Unit 1

Polynomial and Rational Functions: the one-page sheet

1.1 Change in Tandem

Two quantities that change together are described by direction and by how fast. A function is increasing on an interval when larger inputs give larger outputs, decreasing when they give smaller ones. Its rate of change says how fast the output moves; when the rate of change increases the graph is concave up, and when it decreases the graph is concave down.

Increasing on an interval: larger inputs give larger outputs there. Decreasing: larger inputs give smaller outputs.
The rate of change compares the change in output with the change in input: how fast the output moves.
Concave up: the rate of change is increasing (the graph bends upward). Concave down: the rate of change is decreasing.
Direction and concavity are separate: a function can be decreasing and concave up, like cooling coffee.

f is increasing on that interval (and decreasing if f(a) > f(b) every time) the rate of change is increasing and the graph is concave up the rate of change is decreasing and the graph is concave down its outputs reach a lowest value there, as the pond does at t = 2

increasing function: $f$ is increasing on an interval if larger inputs give larger outputs there: $f(a) < f(b)$ whenever $a < b$ in the interval. It is decreasing if $f(a) > f(b)$ whenever $a < b$ .
covary: Two quantities covary when a change in one goes with a change in the other, as the input and output of a function do.
rate of change: How fast a function's output changes compared with its input. A positive rate means the output rises as the input rises; a negative rate means it falls.
concave up: A graph is concave up on an interval when its rate of change is increasing there (it bends upward), and concave down when its rate of change is decreasing (it bends downward).

1.2 Rates of Change

The average rate of change of f on [a, b] is (f(b) − f(a))/(b − a) , the slope of the line through (a, f(a)) and (b, f(b)) , in output units per input unit. Over a short interval around a point it estimates how fast f changes at that point, and comparing those estimates tells where f changes faster.

The average rate of change of $f$ on $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$ : the slope of the line through $(a, f(a))$ and $(b, f(b))$ .
Units: output units per input unit, like miles per hour or dollars per year.
Positive: the output ends higher. Negative: it ends lower. Zero: the outputs at the ends are equal.
To compare rates at two points, use short intervals of the same length around each.

the average rate of change is (f(b) − f(a))/(b − a) , in output units per input unit its slope is that average rate of change the average rate of change estimates how fast f changes at c the larger one tells where f is changing faster

average rate of change: The average rate of change of $f$ on $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$ , the slope of the line through $(a, f(a))$ and $(b, f(b))$ , in output units per input unit.

1.3 Rates of Change in Linear and Quadratic Functions

A linear function has one rate of change. A quadratic's average rates over consecutive equal intervals change by a constant amount, so its second differences are constant; increasing rates mean concave up, decreasing rates concave down. The average rate of change on [a, b] is the slope of the secant line, and a secant estimate is too big where the graph is concave up and too small where it is concave down.

The secant line through $(a, f(a))$ and $(b, f(b))$ has slope $\frac{f(b) - f(a)}{b - a}$ , the average rate of change of $f$ on $[a, b]$ .
Linear: constant rate of change. Quadratic: rates over consecutive equal intervals change linearly, so second differences are constant.
Concave up: graph below the secant line, estimate too big. Concave down: graph above it, estimate too small.

its average rate of change is the same on every interval: the slope the first differences change by a constant amount: the second differences are constant the graph is concave up (or concave down) there the graph lies above the secant line, so the estimate is an underestimate

secant line: A line through two points $(a, f(a))$ and $(b, f(b))$ of a graph. Its slope, $\frac{f(b) - f(a)}{b - a}$ , is the average rate of change of $f$ on $[a, b]$ .
second differences: For outputs at equal input steps, the changes in the first differences (the changes in output). Constant second differences mean the values fit a quadratic function.

1.4 Polynomial Functions and Rates of Change

A polynomial function has a degree and a leading term. Where it switches between increasing and decreasing it has a local maximum or minimum; on a closed interval the endpoints count too, and the greatest and least of all these values are the global maximum and minimum. Where its rate of change switches between increasing and decreasing, the graph has a point of inflection.

