AP Precalculus Unit 2 Notes: Understanding Exponentials Through Patterns, Graphs, and Models

Arithmetic and Geometric Sequences

What a sequence is (and why Unit 2 starts here)

A sequence is an ordered list of numbers—called terms—that follow a rule. In precalculus, sequences matter because they are a “discrete” way to describe patterns of change. Exponential functions are “continuous,” but they often come from the same underlying idea as geometric sequences: repeated multiplication by a constant factor.

When you learn to recognize whether a pattern adds the same amount each step (arithmetic) or multiplies by the same factor each step (geometric), you’re training the exact skill you’ll need to choose and justify an exponential model later.

Arithmetic sequences: constant difference

An arithmetic sequence changes by adding (or subtracting) the same number each step. That constant is the common difference.

If the first term is a_1 and the common difference is d, then each term is:

a_{n} = a_1 + (n-1)d

Why it matters: Arithmetic sequences are the discrete version of linear behavior. If data increases by about the same amount each time period, a linear model is often more appropriate than an exponential one.

How it works: You check consecutive differences:

• Compute a_2-a_1, a_3-a_2, a_4-a_3, etc.
• If they’re (approximately) the same, the pattern is arithmetic.

Example 1: Identify and write an explicit formula

Sequence: 5, 9, 13, 17, ...

The differences are:

• 9-5 = 4
• 13-9 = 4
• 17-13 = 4

So d = 4 and a_1 = 5. The explicit formula is:

a_n = 5 + (n-1)4

You can simplify, but keeping it in this form reduces mistakes.

Geometric sequences: constant ratio

A geometric sequence changes by multiplying by the same number each step. That constant is the common ratio.

If the first term is a_1 and the common ratio is r, then:

a_{n} = a_1 r^{n-1}

Why it matters: This is the discrete version of exponential behavior. If a quantity grows by a constant percent each time period, it forms (approximately) a geometric sequence.

How it works: You check consecutive ratios:

• Compute \frac{a_2}{a_1}, \frac{a_3}{a_2}, \frac{a_4}{a_3}, etc.
• If they’re (approximately) the same, the pattern is geometric.

Example 2: Identify and write an explicit formula

Sequence: 3, 6, 12, 24, ...

Ratios:

• \frac{6}{3} = 2
• \frac{12}{6} = 2
• \frac{24}{12} = 2

So r = 2 and a_1 = 3. The explicit formula is:

a_n = 3 \cdot 2^{n-1}

Arithmetic vs. geometric (how to tell quickly)

A reliable way to decide is to ask: “Is the change additive or multiplicative?”

A common misconception is thinking “if it increases fast, it must be exponential.” Speed alone doesn’t decide it; the type of change does. For example, adding 100 each step can look “fast,” but it’s still arithmetic.

Exam Focus

• Typical question patterns:
  • Given a list or table, decide whether it’s arithmetic or geometric and write an explicit formula.
  • Find a missing term using a common difference or common ratio.
  • Connect a geometric sequence to an exponential function form.
• Common mistakes:
  • Using differences when you should use ratios (or vice versa). If the data involves percent change, ratios are usually the right tool.
  • Mixing up indexing: writing a_n = a_1 r^n instead of a_1 r^{n-1}.
  • Computing ratios when terms can be zero or change sign without checking whether the ratio is consistent.

Exponential Functions and Their Graphs

What an exponential function is

An exponential function is a function where the input appears in the exponent. The most common form in precalculus is:

f(x) = a b^x

Here:

• a is the **initial value** (specifically, f(0)=a).
• b is the base (the constant multiplicative factor).

Why it matters: Exponential functions model situations with constant multiplicative change—like constant percent growth/decay, compound interest, population growth under ideal conditions, depreciation, and radioactive decay.

How the parameters a and b affect the graph

Think of b as the “multiplier per 1 unit of x.”

• If b > 1, the function shows exponential growth.
• If 0 < b < 1, the function shows exponential decay.
• The value a vertically scales the function and sets the y-intercept.

Key graphical features:

• Y-intercept:

f(0) = a b^0 = a

• Horizontal asymptote (basic form): For f(x)=ab^x with b>0, the graph approaches y=0 as x goes in one direction.
• Domain: All real numbers.
• Range: Depends on the sign of a. If a>0, then f(x)>0 for all x.

A frequent misconception is mixing up “initial value” with “value at x=1.” In the form ab^x, the initial value corresponds to x=0.

Exponential functions as continuous versions of geometric sequences

A geometric sequence is defined only for integer step numbers n:

a_n = a_1 r^{n-1}

An exponential function extends that pattern to all real inputs:

f(x) = a b^x

If you sample an exponential function at integer values of x, you get a geometric pattern in the outputs.

Transformations: shifting and changing the asymptote

Many AP-style problems use a transformed exponential:

f(x) = a b^{x-h} + k

Interpretation:

• h shifts the graph horizontally (right if h>0).
• k shifts the graph vertically, changing the horizontal asymptote to:

y = k

Also, a can reflect the graph across the horizontal asymptote if a