Rates of change and behavior of graphs

Lessons

1. Rate of change and average rate of changeI can explain this
2. On a graph, the average rate of change is a slopeI can explain this
3. Average rate of change from a formulaI can explain this
4. When an endpoint is a letter, the answer is an expressionI can explain this
5. The difference quotient: average rate of change from x to x + hI can explain this
6. Difference quotient of a quadraticI can explain this
7. Difference quotient of a fractionI can explain this
8. Increasing, decreasing, and constantI can explain this
9. Local maxima and minimaI can explain this
10. Toolkit behavior, then absolute maximum and minimumI can explain this

Vocabulary

Rate of change rayt of chaynj

How much an output changes per unit change in the input. Units are output units per input unit, such as miles per hour, dollars per year, or amperes per volt. Like: A speedometer: it shows how many miles you add per hour, not how far you have gone.

Average rate of change AV-rij rayt of chaynj

The output change divided by the matching nonzero input change between two inputs. It is the slope of the straight ruler line joining their graph points, regardless of the path between. Like: Splitting a restaurant bill evenly: the total divided by the number of people.

Δ (delta) DEL-tuh

The Greek capital letter delta, read change in. Δx names input change; Δy or Δf names output change. Δf describes a change in values of the same function. Like: The difference between two odometer readings: how far the car moved, not where it is.

Difference quotient DIF-er-ens KWOH-shunt

The average rate from a starting input x to an ending input x + h: f(x + h) − f(x)h$\frac{f(x+h)-f\left(x\right)}{h}$. The signed step h is nonzero, and both inputs must be allowed. Like: Average climb per mile for a directed walk, measured from its two ends.

Increasing in-KREE-sing

A function is increasing on an interval when bigger inputs always give bigger outputs there. Its graph climbs as you move from left to right. Like: Walking uphill on a trail as you head east.

Decreasing dih-KREE-sing

A function is decreasing on an interval when bigger inputs always give smaller outputs there. Its graph drops as you move from left to right. Like: Walking downhill on a trail as you head east.

Constant KON-stunt

A function is constant on an interval when its output stays the same for every input there. Its graph is a flat, horizontal piece. Like: A flat stretch of road: you move forward, but your height never changes.

Open interval OH-pun IN-ter-vul

A stretch of inputs that excludes its finite endpoints, written with parentheses. In this course, intervals of increase and decrease are reported as open stretches. Like: The stretch of road strictly between two mile markers, not counting the markers themselves.

Local maximum LOH-kul MAK-sih-mum

An output at least as high as all outputs in some open neighborhood on both sides of an interior input. A strict hilltop beats its neighbors; flat stretches can satisfy the nonstrict definition. Like: The top of one hump on a roller coaster, even if a taller hump comes later.

Local minimum LOH-kul MIN-ih-mum

An output at most as high as all outputs in some open neighborhood on both sides of an interior input. A strict valley lies below its neighbors; flat stretches can satisfy the nonstrict definition. Like: The bottom of a dip in the road, even if a deeper dip lies ahead.

Local extrema LOH-kul ek-STREE-muh

The local maxima and local minima of a function, taken together. A single one is called an extremum. Many books call them relative extrema instead. Like: All the peaks and valley floors marked on a hiking map.

Absolute maximum AB-suh-loot MAK-sih-mum

The greatest output a function reaches anywhere on its entire domain, if such a value exists. It can occur at more than one x. Like: The tallest mountain in the whole range, not only the tallest one nearby.

Absolute minimum AB-suh-loot MIN-ih-mum

The least output a function reaches anywhere on its entire domain, if such a value exists. It can occur at more than one x. Like: The deepest point of the whole ocean, not only the deepest spot near your boat.

Toolkit functions TOOL-kit FUNK-shunz

The nine basic functions whose graphs you should recognize on sight: constant, identity, absolute value, quadratic, cubic, reciprocal, reciprocal squared, square root, and cube root. Like: The basic shapes an art class practices before drawing anything more complicated.

Input IN-put

The value you feed into a function. On its graph, it is the horizontal coordinate. Like: The item you put into a vending machine.