A polynomial function $p(x)$ $= a_n x^n + \cdots + a_1 x + a_0$ has degree $n$ and leading term $a_n x^n$ .
Local maximum: an output at least as large as at nearby inputs; local minimum: at most as large. Global: over the whole domain.
Between two distinct real zeros there is at least one local extremum. An even-degree polynomial has a global maximum or minimum.
A point of inflection is where the graph changes concavity: the rate of change switches from increasing to decreasing, or back.

it has a local maximum at a (decreasing to increasing gives a local minimum) an included endpoint can also be a local extremum, so check the endpoints the greatest is the global maximum and the least is the global minimum the graph changes concavity: a point of inflection

polynomial function: A function $p(x)$ $= a_n x^n + \cdots + a_1 x + a_0$ with whole-number powers and real coefficients.
degree of a polynomial: The highest power of $x$ in a polynomial with a nonzero coefficient. A nonzero constant has degree $0$ .
leading term: The term with the highest power of $x$ , $a_n x^n$ ; its coefficient $a_n$ is the leading coefficient.
local maximum: A local maximum is an output at least as large as the outputs at all nearby inputs; a local minimum is one at most as large. They occur where a function switches between increasing and decreasing, or at an included endpoint.
global maximum: The global maximum is the greatest output over the whole domain; the global minimum is the least. On a closed interval, compare the local extremum values and the endpoint values.
point of inflection: A point where a graph changes concavity: its rate of change switches from increasing to decreasing, or from decreasing to increasing.

1.5 Polynomial Functions and Complex Zeros

Each factor (x − a)^k of a polynomial gives a zero a of multiplicity k : the graph crosses at odd multiplicity and touches at even multiplicity. A polynomial of degree n has n complex zeros counting multiplicity, and with real coefficients the non-real ones come in conjugate pairs. Constant n th differences point to degree n ; even and odd functions show mirror symmetry.

Multiplicity: the power of the factor $(x - a)$ . Odd: the graph crosses. Even: it touches (a double zero bounces off the axis).
Degree $n$ means $n$ complex zeros counting multiplicity; non-real ones come in conjugate pairs when the coefficients are real.
Equally spaced inputs with constant $n$ th differences fit degree $n$ .
Even function: $f(-x) = f(x)$ , symmetric about the $y$ -axis. Odd function: $f(-x) = -f(x)$ , symmetric about the origin.

a is a zero of multiplicity k p keeps its sign near a : the graph touches the axis without crossing it has exactly n complex zeros counting multiplicity, real or not its conjugate a − bi is a zero too

multiplicity: If $(x - a)^k$ is a factor of a polynomial and no higher power is, $a$ is a zero of multiplicity $k$ . The graph crosses the $x$ -axis at a zero of odd multiplicity and touches it at a zero of even multiplicity (a double zero has multiplicity $2$ ).
complex zero: A zero of a polynomial that may be real or non-real. A polynomial of degree $n$ has $n$ complex zeros counting multiplicity; with real coefficients, non-real zeros come in conjugate pairs $a \pm bi$ .
even function: $f$ is an even function if $f(-x) = f(x)$ (graph symmetric about the $y$ -axis) and an odd function if $f(-x) = -f(x)$ (graph symmetric about the origin).

1.6 Polynomial Functions and End Behavior

As |x| grows, a polynomial behaves like its leading term a_nx^n . Even degree sends both ends the same way and odd degree sends them opposite ways; the sign of a_n picks the direction. State end behavior in full limit notation, for example lim as x → −∞, p(x) = ∞ and lim as x → ∞, p(x) = −∞ .

