NY Regents Algebra II flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Imaginary unit i

The imaginary unit i is defined by i squared equals negative one, so i is a square root of negative one, and powers of i repeat every four exponents.

Complex conjugate

For a complex number a plus bi, the conjugate is a minus bi; multiplying a number by its conjugate produces the real number a squared plus b squared.

Rational exponent rule

A raised to the power m over n equals the nth root of a to the m, which also equals the nth root of a raised to the m when that root is real.

Discriminant of a quadratic

For a x squared plus b x plus c, the discriminant b squared minus four a c tells whether the roots are two real, one repeated real, or a complex conjugate pair.

Rationalizing a denominator

Rewrite a fraction so no radical remains in the denominator by multiplying numerator and denominator by the radical, or by the conjugate when the denominator is a binomial.

Extraneous solutions

Squaring both sides of a radical equation can create solutions that do not satisfy the original equation, so each candidate must be substituted back and rejected if false.

Remainder theorem

When a polynomial p of x is divided by x minus a, the remainder equals p evaluated at a, so p of a equal to zero means x minus a is a factor.

Quadratic formula

The solutions of a x squared plus b x plus c equals zero are x equals negative b plus or minus the square root of b squared minus four a c, all divided by two a.

Vertex form of a parabola

The form y equals a times x minus h squared plus k shows the vertex at h comma k, the axis of symmetry x equals h, and the opening direction from the sign of a.

Sum and product of roots

For a quadratic equation a x squared plus b x plus c equals zero, the two roots add to negative b over a and multiply to c over a.

Logarithm product and quotient rules

The logarithm of a product equals the sum of the logarithms of the factors, and the logarithm of a quotient equals the difference of the logarithms.

Change of base formula

Log base b of a equals log base c of a divided by log base c of b, which allows any logarithm to be computed with common or natural logs.

Solving exponential equations

Rewrite both sides with the same base when possible; otherwise take a logarithm of both sides and apply power rules to isolate the variable in the exponent.

Composition of functions

The composition f of g of x means f evaluated at g of x; the inner function is computed first, and the domain excludes inputs where g of x is undefined.

Inverse function procedure

To find an inverse, swap x and y in the equation and solve for y; the inverse graph is the original graph reflected across the line y equals x.

Domain restrictions

Exclude inputs that make a denominator zero, keep radicands of even roots nonnegative, and require arguments of logarithms to be positive when stating a function's domain.

Even and odd functions

A function is even when f of negative x equals f of x, giving y-axis symmetry, and odd when f of negative x equals negative f of x, giving origin symmetry.

Exponential growth formula

A equals P times one plus r raised to t models growth at rate r per period, while A equals P e to the r t models continuous growth or decay.

Geometric sequence nth term

In a geometric sequence, the nth term equals the first term times the common ratio raised to n minus one, and each term is the previous term times that ratio.

Arithmetic series sum

The sum of the first n terms of an arithmetic sequence equals n over two times the quantity first term plus nth term, giving the total of the progression.

Circle equation standard form

A circle with center h comma k and radius r has equation x minus h squared plus y minus k squared equals r squared; complete the square to convert from general form.

Law of cosines

For a triangle with sides a, b, and c and angle C opposite side c, c squared equals a squared plus b squared minus two a b cosine C.

Law of sines

In any triangle, the ratio of a side to the sine of its opposite angle is constant: a over sine A equals b over sine B equals c over sine C.

Radian measure

One radian is the angle that subtends an arc equal in length to the radius, and conversion uses pi radians equals one hundred eighty degrees.

Arc length and sector area

With central angle theta measured in radians and radius r, arc length equals r times theta and sector area equals one half r squared times theta.

Pythagorean identity

For any angle theta, sine squared theta plus cosine squared theta equals one, and tangent theta equals sine theta divided by cosine theta.

Conditional probability formula

The probability of A given B equals the probability of A and B divided by the probability of B, assuming the probability of B is nonzero.

Independent events product rule

Events A and B are independent exactly when the probability of both occurring equals the product of their individual probabilities; otherwise the events are dependent.

Binomial probability formula

For n independent trials with success probability p, the probability of exactly k successes is n choose k times p to the k times one minus p to the n minus k.

Combinations versus permutations

A combination counts groups where order does not matter, using n factorial over r factorial times n minus r factorial, while a permutation counts arrangements where order matters.

Z-score

A z-score equals the data value minus the mean, divided by the standard deviation, and measures how many standard deviations the value lies from the mean.

Expected value

The expected value of a discrete random variable is the sum of each outcome multiplied by its probability, representing the long-run average result of many trials.