Output OUT-put

The value a function returns for an allowed input. On its graph, it is the vertical coordinate. Like: The item a vending machine returns.

Function FUNK-shun

A rule that assigns exactly one output to each allowed input. Like: A machine with one result for each accepted setting.

Function notation FUNK-shun noh-TAY-shun

Notation such as f(x) that names a function and puts its input in parentheses. It does not mean multiplication. Like: A label on a machine plus the item being fed in.

Evaluate ih-VAL-yoo-ayt

Find an output by substituting an allowed input into every variable position and following the operations. Like: Run a recipe with the chosen ingredient amount.

Solve solv

Find every allowed input that makes an equation true. Like: Find the setting that produces a requested result.

Ordered pair OR-derd pair

Two coordinates written (x, y): the input first and its output second. It names one graph point. Like: An address with an across-position and a height.

Coordinate koh-OR-dih-nut

A number that locates a point along one axis. The x-coordinate is horizontal and the y-coordinate is vertical. Like: One direction in a two-part address.

Horizontal axis hor-ih-ZON-tul AK-sis

The graph’s left-to-right reference line. It usually shows inputs, such as x or time t. Like: The across direction on a map.

Vertical axis VUR-tih-kul AK-sis

The graph’s up-and-down reference line. It usually measures outputs, such as y or distance. Like: The height markings beside a ladder.

Scale skayl

The amount represented by one marked spacing on an axis. Read its labels before counting spaces. Like: A ruler can mark inches or half inches.

Domain doh-MAYN

The entire set of inputs a function accepts, including any stated restrictions. A chosen averaging interval can be only part of it. Like: All settings a machine accepts.

Range raynj

The set of output values the function actually reaches. A nearby height or a limiting height need not belong to it. Like: All items a machine can return.

Interval IN-ter-vul

A stretch of real numbers between boundaries. It may include or exclude a finite endpoint and may continue without bound. Like: A continuous stretch of road.

Interval notation IN-ter-vul noh-TAY-shun

A compact way to name a number-line stretch using its ends, parentheses for exclusions, and square brackets for inclusions. Like: Write a road’s start, end, and gate status.

Closed interval klohzd IN-ter-vul

A finite interval that includes both endpoints, written with square brackets. Like: A stretch of road with both gates admitted.

Endpoint END-point

A boundary of a graph piece or interval. A finite endpoint can be included or excluded. Course local comparisons need inputs on both sides. Like: The gate at the start or end of a road.

Hollow dot HOL-oh dot

An unfilled marker showing that a displayed endpoint or point is excluded from the graph or set. Like: A gate you can approach but cannot enter.

Filled dot fild dot

A solid marker showing that the displayed point or endpoint is included. Like: An admitted gate on a road.

Union YOON-yun

Combining sets so a number belongs if it belongs to at least one piece. The symbol is ∪. Like: Gather the people from either of two rooms.

Infinity in-FIN-ih-tee

The symbol ∞ describes continuing without a finite end. It is not a real endpoint that can be included. Like: A road with no last mile marker.

Slope slohp

A line’s vertical change divided by the matching horizontal change. It is undefined when the horizontal change is zero. Like: A road’s climb per step forward.

Rise ryz

The signed vertical change between two points, using the same end-minus-start order as the run. Like: How much higher or lower the road ends.

Run run

The signed horizontal change between two points, using the same end-minus-start order as the rise. Like: How far right or left the trip ends.

Subscript SUB-skript

A small lower label naming a quantity. It is a name tag, not a power or multiplication. Like: A numbered name badge.

Superscript SOO-per-skript

A small raised symbol. When it is an exponent, it tells how many equal factors are multiplied. Like: A count written above a repeated recipe.

Δx DEL-tuh eks

The signed change in input: ending input minus starting input. It must be nonzero in an average-rate denominator. Like: The move between two mile markers.

Δy DEL-tuh wy

The signed change in output: ending output minus starting output. Like: The difference between two road heights.

Δf DEL-tuh ef

Another name for the change in a function’s output values. It refers to the same function at two inputs. Like: Compare two receipts from the same machine.

