

Step 2: Find the base b. Losing 15% each year means the car retains 85% of its value, so b = 1 − 0.15 = 0.85. Since 0 < 0.85 < 1, this is exponential decay.

b = 1 - 0.15 = 0.85

Step 3: Write the exponential function.

V = 20{,}000 \cdot 0.85^{t}

Step 4: Substitute t = 3.

V = 20{,}000 \cdot 0.85^{3} = 20{,}000 \cdot 0.614125 = 12{,}282.50

Answer: The function is V = 20,000 · 0.85ᵗ. After 3 years, the car is worth $12,282.50.

Frequently Asked Questions

What is the difference between an exponential function and a linear function?

A linear function changes by a constant amount each step (e.g., adding 5 every time), while an exponential function changes by a constant ratio or multiplier each step (e.g., multiplying by 2 every time). This is why exponential functions grow much faster than linear functions for large values of x. Graphically, a linear function is a straight line, whereas an exponential function is a curve that gets steeper (growth) or flatter (decay) over time.

Why can't the base b equal 1 or be negative in an exponential function?

If b = 1, then bˣ = 1 for every x, making y = a · 1 = a — just a constant, not an exponential function. If b were negative, bˣ would be undefined for many values of x (for example, (−2)^(1/2) is not a real number), so we restrict b to positive values other than 1.

How do you find the equation of an exponential function from two points?

Set up two equations using y = a · bˣ with the coordinates of each point. Divide one equation by the other to eliminate a and solve for b. Then substitute b back into either equation to find a. For instance, given (0, 5) and (2, 45): from the first point, a = 5; from the second, 45 = 5 · b², so b² = 9 and b = 3.

Exponential Function vs. Linear Function

	Exponential Function	Linear Function
General form	y = a · bˣ	y = mx + c
Type of change	Constant ratio (multiplicative)	Constant difference (additive)
Graph shape	Curved (J-shape for growth, decreasing curve for decay)	Straight line
Rate of change	Increases (or decreases) as x increases	Constant slope m
Example	y = 100 · 2ˣ (doubles each step)	y = 100 + 50x (adds 50 each step)

Why It Matters

Exponential functions appear throughout algebra, precalculus, and science courses whenever quantities grow or shrink by a fixed percentage. You will use them to model population growth, radioactive decay, compound interest on savings accounts, and the spread of diseases. Mastering them is also essential preparation for logarithms, since logarithmic functions are the inverses of exponential functions.

Common Mistakes

Mistake: Confusing the initial value a with the base b, or placing the variable x as the base instead of the exponent (writing y = xᵇ instead of y = a · bˣ).

Correction: Remember that in an exponential function the variable is always in the exponent. The base b is a fixed positive number, and a is the y-value when x = 0.

Mistake: Using b = 0.15 for a 15% decrease instead of b = 0.85.

Correction: A 15% decrease means you keep 85% of the value each period, so b = 1 − 0.15 = 0.85. Similarly, a 15% increase gives b = 1 + 0.15 = 1.15. The base represents the fraction retained, not the fraction lost or gained.

Related Terms

Function — General concept that exponential functions belong to
Exponential Growth — Occurs when the base b > 1
Exponential Decay — Occurs when 0 < b < 1
Compound Interest — Key financial application of exponential functions
Continuously Compounded Interest — Uses the natural base e in the exponent
e — Natural base ≈ 2.718 used in many exponential models
Variable — The exponent x is the independent variable
Model — Exponential functions model real-world growth and decay