NY Regents Algebra II: a complete guide to exponential, logarithmic, and trigonometric functions on the exam

What this strand demands

The exponential, logarithmic, and trigonometric strand carries the function-modeling core of NY Regents Algebra II. It rewards fluency with the inverse log-exponential relationship and the log properties, confident solving of exponential and logarithmic equations, the radian and unit-circle foundation of trigonometry, sinusoidal graphing and modeling, and arithmetic and geometric sequences and series. This guide ties together the dot-point pages, each with its own practice: exponential and logarithmic functions, solving exponential and logarithmic equations, radian measure and the unit circle, graphing sinusoidal functions, and sequences and series.

Exponential and logarithmic functions

A logarithm is an exponent: $\log_b y = x$ means $b^x = y$, so the two functions are inverses. The log properties mirror the exponent laws: products become sums, quotients become differences, and powers become coefficients. The natural base $e$ pairs with $\ln$. A logarithm's argument must be positive, which forces domain checks when solving.

Solving equations and modeling

Solve an exponential equation by matching bases (set exponents equal) or, failing that, by taking a logarithm and using the power property. Solve a logarithmic equation by condensing to one log, rewriting in exponential form, and checking the domain. Growth and decay use $A = a(1 \pm r)^t$; compound interest uses $A = P(1 + \frac{r}{n})^{nt}$; finding when a target is reached requires logarithms.

Radian measure and the unit circle

A full circle is $2\pi$ radians (360 degrees), so $180$ degrees $= \pi$ radians; convert with $\frac{\pi}{180}$ or $\frac{180}{\pi}$. On the unit circle the point at angle $\theta$ is $(\cos\theta, \sin\theta)$, which defines sine and cosine for any angle, with the quadrant setting the sign. The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ relates them; it gives the magnitude while the quadrant gives the sign.

Graphing sinusoidal functions

For $y = A\sin(B(x - C)) + D$: amplitude $|A|$, period $\frac{2\pi}{B}$, midline $y = D$, phase shift $C$. The maximum is $D + |A|$ and the minimum is $D - |A|$. Model a periodic situation by setting $D$ to the average of max and min, $A$ to half their difference, and $B = \frac{2\pi}{\text{period}}$.

Sequences and series

Arithmetic sequences add $d$ ($a_n = a_1 + (n-1)d$); geometric sequences multiply by $r$ ($a_n = a_1 r^{n-1}$). The arithmetic series sums to $\frac{n}{2}(a_1 + a_n)$ and the finite geometric series to $\frac{a_1 - a_1 r^n}{1 - r}$. Identify the type first, then select the matching formula.

How this strand is examined

• Part I (2 credits). A log-to-exponential conversion, a base-matching solve, a degree-radian conversion, a period, or a sequence term.
• Part II (2 credits). A log expansion or condensation, a Pythagorean-identity value with the correct sign, or a recursive formula. Show the property used.
• Part III and IV (4 to 6 credits). A compound-interest or growth model with a logarithmic part, a sinusoidal model with amplitude, period, and a maximum, or a sequence-and-series application. Show every step and round as asked.

Check your knowledge

Work these as you would for credit.

1. Rewrite $\log_3 81 = x$ in exponential form and find $x$. (2 credits)
2. Expand $\log\frac{a^2}{b}$ completely. (2 credits)
3. Solve $4^{x} = 64$ by matching bases. (2 credits)
4. Solve $2^x = 30$, leaving an exact logarithmic answer. (2 credits)
5. Convert $\frac{5\pi}{6}$ radians to degrees. (2 credits)
6. State the amplitude, period, and midline of $y = 6\sin(4x) - 2$. (2 credits)
7. Find the 5th term of a geometric sequence with $a_1 = 3$ and $r = 2$. (2 credits)
8. Find the sum of the first 10 terms of an arithmetic sequence with $a_1 = 4$ and $d = 3$. (4 credits)