AP precalculus

What is the general form of a logarithmic function

f(x) = a log_b(x), where a ≠ 0, b > 0, b ≠ 1.

What defines a function in terms of input and output values

A function is a relation where each input (x) maps to exactly one output (y), with input values as the domain and independent variable, and output values as the range and dependent variable.

How does the average rate of change behave in quadratic functions

It changes at a linear rate and represents the slope of the secant line over an interval.

How are zeros of a rational function found

Set numerator equal to zero for real zeros; points where denominator is zero are vertical asymptotes, not zeros.

How are inverse trigonometric functions restricted

Their domains are limited to specific quadrants: sine and tangent to quadrants 1 and 4, cosine to quadrants 1 and 2, to ensure they are one-to-one.

What does the slope of a graph represent

The slope of a graph represents its rate of change.

In linear functions, how does the rate of change behave

The rate of change is constant, so the average rate of change is the same over any interval.

What is a polynomial function, and what are its key characteristics

A polynomial function is expressed as a sum of terms with coefficients and decreasing degrees, with no negative degrees, imaginary coefficients, or division within the polynomial.

How are vertical asymptotes identified for rational functions

Solve where the denominator equals zero, then exclude any zeros that cancel with the numerator (holes).

What are the key properties of tangent functions

Period is π, undefined where cosine is zero, and they have asymptotes at those points.

What is a multiplicative transformation of a function

A vertical dilation (a·f(x)) stretches or compresses the graph; if a is negative, it reflects over the x-axis; a·f(bx) horizontally dilates or reflects over the y-axis.

What is the significance of the phase shift C in sinusoidal functions

It moves the graph horizontally; a positive C shifts left, a negative C shifts right.

What does the degree of a polynomial indicate about its zeros

The degree equals the highest exponent and indicates the total number of real and imaginary zeros.

What is the significance of the sign of the rate of change

A positive rate indicates both quantities increase or decrease together; a negative rate indicates one increases while the other decreases.

How do you find the end behavior of a rational function

Compare the degrees of numerator and denominator: if bottom-heavy, y approaches 0; if same degree, y approaches the ratio of leading coefficients; if top-heavy, y approaches infinity or negative infinity.

What are the zeros of a function

The zeros are the x-values where the graph intersects the x-axis, meaning y = 0.

What are the key transformations of functions

Vertical/horizontal translations, stretches, compressions, and reflections based on algebraic modifications.

When is a function increasing or decreasing

A function is increasing if output values increase as input increases, and decreasing if output decreases as input increases.

How do you model periodic functions like sine and cosine

Using the skeleton equation y = A sin(B(x + C)) + D, where A is amplitude, B relates to period, C is phase shift, and D is vertical shift.

What is the general form of a sinusoidal function

y = A sin(B(x + C)) + D, where A is amplitude, B relates to period, C is phase shift, and D is vertical shift.

How do you find the average rate of change between two points

Label the points as (x1, y1) and (x2, y2), then calculate (y2 - y1) / (x2 - x1).

What does the natural logarithm (ln) represent

The logarithm base e, used to solve for exponents in natural growth or decay processes.

What characterizes an even function versus an odd function

An even function satisfies f(x) = f(-x) and is symmetric across the y-axis; an odd function satisfies f(-x) = -f(x) and is symmetric under 180° rotation.

What is the general form of a sinusoidal function

$y = A imes ext{sin} (B(x + C)) + D$, where $A$ is amplitude, $B$ relates to period, $C$ is phase shift, and $D$ is vertical shift.

What is the period of sine and cosine functions

The length of one complete cycle, which is 2π for the parent functions.

How do you find the real zeros of a rational function

Set numerator equal to zero and solve; also set denominator equal to zero and solve, but zeros that match are holes, not zeros.

How do you convert from Cartesian to polar coordinates

Calculate r = sqrt(x^2 + y^2) and θ = arctan(y/x), adjusting θ based on the quadrant.

What is Pascal's triangle used for in algebra

To find coefficients for binomial expansions like (a + b)^n, providing a shortcut to binomial theorem calculations.

How does the amplitude of a sinusoidal function relate to its graph

Amplitude $A$ is the distance from the midline $D$ to the maximum or minimum point.

What is concave up and concave down in a graph

Concave up means the rate of change is increasing (U-shape), while concave down means the rate of change is decreasing (upside-down U-shape).

What is a point of inflection in a polynomial

A point where the concavity of the graph changes from increasing to decreasing or vice versa.

What does polynomial long division do

It simplifies a polynomial divided by another, often to find slant asymptotes or simplify rational functions.

What characterizes exponential decay

When $a > 0$ and $0 < b < 1$, the graph decreases and approaches zero as $x$ increases.

