Pre-Calculus Comprehensive Review: Systems, Transformations, Inverse, Polynomial, and Rational Functions

A set of vocabulary flashcards reviewing systems of linear and circular equations, graph transformations, inverse functions, polynomial functions, and rational functions.

Tangent of a Circle

A straight line that touches the circumference of a circle at exactly one point and is perpendicular to the radius at the point of tangency.

Radius-Tangent Theorem

A theorem stating that the angle between a tangent line and a radius at the point of tangency is $90^\circ$.

Two-Tangent Theorem

A theorem stating that tangents to a circle that meet at the same point outside the circle are equal in length.

Secant of a Circle

A line that intersects a circle at exactly two points, derived from the Latin word secare, meaning to cut.

Substitution Method for Systems of Equations

A technique to find intersection points by substituting the linear equation $y = mx + b$ into the circle equation $(x - h)^2 + (y - k)^2 = r^2$ to yield a quadratic equation in terms of $x$.

Discriminant Criteria (Line and Circle System)

The value $\Delta = b^2 - 4ac$ from the resulting quadratic equation used to determine solutions: two solutions if $\Delta > 0$, one solution if $\Delta = 0$, and no real solution if $\Delta < 0$.

Transformation

The change in position, orientation, or size of the graph of a function.

Translation

A rigid transformation that shifts a graph vertically or horizontally without changing its shape.

Vertical Shift

A translation that moves a graph up or down by adding or subtracting a constant $k$ to the function, written as $y = f(x) + k$ or $y = f(x) - k$.

Horizontal Shift

A translation that moves a graph left or right by replacing input $x$ with $x + h$ or $x - h$, written as $y = f(x - h)$.

Transformation Form

The general formula for a transformed function expressed as $y = f(x - h) + k$.

Vertical Reflection

A transformation across the x-axis produced by multiplying all function outputs by $-1$, yielding $y = -f(x)$.

Horizontal Reflection

A transformation across the y-axis produced by multiplying all function inputs by $-1$, yielding $y = f(-x)$.

Vertical Stretch and Compression

Transformations represented by $y = a f(x)$, where the graph stretches vertically if $a > 1$ and compresses vertically if $0 < a < 1$.

Horizontal Stretch and Compression

Transformations represented by $g(x) = f(bx)$, where the graph compresses horizontally by $\frac{1}{b}$ if $b > 1$ and stretches horizontally by $\frac{1}{b}$ if $0 < b < 1$.

One-to-One Function

A function in which each output corresponds to exactly one input, which is a required condition for a function to have an inverse function.

Properties of Inverse Functions

Key properties stating that domain and range are interchanged, ordered pairs $(a, b)$ are reversed to $(b, a)$, and the graph reflects across the line $y = x$.

Algebraic Inverse Test

The procedure of verifying that two functions $f(x)$ and $g(x)$ are inverses of each other by confirming that both $f(g(x)) = x$ and $g(f(x)) = x$ are true.

Restricting the Domain

The process of limiting the domain of a non-one-to-one function so that it becomes one-to-one, allowing an inverse function to exist.

Polynomial

An expression consisting of variables and coefficients involving addition, subtraction, multiplication, and non-negative integer exponents.

Degree of a Polynomial

The highest degree of its individual terms when expressed in canonical form.

Polynomial Function

A function represented as $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where coefficients are real numbers and $n$ is a whole number.

Leading Coefficient

The coefficient of the polynomial term that possesses the highest degree.

Quintic Polynomial Function

A polynomial function with a degree of $5$.

Synthetic Substitution

The use of synthetic division as a shortcut procedure to evaluate a polynomial function at a given value.

Remainder Theorem

A theorem stating that if a polynomial $f(x)$ is divided by $x - a$, the remainder is the constant $f(a)$.

Depressed Polynomial

The quotient polynomial obtained when a polynomial function is divided by one of its linear binomial factors.

Factor Theorem

A theorem stating that a binomial $x - a$ is a factor of polynomial $f(x)$ if and only if $f(a) = 0$.

Zero of a Polynomial

A real number $a$ such that $f(a) = 0$, which corresponds to an x-intercept of the graph and a solution to $f(x) = 0$.

Rational Expression

An algebraic fraction in which both the numerator and the denominator are polynomials.

Vinculum

The horizontal fraction bar separating the numerator and denominator in a rational expression.

Rational Equation

An equation containing at least one fraction whose numerator and denominator are polynomials.

Rational Function

A function defined in the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.

Extraneous Solution

A value derived from solving an equation algebraically that does not satisfy the original equation because it creates an undefined denominator.

Asymptote

An imaginary line that a graph continually approaches but does not intersect.

Vertical Asymptote

A vertical line $x = c$ found by setting the simplified denominator of a rational function equal to zero ($Q(c) = 0$).

Horizontal Asymptote

A horizontal line $y = d$ determined by comparing the leading degree $n$ of the numerator with degree $m$ of the denominator.

Oblique Asymptote

A slanted non-horizontal asymptotic line that occurs in a rational function when the numerator degree $n$ is greater than the denominator degree $m$.

Hole (Removable Discontinuity)

A point discontinuity in the graph of a rational function that occurs when a common zero factor is canceled from both numerator and denominator.

Domain of a Rational Function

The set of all real numbers except for values that cause the denominator polynomial $Q(x)$ to equal zero.