Exponential Functions: Properties, Graphs, Transformations and Asymptotes

An exponential function has the form f(x) = a · b^x with b > 0 and b ≠ 1, where the variable is in the exponent and the graph has a horizontal asymptote. In brief: f(x) = 2^x doubles whenever x increases by 1, passes through (0, 1), is always positive and approaches the x-axis as x goes to −∞. For 0 < b < 1, as in (1/2)^x, the function decreases.

Exponential functions model growth and decay in which the rate of change is proportional to the current amount: populations, money with compound interest, radioactive material. Their graphs have a typical shape, and shifts and reflections move the asymptote and change the range. This guide covers the properties, the graphs for base greater and less than 1, the transformations, a value table and common mistakes.

What is an exponential function?

The general form is f(x) = a · b^x. The base b is a positive constant different from 1, and a is the initial value, f(0) = a. If b > 1 the function describes growth, and b is the growth factor. If 0 < b < 1 it describes decay, and b is the decay factor. The base is excluded for b = 1, since 1^x = 1 is a constant, and for b ≤ 0, where powers are not defined for all real x.

Do not confuse it with a power function x², where the base is the variable. In 2^x the base is fixed and the exponent changes.

What are the properties of f(x) = bˣ?

• The domain is all real numbers.
• The range is y > 0, because b^x is always positive.
• The graph passes through (0, 1) and (1, b).
• There is a horizontal asymptote y = 0: for b > 1, b^x → 0 as x → −∞, and for 0 < b < 1, b^x → 0 as x → +∞.
• The function is strictly increasing for b > 1 and strictly decreasing for 0 < b < 1, so it is one-to-one and has an inverse, the logarithm.
• Key rule: b^{x + y} = b^x · b^y. Adding to x multiplies the output.
• The graphs of b^x and (1/b)^x = b^−x are mirror images in the y-axis.

What does the graph look like?

For b > 1, the graph rises slowly at first (for negative x), crosses the y-axis at 1 and then climbs quickly. A larger base gives a steeper graph on the right. For x > 0 the curve eventually grows faster than any polynomial.

The curve never touches the x-axis. Moving to the left the values get smaller but remain positive: 2⁻¹⁰ = 1/1024 ≈ 0.001.

How do transformations change the graph and the asymptote?

For g(x) = a · b^{x − h} + k:

• k shifts the graph vertically, and the asymptote moves to y = k.
• h shifts the graph horizontally and does not change the asymptote.
• a < 0 reflects the graph in the x-axis, and the range becomes y < k.
• |a| stretches the graph vertically and changes the y-intercept.

Example 1: g(x) = 2^{x − 1} + 3

The asymptote is y = 3, the range is y > 3 and the point (0, 1) moves to (1, 4). The y-intercept is g(0) = 1/2 + 3 = 3.5.

Example 2: g(x) = −3 · 2^x + 5

The graph opens downward from the asymptote y = 5, so the range is y < 5. The y-intercept is g(0) = −3 + 5 = 2. The zero: 3 · 2^x = 5, 2^x = 5/3, so x = log₂(5/3) ≈ 0.737.

Example 3: finding a formula from two points

An exponential function f(x) = a · b^x passes through (1, 6) and (3, 24). The ratio f(3)/f(1) = b² = 4, so b = 2. Then a = 6/2 = 3. The function is f(x) = 3 · 2^x. Check: f(3) = 3 · 8 = 24 ✓.

Value table for 2ˣ and (1/2)ˣ

Where does the exponential function come from?

Babylonian tablets contain problems about compound interest, which are essentially exponential. The function became a central object with Euler’s work in the eighteenth century, who introduced the number e and studied exponential functions systematically. The connection between exponentials and logarithms, introduced by Napier.

Where are exponential functions used today?

A bank account of 1,000 dollars with 5% yearly interest has the value A(t) = 1000 · 1.05^t. After 10 years it is about 1,628.89 dollars. A medicine with a decay factor of 0.8 per hour leaves 0.8³ = 51.2% after 3 hours. Computing, epidemics, and sound intensity in decibels all involve exponentials.

Which mistakes are common?

Confusing exponential and power functions

2^x and x² are different. At x = 5, 2⁵ = 32 and 5² = 25, but at x = 3, 2³ = 8 and 3² = 9.

Thinking the graph reaches zero

The graph approaches the asymptote but never touches it, so there is no x with b^x = 0.

Moving the asymptote with a horizontal shift

Only the vertical shift k changes the asymptote.

Mixing up growth and decay factors

A decrease of 20% corresponds to the factor 0.8, not 0.2.