What is end behavior in polynomial functions

It describes how the graph behaves as x approaches positive or negative infinity, determined by the degree and leading coefficient.

What is the period of tangent functions

The period of tangent functions is $ ext{pi}$.

What is the equation of a line that describes the relationship in a function

The equation is y = mx + b, where m is the slope and b is the y-intercept.

How does the multiplicity of a zero affect the graph

An odd multiplicity zero is where the graph passes through; an even multiplicity zero is where the graph bounces off.

What are vertical and horizontal asymptotes

Vertical asymptotes are lines the graph approaches but never touches, where the denominator is zero; horizontal asymptotes describe the end behavior as x approaches infinity.

What is an additive transformation of a function

A vertical translation (f(x) + k) shifts the graph up or down; a horizontal translation (f(x + k)) shifts left or right.

How do you determine end behavior from an equation

Identify the degree and the sign of the leading coefficient and use the end behavior table to find the limits as x approaches infinity and negative infinity.

What is the product property of exponents

$b^m imes b^n = b^{m+n}$, indicating that multiplying two powers with the same base adds their exponents.

What describes a rational function where the degree of numerator is higher than the degree of denominator

The function is top-heavy, with limits approaching positive or negative infinity as x approaches positive or negative infinity.

What is the period of sine and cosine functions

2π, representing the length of one full cycle of the wave.

What is an even function

A function satisfying $f(x) = f(-x)$, symmetric across the y-axis, e.g., cosine.

What are local and global extrema in polynomial functions

Local maxima and minima are the highest or lowest points in a specific interval; global extrema are the highest or lowest across the entire function.

What is a hole in a rational function

A point where numerator and denominator share a factor, resulting in an open circle on the graph.

What is a rational function

A function expressed as the division of two polynomials, which introduces asymptotes.

How is the horizontal asymptote determined when the degrees of numerator and denominator are equal

It is the ratio of the leading coefficients of numerator and denominator.

What are the key properties of exponential functions

Exponential functions have a skeleton form of $f(x) = a imes b^x$, with constraints: $a eq 0$, $b > 0$, and $b eq 1$.

What is the phase shift in a sinusoidal function

A horizontal shift given by $-C$, moving the graph left or right depending on the sign of $C$.

What is the binomial theorem used for

Expanding expressions of the form (a + b)^n using coefficients from Pascal's triangle.

How are vertical asymptotes of a rational function identified

Set numerator and denominator equal to zero, solve for x, cross out matching zeros (holes), and remaining zeros are the vertical asymptotes.

What is an odd function

A function satisfying $f(-x) = -f(x)$, symmetric about the origin.

What is the exponent root property

$b^{1/k} = ext{the }k ext{-th root of }b$, relating fractional exponents to roots.

What is the purpose of the exp regression on a calculator

To determine if data fits an exponential model and to find the best exponential regression equation.

What is the significance of the constant $e \\approx 2.718$ in exponential functions

It's the base of natural exponential functions, modeling continuous growth or decay.

What is a hole in a rational function

A point where numerator and denominator share a common factor, resulting in an open circle on the graph, with the function approaching a specific limit.

What is the limit of a rational function as x approaches infinity when the degree of the denominator is higher than the numerator

The limit is zero, and the graph has a horizontal asymptote at y = 0.

What is the midline in a sinusoidal function

The horizontal line halfway between the maximum and minimum, given by D in the equation.

How does the sign of the rate of change affect the graph of a function

A positive rate indicates the function is increasing; a negative rate indicates decreasing.

What is the power property of exponents

$(b^m)^n = b^{mn}$, meaning raising a power to another power multiplies the exponents.

How do you determine the period of a sine or cosine function

Period = $2 ext{pi} / B$, where $B$ is the coefficient inside the sine or cosine function.

What is an odd function

A function satisfying $f(-x) = -f(x)$, symmetric about the origin, e.g., sine.

What is the midline of a sinusoidal function

The horizontal line $y = D$, located halfway between the maximum and minimum of the graph.

What is the key condition for a function to have an inverse

A function must be one-to-one, meaning each output is produced by exactly one input, to have an inverse.

How are inverse sine, cosine, and tangent functions restricted

They are restricted to specific domains: $ ext{arcsin}$ and $ ext{arctan}$ typically to $[- ext{pi}/2, ext{pi}/2]$, and $ ext{arccos}$ to $[0, ext{pi}]$, to ensure they are functions.

How do you find slant or oblique asymptotes of a rational function

Use polynomial long division; the quotient is the equation of the slant or oblique asymptote.

