Algebra 2 Spring Final Review

This set of vocabulary flashcards covers essential Algebra 2 concepts including function behaviors, asymptotes, complex equations, and growth/decay formulas derived from the final review transcript.

Relative Minima and Maxima

The low and high points within a specific range of a function; for the function $f(x) = -x^3 + 2x^2 + 3$, the approximate minimum is $(0, 3)$ and the maximum is $(1.333, 4.185)$.

Distinct Real Zeros

The points where a function's graph crosses the x-axis, also referred to as x-intercepts.

Inverse Function

A function that reverses the operation of the original function, denoted as $g^{-1}(x)$; for the function $y = \frac{x-2}{3}$, the inverse is discovered by swapping variables to get $g^{-1}(x) = 3x + 2$.

Horizontal Asymptote

A horizontal line that a graph approaches as $x$ goes to infinity; in $f(x) = \frac{5x+4}{x-3}$, the asymptote is $y=5$ because the degrees of the numerator and denominator are equal ($n=d$), whereas in $f(x) = \frac{1}{x+5}$, the asymptote is $y=0$ because $n < d$.

Vertical Asymptote

A vertical line where a function is undefined, typically found by setting the denominator to zero; for $f(x) = \frac{x}{(x+3)(x-4)}$, the vertical asymptotes are $x=-3$ and $x=4$.

Extraneous Solutions

Solutions derived during algebraic manipulation that do not satisfy the original equation, often requiring validation through checking, such as in rational equations like $\frac{1}{x-6} + \frac{2}{x^2-36} = \frac{x+2}{x+6}$.

Exponential Depreciation Formula

A mathematical model used to calculate decreasing value over time, represented by $f(t) = a(1 - r)^t$; applied to a car purchased for $\$22,400$ at an $11\%$ rate, the value after $6$ years is approximately $\$11,132.38$.

Continuous Compounding Formula

A growth model represented by $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate, and $t$ is time; for an account growing from $\$5,000$ to $\$6,880$ at $3.9\%$, the time required is $8.2$ years.

Exponential Growth Formula

A model used to calculate population or value increases over time, expressed as $f(t) = a(1 + r)^t$; used when a town of $17,000$ people grows at $3\%$ for $10$ years to reach $22,846$ people.

Leading Coefficient

The coefficient of the term with the highest degree in a polynomial, which determines the end behavior of the graph along with the degree being even or odd.

Function Composition

The application of one function to the results of another, such as finding $f(g(0))$ by substituting the value from $g(x)$ into $f(x)$; for $f(x) = 3x-5$ and $g(x) = x^2-9$, the result is $-32$.