New York State Regents Examinations Algebra II (Regents Examination) Syllabus

If a polynomial with real coefficients has the root 2 + 3i, what other root must it have?

Its complex conjugate, 2 - 3i (complex roots occur in conjugate pairs).

Give the general form of an exponential function and state the condition on the base.

f(x) = a·b^x, where a ≠ 0 and the base b > 0, b ≠ 1.

In y = a·b^x, how do you tell exponential growth from decay?

Growth when b > 1; decay when 0 < b < 1.

For exponential growth/decay y = a(1 ± r)^t, what do a and r represent?

a is the initial amount; r is the growth (+) or decay (−) rate per time period (as a decimal).

Convert to logarithmic form: b^y = x.

log_b(x) = y.

State the product, quotient, and power properties of logarithms.

log_b(MN) = log_b M + log_b N; log_b(M/N) = log_b M − log_b N; log_b(M^p) = p·log_b M.

State the change-of-base formula for log_b(x).

log_b(x) = log(x)/log(b) = ln(x)/ln(b) (any common base).

What is the natural logarithm, and what are ln(e) and ln(1)?

ln is the logarithm with base e; ln(e) = 1 and ln(1) = 0.

Outline the method to solve an exponential equation like b^x = c when bases can't be matched.

Take the logarithm of both sides, use the power rule to bring the exponent down, then solve for x.

Solve for x: log_2(x) = 5.

x = 2^5 = 32.

When solving a logarithmic equation, why must answers be checked?

Because the argument of a logarithm must be positive; solutions making any argument ≤ 0 are extraneous and rejected.

Define radian measure of an angle in terms of arc length and radius.

A radian is the central angle that subtends an arc equal in length to the radius; θ (radians) = arc length / radius.

Give the conversion factors between degrees and radians.

Multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees. (180° = π radians.)

On the unit circle, what do the coordinates (x, y) of the terminal point of angle θ represent?

x = cos θ and y = sin θ; also tan θ = y/x.

Give the exact values of sin, cos, and tan for θ = π/6 (30°).

sin = 1/2, cos = √3/2, tan = √3/3 (=1/√3).

Give the exact values of sin, cos, and tan for θ = π/4 (45°).

sin = √2/2, cos = √2/2, tan = 1.

How does the sign of the trig functions vary by quadrant (the 'ASTC' rule)?

Quadrant I: all positive; II: sine (and csc) positive; III: tangent (and cot) positive; IV: cosine (and sec) positive.

For y = A·sin(Bx + C) + D, identify amplitude, period, phase shift, and midline.

Amplitude = |A|; period = 2π/|B|; phase shift = −C/B; midline = y = D.

What are the period and range of the basic sine and cosine functions?

Both have period 2π and range [−1, 1].

What is the period of the basic tangent function, and where are its vertical asymptotes?

Period π; vertical asymptotes at x = π/2 + πk (where cosine = 0).

State the Pythagorean identity relating sine and cosine.

sin^2 θ + cos^2 θ = 1.

Express tan θ and the reciprocal identities (csc, sec, cot) in terms of sine and cosine.

tan θ = sin θ/cos θ; csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = cos θ/sin θ.

State the double-angle formula for sine.

sin(2θ) = 2 sin θ cos θ.

Describe how the transformation g(x) = a·f(b(x − h)) + k changes the graph of f(x).

a: vertical stretch/compression (and reflection if a<0); b: horizontal stretch/compression (reflection if b<0); h: horizontal shift; k: vertical shift.

Compare f(x − h) versus f(x) − h on the graph.

f(x − h) shifts the graph horizontally (right if h>0); f(x) − h shifts it vertically (down by h).

How are the graphs of a function and its inverse related, and what algebraic step finds the inverse?

They are reflections across the line y = x; to find the inverse, swap x and y and solve for y.

What condition must a function meet to have an inverse that is also a function (use a graphical test)?

It must be one-to-one, i.e., pass the horizontal line test (each y-value used at most once).

Give the recursive and explicit formulas for an arithmetic sequence with first term a_1 and common difference d.

Recursive: a_n = a_(n−1) + d; explicit: a_n = a_1 + (n − 1)d.

Give the explicit formula for a geometric sequence and the formula for the sum of its first n terms.

a_n = a_1·r^(n−1); sum S_n = a_1(1 − r^n)/(1 − r) for r ≠ 1.

What key property of the normal distribution is described by the empirical (68-95-99.7) rule?

For a normal distribution, about 68% of data lies within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.

How do you compute a z-score, and what does it measure?

z = (x − μ)/σ; it measures how many standard deviations a value x is above (+) or below (−) the mean.

Distinguish a population parameter from a sample statistic.

A parameter is a fixed numerical value describing an entire population; a statistic is a value computed from a sample used to estimate that parameter.

Distinguish a sample survey, an observational study, and a controlled experiment.

A survey collects responses from a sample; an observational study records data without imposing treatment; an experiment deliberately imposes a treatment to establish cause and effect.

What is a simulation, and what is its purpose in statistical inference?

A simulation uses random trials (e.g., random numbers) to model a chance process, generating an empirical distribution to estimate probabilities or assess whether observed results are statistically significant.

In comparing two treatment groups, when is an observed difference considered statistically significant via simulation?

When the observed difference is unlikely (e.g., rarely occurs) under simulated random re-assignments assuming no real difference, indicating it is probably not due to chance.

State the formula for the conditional probability P(A | B).

P(A | B) = P(A and B) / P(B), provided P(B) ≠ 0.

How do you test whether two events A and B are independent?

They are independent if P(A | B) = P(A) (equivalently P(A and B) = P(A)·P(B)).