End behavior: what the outputs do as $x \to \infty$ and as $x \to -\infty$ . For a polynomial, the leading term decides it.
Limit notation: $\lim_{x \to \infty} f(x) = \infty$ reads "as $x$ increases without bound, $f(x)$ increases without bound."
Even degree: ends match. Odd degree: ends opposite. The sign of the leading coefficient picks which way.

the leading term a_nx^n outgrows every other term both ends go the same way: up if a_n > 0 , down if a_n < 0 the ends go opposite ways: left down and right up if a_n > 0 , the reverse if a_n < 0 write both limits in full, such as lim as x → ∞, p(x) = −∞

end behavior: What the outputs of a function do as $x \to \infty$ and as $x \to -\infty$ . For a polynomial it matches its leading term: even degree, both ends the same way; odd degree, opposite ways.
limit notation: Limit notation describes where outputs go: $\lim_{x \to \infty} f(x) = \infty$ means that as $x$ increases without bound, $f(x)$ increases without bound. Write the function, its input, where the input goes and where the outputs go.

1.7 Rational Functions and End Behavior

A rational function's end behavior matches the quotient of its leading terms. Equal degrees give a horizontal asymptote at the ratio of the leading coefficients; a larger denominator degree gives y = 0 ; a numerator one degree larger gives a slant asymptote. State the end behavior in limit notation, such as lim as x → ∞, r(x) = 3 .

A rational function is $r(x) = \frac{p(x)}{q(x)}$ with polynomials $p$ and $q$ .
End behavior comes from the quotient of the leading terms, written in limit notation.
Horizontal asymptote $y = c$ when $\lim_{x \to \infty} r(x) = c$ or $\lim_{x \to -\infty} r(x) = c$ : equal degrees give the ratio of leading coefficients; a bigger bottom degree gives $y = 0$ .
Slant asymptote when the top's degree is one more: rewrite as a line plus a fraction that goes to $0$ .

each polynomial behaves like its leading term, so the function behaves like their quotient lim as x → ±∞, r(x) = c and y = c is a horizontal asymptote the quotient goes to 0 : the horizontal asymptote is y = 0 the graph approaches a slant asymptote y = mx + b

rational function: A function $r(x) = \frac{p(x)}{q(x)}$ where $p$ and $q$ are polynomials and $q$ is not the zero polynomial.
horizontal asymptote: A line $y = c$ with $\lim_{x \to \infty} r(x) = c$ or $\lim_{x \to -\infty} r(x) = c$ . For a rational function: the ratio of leading coefficients when the degrees are equal, $y = 0$ when the denominator's degree is greater.
slant asymptote: A non-horizontal line $y = mx + b$ that the graph approaches as $x \to \pm\infty$ ; a rational function has one when the numerator's degree is one more than the denominator's.

1.8 Rational Functions and Zeros

The real zeros of a rational function are the numerator's zeros that are in its domain. Zeros and excluded inputs split the number line; a test value in each interval gives the sign there. Solve a rational inequality from that sign chart, including zeros only for ≤ or ≥ and never an excluded input.

Real zeros of $\frac{p(x)}{q(x)}$ : the zeros of $p$ that are in the domain (not zeros of $q$ ).
A sign chart splits the line at the zeros of both numerator and denominator; one test value per interval gives its sign.
Solve a rational inequality from the chart; include zeros only for $\le$ or $\ge$ , never an excluded input.
With a calculator: graph, find every zero, then read the sign between them.

that input is a real zero of r that input is not in the domain, so it is never a zero they split it into intervals where r keeps one sign: test one input in each include the zeros with brackets; excluded inputs always get parentheses

rational inequality: An inequality such as $\frac{p(x)}{q(x)} \ge 0$ . Solve it with a sign chart: split the number line at the zeros of the numerator and the denominator, test one input in each interval, include zeros only for $\le$ or $\ge$ , and never include an excluded input.

1.9 Rational Functions and Vertical Asymptotes

A rational function has a vertical asymptote x = a where (x − a) divides the denominator more times than the numerator. Describe it from each side with one-sided limits, lim as x → a⁻, f(x) and lim as x → a⁺, f(x) , each ∞ or −∞ , found from the signs of the factors just beside a .

