Algebra II Final Exam Review Flashcards

Comprehensive practice vocabulary flashcards covering Polynomial Functions, Radicals, Rational Exponents, Logarithmic and Exponential Functions, and Rational Functions based on the Algebra II review transcript.

Zeros of a Function

The values of $x$ for which $f(x) = 0$, representing the points where the graph of the polynomial function intersects the $x$-axis.

Synthetic Substitution

The process of using synthetic division to evaluate a polynomial function $f(x)$ for a specific value by using the coefficients of the polynomial.

Remainder Theorem

A theorem stating that if a polynomial $f(x)$ is divided by $(x-k)$, the resulting remainder is equal to the value of $f(k)$.

Factoring by Grouping

A method used to solve polynomial equations, such as $2x^{3}+x^{2}-16x-8=0$, by factoring common terms out of specific pairs, resulting in $(x^{2}-8)(2x+1)=0$.

Inverse Function

A function that reverses the effects of another function; if $(f \text{ o } g)(x) = x$ and $(g \text{ o } f)(x) = x$, then $f(x)$ and $g(x)$ are inverses.

Rationalizing the Denominator

The procedure of removing radicals from the denominator of a fraction, for example by multiplying the numerator and denominator by the conjugate $5+\sqrt{3}$ to simplify $\frac{1}{5-\sqrt{3}}$.

Rational Exponents

A way to represent radicals as powers, where $\sqrt[n]{x} = x^{\frac{1}{n}}$ and $\sqrt[n]{x^{m}} = x^{\frac{m}{n}}$. For instance, $\sqrt[3]{(xy)^{3/2}} = (xy)^{3/2 \cdot 1/3} = (xy)^{1/2}$.

Exponential Growth

Occurs when a quantity increases by a fixed percent over time, modeled by the formula $f(t) = a(1+r)^{t}$, where $a$ is the initial amount and $r$ is the growth rate.

Exponential Decay

Occurs when a quantity decreases by a fixed percent over time, often used for depreciation and modeled by $f(t) = a(1-r)^{t}$.

Compound Interest

Interest calculated on both the principal and the accrued interest, modeled by $A = P(1 + \frac{r}{n})^{nt}$, where $n$ is the number of times interest is compounded per year.

Logarithm

The exponent to which a base must be raised to produce a given number; the logarithmic form $\log_{b}(y) = x$ is equivalent to the exponential form $b^{x} = y$.

Natural Logarithm

A logarithm with base $e$, denoted as $\ln(x)$. It is the inverse of the natural exponential function $e^{x}$.

Change of Base Formula

A formula used to evaluate logarithms with any base by converting them to common or natural logarithms: $\log_{b}(a) = \frac{\log(a)}{\log(b)}$.

Product Rule of Logarithms

A property used to condense or expand logs stating that $\log_{b}(m \cdot n) = \log_{b}(m) + \log_{b}(n)$. For natural logs, $\ln(3 \cdot 16) = \ln(48)$.

Quotient Rule of Logarithms

A property stating that $\log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n)$. In natural logs, $\ln(63) - 2\ln(3) = \ln(\frac{63}{3^{2}}) = \ln(7)$.

Power Rule of Logarithms

A property that allows an exponent to be moved in front of a logarithm: $\log_{b}(m^{n}) = n \cdot \log_{b}(m)$.

Vertical Asymptote

A vertical line $x = c$ that a rational function's graph approaches but never reaches, found where the simplified denominator equals zero.

Horizontal Asymptote

A horizontal line indicating the value that a function's graph approaches as $x$ goes to positive or negative infinity.

Hole (in Rational Functions)

A removable discontinuity appearing on a graph as an open circle, occurring when a factor is present in both the numerator and denominator.

Transformation Variable k

In the general form $a \cdot f(x \pm h) \pm k$, this value represents a vertical shift, moving the graph up if positive and down if negative.

Transformation Variable h

In the general form $a \cdot f(x \pm h) \pm k$, this value represents a horizontal shift, moving the graph left if represented as $(x+h)$ and right if $(x-h)$.

Transformation Variable a

In the parent function transformation formula, this coefficient determines if the graph is reflected across the $x$-axis and if it undergoes a vertical stretch ($|a|>1$) or shrink ($0<|a|<1$).