NY Regents Algebra II common mistakes
NY Regents Algebra II · Understand the traps and how to avoid them.
Number and Quantity: rewrites sqrt(x^2) as x, or splits sqrt(a+b) into sqrt(a)+sqrt(b), or treats a rational exponent denominator as a multiplier rather than a root.
The student overgeneralizes distributive and integer-exponent rules and forgets that the principal square root is nonnegative and that radical rules apply to products/quotients, not sums.
What to do instead
Substitute a negative and a positive number to test the simplification; state sqrt(x^2)=|x| and use sqrt(ab)=sqrt(a)sqrt(b) only for nonnegative factors; convert rational exponents by identifying root first, then power.
Algebra: solves a rational equation by multiplying by the least common denominator but does not reject solutions that make any original denominator zero.
The student treats the rational equation like a polynomial equation and checks only whether the arithmetic works, not whether the solution is in the domain.
What to do instead
List excluded values before clearing denominators; after solving, substitute each candidate into every original denominator and discard any that give zero; state the restrictions in the answer.
Algebra: applies logarithm laws incorrectly, such as log(a+b)=log(a)+log(b), log(a/b)=log(a)/log(b), or omits the base-change formula.
The student confuses logarithms with linear operations and exponent rules, especially when the expression inside the log is a sum or quotient.
What to do instead
Use only the product, quotient, and power rules for logs; verify with simple numbers; memorize and apply base change as log_b(x)=log(x)/log(b).
Functions: reads f(x-3) as f(x)-3, shifts a graph in the wrong horizontal direction, or writes the inverse of f as 1/f(x).
The student focuses on the symbols rather than on input-output meaning and transformation order, and confuses inverse function notation with reciprocal notation.
What to do instead
Build a small input-output table to see the shift; remember horizontal shifts are opposite the sign inside the function; find inverses by swapping x and y and solving, and check that f(f^{-1}(x))=x on the restricted domain.
Functions: composes or inverts functions without checking domain restrictions, such as reporting an inverse of a quadratic without restricting its domain or composing with a value outside the inner function's domain.
The student treats domain as an afterthought and works only with algebraic formulas, missing that radicals, rationals, and even powers impose restrictions.
What to do instead
Determine the domain of each function before composing or inverting; for a composition, require the input to be in the inner domain and the output to be in the outer domain; restrict the original function so its inverse is a function.
Geometry: completes the square for a circle equation with sign errors, misidentifies the center from (x-h)^2+(y-k)^2=r^2, or uses the diameter as the radius.
The student memorizes the circle form loosely and does not connect the signs in the equation to the coordinates of the center, and confuses diameter and radius in word problems.
What to do instead
Rewrite the equation into standard circle form step by step; read the center as (h,k) with signs opposite those inside the parentheses; take the square root of the constant for the radius and divide a given diameter by two.
Statistics and Probability: computes a normal-curve probability without standardizing, uses a z-score as if it were an area, or chooses the wrong tail for a greater-than/less-than question.
The student mixes up the z-score formula, table/calculator output, and the geometry of the normal curve, especially under time pressure.
What to do instead
Always write z=(x-mu)/sigma, sketch the curve, shade the requested region, and use calculator normalcdf with correct lower and upper bounds; interpret the result as an area, not as a z-value.
Statistics and Probability: uses P(A and B) for P(A|B), or assumes independence without checking whether P(A|B)=P(A).
The student conflates joint and conditional probability language and does not distinguish the sample space being restricted by the given condition.
What to do instead
Write the conditional formula P(A|B)=P(A and B)/P(B); organize data in a two-way table when possible; test independence by comparing P(A|B) with P(A), not by intuition.
Algebra/Functions: models exponential growth or decay with r instead of 1+r or 1-r, mismatches the time unit, or misuses a negative exponent.
The student confuses the percent rate with the growth factor and does not track how often compounding occurs or what the variable t represents.
What to do instead
Identify the initial value and the multiplier per period; use 1+r for growth and 1-r for decay; convert years/months to the same period as the rate; check t=0 gives the initial value and that the trend matches the context.
Geometry/Functions: evaluates trigonometric expressions in the wrong calculator mode, swaps sine and cosine coordinates on the unit circle, or ignores reference angles and quadrant signs.
The student relies on calculator output without checking degree/radian mode and memorizes unit-circle values without connecting them to coordinates and quadrant location.
What to do instead
Check the calculator mode before evaluating; draw the angle and label the point (cos theta, sin theta); use reference angles and quadrant signs for exact values, and verify a known value such as sin(pi/6)=1/2.