Vertical asymptote $x = a$ : the outputs increase or decrease without bound as $x$ approaches $a$ from one side or both.
For a rational function: a zero of the denominator with greater multiplicity than in the numerator.
One-sided limits: $\lim_{x \to a^-}$ from the left, $\lim_{x \to a^+}$ from the right, each $= \infty$ or $= -\infty$ .
Find the sign with a test input just beside $a$ , such as $a - 0.1$ and $a + 0.1$ .

x = a is a vertical asymptote: outputs grow without bound in size near a it cancels and there is no vertical asymptote at a you get each one-sided limit, ∞ or −∞ both one-sided limits have the same sign

vertical asymptote: A vertical line $x = a$ near which a function's outputs increase or decrease without bound. A rational function has one where $(x - a)$ divides the denominator more times than the numerator.
one-sided limit: The behavior of $f(x)$ as $x$ approaches $a$ from one side: $\lim_{x \to a^-} f(x)$ from the left, $\lim_{x \to a^+} f(x)$ from the right. At a vertical asymptote each is $\infty$ or $-\infty$ .

1.10 Rational Functions and Holes

A rational function has a hole at x = a when (x − a) divides the numerator at least as often as the denominator. Cancel the factor and evaluate at a to get the hole's height L ; then lim as x → a, r(x) = L while r(a) is undefined. Near a hole the outputs approach a number, unlike near a vertical asymptote.

Hole at $x = a$ : the multiplicity of $(x - a)$ in the numerator is at least its multiplicity in the denominator.
Its height is $\lim_{x \to a} r(x)$ : cancel, then evaluate at $a$ .
Near a hole the outputs approach a number; near a vertical asymptote they grow without bound.
Write $\lim_{x \to a} r(x) = L$ , never $r(a) = L$ : $r(a)$ is undefined.

it cancels completely from the denominator, and the graph has a hole at x = a you get the hole's y -coordinate L , and lim as x → a, r(x) = L the point (a, L) is drawn as an open circle, and a is not a zero even if L = 0

hole in the graph: A single point missing from a graph. A rational function has a hole at $x = a$ when $(x - a)$ divides the numerator at least as many times as the denominator; its height is $\lim_{x \to a} r(x)$ , found by cancelling the factor and evaluating.

1.11 Equivalent Representations of Polynomial and Rational Expressions

Factored form shows zeros, holes, vertical asymptotes and the domain; standard form shows end behavior; division writes a rational function as quotient plus remainder over divisor, and a linear quotient is the slant asymptote. Pascal's triangle expands (a + b)^n .

Equivalent forms give the same output for every input in the domain; each form makes different features easy to read.
Polynomial long division writes dividend $=$ divisor $\cdot$ quotient $+$ remainder, so $\frac{p(x)}{d(x)}$ $= q(x)$ $+ \frac{r(x)}{d(x)}$ .
When the quotient is linear, the line $y = q(x)$ is a slant asymptote: the remainder term shrinks toward $0$ for large inputs.
Row $n$ of Pascal's triangle gives the coefficients of $(a + b)^n$ ; each entry is the sum of the two above it.

read zeros, holes, vertical asymptotes and the domain read the leading term and so the end behavior get quotient + remainder over divisor, and the quotient is the slant or horizontal asymptote expand (a + b)^n without multiplying it out

polynomial long division: A way to divide one polynomial by another, like long division of numbers: divide the leading terms, multiply, subtract, repeat. It writes the dividend as divisor times quotient plus remainder, where the remainder has lower degree than the divisor.
binomial theorem: The rule for expanding a power of a sum, $(a + b)^n$ : the terms run from $a^n$ down to $b^n$ , and their coefficients are row $n$ of Pascal's triangle, where each entry is the sum of the two entries above it.
factored form: A polynomial written as a product of factors, such as $2x(x - 3)(x + 1)$ . Each linear factor shows a zero. Standard form, $2x^3 - 4x^2 - 6x$ , lists the terms by degree and shows the leading term and so the end behavior.

