AP Pre-Calc Unit 2.1-2.4

What is a sequence?

A function whose domain is a set of consecutive integers, usually starting at 0 or 1.

What is the domain of a sequence?

Discrete integer values, usually n = 0, 1, 2, 3, ...

Why are sequence graphs made of separate dots?

Sequences have discrete domains, so values between the terms are not included.

Should you connect dots on a sequence graph?

No. Only plot the individual terms because sequences are not continuous.

What does aₙ mean?

The value of the nth term in an arithmetic sequence.

What does gₙ mean?

The value of the nth term in a geometric sequence.

What does a₀ or g₀ represent?

The initial term, at index n = 0.

What does aₖ or gₖ represent?

The value of the sequence at index k.

What does n represent in a sequence?

The index or term number.

What is an arithmetic sequence?

A sequence where consecutive terms have a constant difference.

How does an arithmetic sequence change?

By adding the same common difference each step.

What is the common difference d?

The constant amount added to obtain the next term.

How do you find the common difference using consecutive terms?

d = aₙ₊₁ − aₙ.

How do you find d using two nonconsecutive terms?

d = (aₙ − aₖ)/(n − k).

What is the arithmetic sequence equation using a₀?

aₙ = a₀ + dn.

What is the other arithmetic sequence equation?

aₙ = aₖ + d(n − k).

What does d mean in an arithmetic equation?

The common difference or constant rate of change.

How does a positive common difference affect a sequence?

Its terms increase by the same amount each step.

How does a negative common difference affect a sequence?

Its terms decrease by the same amount each step.

What happens when d = 0?

Every term is equal, making a constant sequence.

How do you find a₀ from a₁?

a₀ = a₁ − d.

How do you find a₀ if the common difference is positive?

Subtract the positive difference from a₁.

How do you find a₀ if the common difference is negative?

Subtract the negative difference, which means adding its absolute value.

How do you find a₀ from aₖ?

a₀ = aₖ − dk.

Why is using a known term helpful when writing a sequence equation?

You can use aₖ + d(n − k) without first finding a₀.

What type of function behaves like an arithmetic sequence?

A linear function, except the sequence has a discrete domain.

What does an arithmetic sequence graph look like?

Separate points lying on a straight line.

How does an arithmetic sequence's output change over equal steps?

By the same amount, not the same percentage.

What is linear slope-intercept form?

f(x) = b + mx.

What is linear point-slope form?

f(x) = y₁ + m(x − x₁).

How does linear slope m relate to arithmetic difference d?

They both represent constant additive change.

How do you write an arithmetic equation when a₂ = 7 and a₆ = 9?

d = (9 − 7)/(6 − 2) = 1/2; aₙ = 9 + (1/2)(n − 6).

For a₂ = 7 and a₆ = 9, what is a₂₄?

9 + (1/2)(24 − 6) = 18.

If aₙ = 8 − 2n, what is d?

−2.

If aₙ = 8 − 2n, what is a₁₇?

8 − 2(17) = −26.

How do you recognize an arithmetic sequence in a table?

Equal changes in x produce equal differences in y.

What is a geometric sequence?

A sequence where consecutive terms have a constant ratio.

How does a geometric sequence change?

By multiplying each term by the same number.

What is the common ratio r?

The constant multiplier between consecutive terms.

How do you find r using consecutive nonzero terms?

r = gₙ₊₁/gₙ.

What is the geometric sequence formula using g₀?

gₙ = g₀rⁿ.

What is the other geometric sequence formula?

gₙ = gₖr⁽ⁿ⁻ᵏ⁾.

What does r represent in the geometric sequence equation?

The common ratio or multiplication factor.

How do you find g₀ from g₁?

g₀ = g₁/r.

How do you find g₀ if r is positive?

Divide g₁ by r, regardless of whether r is above or below 1.

How do you find g₀ if r is negative?

Divide g₁ by the negative ratio; the sign changes if g₁ is nonzero.

How do you find g₀ from gₖ?

g₀ = gₖ/rᵏ, when r ≠ 0.

What does 0 < r < 1 indicate when g₀ > 0?

Geometric decay; each term is a smaller positive value.

What does r > 1 indicate when g₀ > 0?

Geometric growth; each term increases.

What happens when r < 0?

The terms alternate signs if the initial term is nonzero.

How does a geometric sequence change proportionally?

It has the same percentage change each step when r is positive.

What type of function behaves like a geometric sequence?

An exponential function, except the sequence is discrete.

What does a geometric sequence graph look like?

Separate points following an exponential pattern.

Why shouldn't you connect geometric sequence points?