What is the process of polynomial long division used for in rational functions

To simplify polynomials, find the quotient and remainder when dividing, which helps identify asymptotes and simplify the function.

How are inverse sine, cosine, and tangent functions restricted

They have limited domains: inverse sine and cosine are restricted to quadrants 1 and 4, tangent to quadrants 1 and 4 with domain (-π/2, π/2), to ensure they are one-to-one.

What does the natural logarithm $ ext{ln}(x)$ represent

The logarithm base $e$, indicating the power to which $e$ must be raised to get $x$.

How do you model a sinusoidal function from a graph

Identify the midline (D), amplitude (A), period to find B, and phase shift (C) to construct the equation y = A sin(B(x + C)) + D.

What is the key form of a sinusoidal function's skeleton equation

The key form is $y = A imes ext{sin}(B(x + C)) + D$, where $A$ is amplitude, $B$ relates to period, $C$ is phase shift, and $D$ is midline.

What does the amplitude $A$ of a sinusoidal function represent

Amplitude is the vertical distance from the midline to the maximum (or minimum) point.

How do you convert from Cartesian to polar coordinates

Use $r = ext{sqrt}(x^2 + y^2)$ and $ heta = ext{arctan}(y/x)$ (adjusting for quadrant).

What characterizes a tangent function's graph

It has asymptotes where cosine is zero, a period of π, and repeats every π units.

What is a vertical asymptote in a rational function

A vertical line where the function approaches infinity, found where the denominator equals zero and the numerator does not.

What is the effect of the phase shift $C$ in a sinusoidal function

The phase shift moves the graph horizontally by $-C$ units; positive $C$ shifts left, negative shifts right.

What is an even function

A function satisfying $f(x) = f(-x)$, symmetric across the y-axis.

What is the domain and range of a logarithmic function

Domain is (0, ∞), range is all real numbers, with a vertical asymptote at x=0.

What is the effect of a phase shift C in a sinusoidal function

It shifts the graph horizontally by -C units.

What does the amplitude of a sinusoidal function represent

The vertical distance from the midline to the maximum or minimum point.

How are inverse tangent functions restricted in their domain

Inverse tangent functions are restricted to outputs where $x$ is between $-rac{ ext{pi}}{2}$ and $rac{ ext{pi}}{2}$.

What is the general form of a logarithmic function

The general form is $y = a imes ext{log}_b(x)$ where $a eq 0$, $b > 0$, and $b eq 1$.

What characterizes exponential growth

When $a > 0$ and $b > 1$, the graph increases and approaches infinity as $x$ increases.

What is the significance of the base $b$ in a logarithmic function

The base $b$ determines the rate of change of the log function and must be positive and not equal to 1.

How do you determine the end behavior of a rational function when the degrees of numerator and denominator are equal

The end behavior approaches the ratio of the leading coefficients of the numerator and denominator.

How do you model exponential functions from real-world scenarios

Identify scenarios based on multiplication, select two points, and fit the exponential model using regression or algebra.

What is the period of tangent functions

The period is $ ext{pi}$, due to the function's zeros and asymptotes repeating every $ ext{pi}$.

What does the base b in a logarithmic function determine

It determines the scale of the logarithm; common bases include 10 (common log) and e (natural log).

How do you identify holes in a rational function

Holes occur at points where the numerator and denominator share a common factor, which cancels out, leaving a discontinuity.

What is the parent function of exponential functions

$b^x$, with $b > 1$ for growth or $0 < b < 1$ for decay, has a point at (0,1) and a horizontal asymptote at $y=0$.

What is the process of polynomial long division used for in rational functions

To simplify rational functions, find slant or oblique asymptotes, or divide polynomials to analyze end behavior.

What are the rules for the base $b$ of an exponential function

In an exponential function $f(x) = a imes b^x$, $b$ must be positive and not equal to 1.

How do you model exponential growth or decay scenarios

Use the form $f(x) = a imes b^x$, where $b > 1$ for growth and $0 < b < 1$ for decay.

What is the midline of a sinusoidal function

The midline is the horizontal line $y = D$, representing the central value around which the function oscillates.

How is the period of a sine or cosine function calculated from B

Period = 2π / B, where B is the coefficient in the sine or cosine function.

What is the key characteristic of the graph of a logarithmic function regarding its asymptote

Logarithmic functions are vertically asymptotic to the line $x = 0$.

How do you find the inverse of a logarithmic function

Swap x and y in the equation and solve for y to find the inverse function.

What is the relationship between the period and the coefficient B in a sine or cosine function

Period = 2π / B, where B affects the frequency of the wave.

What is a hole in a rational function

A hole occurs where a zero of the numerator and denominator cancel out, creating a discontinuity.