1.12 Transformations of Functions

g(x) = af(b(x − h)) + k dilates and reflects f by a and b and translates it h right and k up. Outside changes act on outputs and the range; inside changes act on inputs and the domain, in reverse.

Changes outside the function act on outputs and move the graph up, down, or stretch it vertically. Changes inside act on inputs and work in reverse.
$f(x - 3)$ moves the graph $3$ right: the old output at $0$ now appears at $x = 3$ .
In $a f(x) + k$ , dilate first, then shift. Domain changes come from inside changes; range changes come from outside changes.

every output moves by k (a vertical translation) the graph moves h units left when h > 0 (a horizontal translation) outputs are multiplied by a (a vertical dilation; reflection across the x -axis if a < 0 ) inputs are scaled by 1/b (a horizontal dilation; reflection across the y -axis if b < 0 )

function transformation: A change to a function that moves, stretches or flips its graph, made by adding to or multiplying the input or the output. The new function is written in terms of the old one, such as $g(x) = 2f(x - 3) + 1$ .
vertical translation: A shift of a graph without changing its shape. A vertical translation $g(x) = f(x) + k$ moves every output $k$ units up; a horizontal translation $g(x) = f(x - h)$ moves the graph $h$ units right.
vertical dilation: A stretch or compression of a graph. A vertical dilation $g(x) = af(x)$ multiplies every output by $a$ ; a horizontal dilation $g(x) = f(bx)$ multiplies every input by $\frac{1}{b}$ .
reflection across the x-axis: A flip of a graph. $g(x) = -f(x)$ is the reflection across the $x$ -axis (outputs change sign); $g(x) = f(-x)$ is the reflection across the $y$ -axis (inputs change sign).

1.13 Function Model Selection and Assumption Articulation

Choose a model type from the changes over equal input steps and from the context, then state the domain and range the context allows and the assumptions the model makes.

A function model is chosen from the data (the pattern of changes) and from the context (what the quantity measures).
Linear: constant rate of change. Quadratic: the rate of change changes at a constant rate, or one extremum with symmetry. Piecewise: different behavior on different intervals.
A model is only trusted for inputs the context allows: no negative times, no heights below the ground, no cuts bigger than the sheet.

constant first differences suggest linear; constant second differences suggest quadratic area suggests degree 2, volume degree 3, different rules on different intervals suggest piecewise restrict the domain and range, and state the assumptions the model makes

function model: A function chosen to describe how one quantity in a real situation depends on another, such as $h(t)$ for height after $t$ seconds. It is chosen from the pattern in the data and from the context, and used to predict and explain.
model assumption: A condition a function model takes for granted, such as a sheet having no thickness or a rate staying constant. Assumptions include the restricted domain (or contextual domain) and range: only the inputs and outputs the situation allows.

1.14 Function Model Construction and Application

Build a model from data points (a system), from features and transformations, or by regression; store its constants; use it to predict and to find average rates of change with units, only for inputs the context allows.

A model is built from points (a system), from a known feature and transformations (vertex form), or from all the data at once (a regression).
Inverse proportion: $y = \frac{k}{x}$ ; inverse-square laws: $y = \frac{k}{x^2}$ . One data point gives $k$ .
Report predictions and rates with units: an average rate of change is output units per input unit, such as hundred plants per month.

a form with unknown constants, such as at² + bt + c the constants, stored at full precision predictions with units, inside the domain the context allows

regression: Fitting a model of a chosen type (linear, quadratic, cubic, quartic, and later exponential and others) to all the data at once with a calculator, which picks the constants that make the model closest to the data. Linear regression and quadratic regression are the common cases.
inverse proportion: Inverse proportion: $y = \frac{k}{x}$ , so the product $xy$ stays constant and doubling $x$ halves $y$ . An inverse-square law, $y = \frac{k}{x^2}$ , divides $y$ by $4$ when $x$ doubles, as for light intensity and distance.