PreCalculus Review Unit 2 (2.1-2.7)

What is an arithmetic sequence?

A sequence where consecutive terms change by a constant difference (d).

What is the arithmetic sequence formula?

an = ak + d (n-1) It finds any term in an arithmetic sequence.

In the arithmetic formula, what do the variables mean?

an = target term; a1 = first term; n = term position; d = common difference.

How do you identify the common difference?

Subtract a term from the term immediately after it. The difference stays constant.

What is a geometric sequence?

A sequence where consecutive terms change by a constant multiplicative ratio (r).

What is the geometric sequence formula?

gn = gor^n It finds any term in a geometric sequence.

In the geometric formula, what do the variables mean?

gn = target term; go = initial term; r = common ratio; n = term position.

How do you identify the common ratio?

Divide a term by the term immediately before it. The ratio stays constant.

What is the difference between linear and exponential change?

Linear: constant additive change. Exponential: constant multiplicative change or percentage rate.

What is the linear function formula?

f(x) = mx + b. It represents a constant rate of additive change.

In f(x) = mx + b, what do m and b represent?

m = slope or rate of change; b = y-intercept or initial value.

What is the general exponential function formula?

f(x) = a(b)ˣ. It represents repeated multiplication by a constant factor.

In f(x) = a(b)ˣ, what do a and b represent?

a = initial value; b = growth or decay factor; x = number of time periods.

What makes an exponential function show growth?

Its growth factor is greater than 1 (b > 1).

What makes an exponential function show decay?

Its decay factor is between 0 and 1 (0 < b < 1).

How do you convert a percentage growth rate into a growth factor?

Add the decimal rate to 1: b = 1 + r.

How do you convert a percentage decay rate into a decay factor?

Subtract the decimal rate from 1: b = 1 − r.

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

y = 0, provided there is no vertical shift.

What does the y-intercept represent in an exponential function?

The initial value, because f(0) = a.

What is an equivalent exponential expression?

A rewritten expression that has the same value or represents the same function.

What is the exponent product rule?

When multiplying powers with the same base, add exponents: bᵐ · bⁿ = bᵐ⁺ⁿ.

What is the exponent quotient rule?

When dividing powers with the same base, subtract exponents: bᵐ / bⁿ = bᵐ⁻ⁿ.

What happens when an exponential graph is shifted vertically?

Adding k outside the function shifts it up k; subtracting k shifts it down k.

What is exponential data modeling?

Using an exponential function to represent data that changes by a consistent percentage or factor.

What is exponential regression used for?

Finding an exponential function that best fits a given data set.

What is the difference between interpolation and extrapolation?

Interpolation predicts within the observed data range; extrapolation predicts outside it.

What is a residual?

The difference between an observed value and a model's predicted value.

What is the residual formula?

Residual = observed value − predicted value.

How do you interpret a residual plot?

Random scatter around zero suggests a suitable model; patterns suggest the model may not fit well.

How do you compare competing function models?

Compare residuals, patterns, context, and how well each model represents the data.

What is function composition?

Combining functions by using the output of one function as the input of another.

What does (f ∘ g)(x) mean?

f(g(x)): evaluate g first, then use its output as the input of f.

What is the correct order for function composition?

Work from the inside outward. In f(g(x)), evaluate g before f.

What is the difference between f(g(x)) and g(f(x))?

They generally produce different results because the order of composition changes.

What is the domain restriction in a composition?

The input must be in g's domain, and g(x) must be in f's domain