The sequence is discrete, even though the corresponding exponential function is continuous.

How do you find r from two geometric terms?

Use gₙ = gₖr⁽ⁿ⁻ᵏ⁾, divide by gₖ, then solve for r.

Which geometric term should you use as gₙ when given two terms?

Use the term with the larger index n.

What is the geometric equation if g₃ = −3 and r = 10?

gₙ = −3(10)⁽ⁿ⁻³⁾.

What is another equation if g₃ = −3 and r = 10?

gₙ = −0.003(10)ⁿ.

What is g₀ when g₃ = −3 and r = 10?

−3/10³ = −3/1000.

What is the equation if g₂ = 24 and r = 1/2?

gₙ = 24(1/2)⁽ⁿ⁻²⁾ = 96(1/2)ⁿ.

What is g₅ if g₂ = 24 and r = 1/2?

24(1/2)³ = 3.

If g₃ = −2 and g₆ = 128, what is r?

128 = −2r³; r³ = −64; r = −4.

What is the equation if g₃ = −2 and g₆ = 128?

gₙ = −2(−4)⁽ⁿ⁻³⁾.

For g₃ = −2 and r = −4, what is g₁₁?

−2(−4)⁸ = −131,072.

If g₂ = 48 and g₇ = 1.5, what is r?

1.5 = 48r⁵; r⁵ = 1/32; r = 1/2.

What is the equation if g₂ = 48 and g₇ = 1.5?

gₙ = 48(1/2)⁽ⁿ⁻²⁾.

If g₂ = 48 and r = 1/2, what is g₁₁?

48(1/2)⁹ = 3/32.

What is the general form of an exponential function?

f(x) = abˣ, with a ≠ 0, b > 0, and b ≠ 1.

What does a represent in f(x) = abˣ?

The initial value, f(0), or y-intercept.

What does b represent in f(x) = abˣ?

The growth or decay factor.

What does b > 1 mean when a > 0?

Exponential growth; the function increases.

What does 0 < b < 1 mean when a > 0?

Exponential decay; the function decreases.

What happens when a < 0 and b > 1?

The function decreases and is concave down.

What happens when a < 0 and 0 < b < 1?

The function increases and is concave down.

What happens when a > 0 in an exponential function?

The graph is above the x-axis and concave up.

What happens when a < 0 in an exponential function?

The graph is below the x-axis and concave down.

What is the domain of f(x) = abˣ?

All real numbers, (−∞, ∞).

What is the range of f(x) = abˣ if a > 0?

(0, ∞).

What is the range of f(x) = abˣ if a < 0?

(−∞, 0).

What is the horizontal asymptote of f(x) = abˣ?

y = 0.

Can a basic exponential function equal zero?

No, if a ≠ 0 and b > 0.

Do basic exponential functions have relative maxima or minima?

No. They are always increasing or always decreasing.

Can a basic exponential function switch concavity?

No. Its concavity stays the same.

Does a basic exponential function have an inflection point?

No, because it never changes concavity.

What does concave up mean?

The graph bends upward and its slopes increase.

What does concave down mean?

The graph bends downward and its slopes decrease.

Can an exponential function decrease while concave up?

Yes. Positive exponential decay is decreasing and concave up.

Can an exponential function increase while concave down?

Yes. A negative exponential with 0 < b < 1 is increasing and concave down.

What is the left-end behavior of f(x) = 2ˣ?

As x → −∞, f(x) → 0.

What is the right-end behavior of f(x) = 2ˣ?

As x → ∞, f(x) → ∞.

What is the left-end behavior of f(x) = (1/2)ˣ?

As x → −∞, f(x) → ∞.

What is the right-end behavior of f(x) = (1/2)ˣ?

As x → ∞, f(x) → 0.

What is the left-end behavior of f(x) = −2ˣ?

As x → −∞, f(x) → 0 from below.

What is the right-end behavior of f(x) = −2ˣ?

As x → ∞, f(x) → −∞.

What does lim as x → ∞ of f(x) = 0 mean?

As x increases without bound, f(x) approaches 0.

What does lim as x → −∞ of f(x) = ∞ mean?

As x decreases without bound, f(x) increases without bound.

What does a horizontal asymptote describe?

A horizontal line the graph approaches as x moves toward an end.

What is the left-end behavior of g(x) = 5(2/3)ˣ?

As x → −∞, g(x) → ∞.

What is the right-end behavior of g(x) = 5(2/3)ˣ?

As x → ∞, g(x) → 0.

How can you tell linear and exponential functions apart?

Linear has constant differences; exponential has constant nonzero ratios over equal x-steps.