3.3 Sequences & Transformations

Arithmetic Sequences

An arithmetic sequence adds a fixed number, the common difference d, to get the next term. The sequence 4, 7, 10, 13, … has d = 3.

The Regents reference sheet gives both forms:

• Explicit: aₙ = a₁ + (n − 1)d — jump straight to any term n.
• Recursive: a₁ given, aₙ = aₙ₋₁ + d — build from the previous term.

Worked example. For 4, 7, 10, … find the 20th term. a₁ = 4, d = 3:

a₂₀ = 4 + (20 − 1)(3) = 4 + 57 = 61.

Watch the (n − 1): the first term uses zero jumps, not one.

The recursive form is handy when a question gives "each term is 3 more than the one before" — you write a₁ = 4 and aₙ = aₙ₋₁ + 3. Recursive rules describe the step; explicit rules jump to any term. The Regents may ask you to match a sequence to its correct recursive or explicit definition, so know both.

Geometric Sequences

A geometric sequence multiplies by a fixed factor, the common ratio r. The sequence 3, 6, 12, 24, … has r = 2 (each term is the previous times 2).

• Explicit: aₙ = a₁ · r^(n − 1)
• Recursive: a₁ given, aₙ = aₙ₋₁ · r

Worked example. For 3, 6, 12, … find the 6th term. a₁ = 3, r = 2:

a₆ = 3 · 2^(6−1) = 3 · 2⁵ = 3 · 32 = 96.

Find r by dividing any term by the one before it (6 ÷ 3 = 2). If dividing consecutive terms gives a constant, the sequence is geometric; if subtracting gives a constant, it is arithmetic.

Connecting Sequences to Functions

Sequences are just functions whose domain is the positive integers (term numbers):

The common difference d plays the role of slope; the common ratio r plays the role of the exponential base b. This is why the Regents treats aₙ = a₁ + (n−1)d as a disguised line and aₙ = a₁·r^(n−1) as a disguised exponential. Classify an unknown sequence by testing for a common difference first, then a common ratio.

Function Transformations

Starting from a parent graph f(x), small changes shift or flip it. The Regents tests these on lines, parabolas, and absolute-value graphs.

The counterintuitive rule: horizontal shifts move opposite the sign. f(x + 3) moves the graph left 3, not right, because the input reaches its old value sooner.

A reliable check: pick the graph's key point (the vertex of a parabola, the y-intercept of a line) and track where the transformation sends it. Vertical changes act on the output (outside the function) and behave normally; horizontal changes act on the input (inside the function) and behave in reverse. Keeping that inside/outside split straight prevents most transformation mistakes.

Transformations: Worked Examples

Let f(x) = x² (a parabola with vertex at the origin).

• g(x) = x² + 4 = f(x) + 4: shifts the vertex up to (0, 4).
• h(x) = (x − 5)² = f(x − 5): shifts the vertex right to (5, 0).
• p(x) = −x² = −f(x): reflects across the x-axis, so it opens downward.
• q(x) = (x + 2)² − 3: combines a left 2 and down 3 shift, vertex at (−2, −3).

Reading the vertex straight from this vertex form y = (x − h)² + k, where the vertex is (h, k), is a frequent Regents shortcut.