Step size (h) step syz; aych

The signed change from start x to end x + h. Its sign gives direction; |h| gives distance. A difference quotient requires h ≠ 0. Like: A directed walk between mile markers.

Difference DIF-er-ens

The result of subtraction, retaining its sign. In a rate it measures an end value minus a start value. Like: Compare two odometer readings.

Quotient KWOH-shunt

The result of division. A rate quotient divides output change by a nonzero input change. Like: Share an amount into equal groups.

Turning point TUR-ning point

An interior point where a graph switches from rising to falling or from falling to rising. A flattening point need not turn. Like: A hilltop or valley floor on a road.

Relative maximum REL-uh-tiv MAK-sih-mum

Another name for a Local maximum: an output that beats or ties all outputs in a two-sided neighborhood of an interior input. Like: The highest nearby hill.

Relative minimum REL-uh-tiv MIN-ih-mum

Another name for a Local minimum: an output below or equal to all outputs in a two-sided neighborhood of an interior input. Like: The lowest nearby valley.

Local maxima LOH-kul MAK-sih-muh

The plural of Local maximum. A function can have several locally highest output values, each compared with nearby inputs. Like: Several nearby hilltop heights along a road.

Local minima LOH-kul MIN-ih-muh

The plural of Local minimum. A function can have several locally lowest output values, each compared with nearby inputs. Like: Several nearby valley-floor heights along a road.

Relative extrema REL-uh-tiv ek-STREE-muh

Another name for Local extrema: local maxima and local minima together. Like: The nearby hill and valley heights marked on a road map.

Local extreme values LOH-kul ek-STREEM VAL-yooz

Another name for the output values of Local extrema. Each value comes with an input location. Like: Heights written beside nearby hills and valleys.

Extremum ek-STREE-mum

One maximum or minimum value. State whether it is local or absolute, and give where it occurs. Like: One marked high or low on a map.

Extrema ek-STREE-muh

The plural of Extremum: maximum and minimum values considered together. They can be local or absolute. Like: Several marked highs and lows on a map.

Strict local maximum strikt LOH-kul MAK-sih-mum

An output greater than every other output in some two-sided neighborhood of an interior input. It does not tie nearby outputs. Like: A single hilltop above both sides.

Strict local minimum strikt LOH-kul MIN-ih-mum

An output less than every other output in some two-sided neighborhood of an interior input. It does not tie nearby outputs. Like: A single valley floor below both sides.

Neighborhood NAY-ber-hood

A small open interval around an input, containing nearby allowed inputs on both sides. Local extrema compare outputs inside one such interval. Like: The nearby road on both sides of your car.

Attain uh-TAYN

Actually reach a value at an allowed input. Approaching a height forever or drawing a hollow dot there does not attain it. Like: Arrive at a destination instead of only getting close.

Estimate ES-tih-mayt

Read or calculate an approximate value when an exact value is unavailable. Mark it with ≈ and state the precision when needed. Like: Read a length between ruler marks.

Approximation uh-prok-sih-MAY-shun

A value close to the exact answer, often obtained by rounding. It need not equal the exact value. Like: A nearby ruler tick used to report a measurement.

Exact value eg-ZAKT VAL-yoo

A value without rounding error, such as a fraction or an unevaluated radical. Like: A precise recipe amount instead of an estimated scoop.

± (plus or minus) plus or MY-nus

A symbol naming two choices, one positive and one negative. It does not change the meaning of the square-root symbol. Like: Two opposite directions from the same starting point.

Parabola puh-RAB-uh-luh

The U-shaped graph of a quadratic function, opening upward or downward. Like: A bowl or an upside-down bowl.

Vertex VUR-teks

The turning point of a parabola, or the corner of a V-shaped absolute-value graph. Its coordinates give both location and height. Like: The bottom of a bowl or tip of a V.

Numerator NOO-mer-ay-ter

The top of a fraction. It counts how many equal pieces are being represented. Like: The number of pizza slices you have.

Denominator dih-NOM-ih-nay-ter

The bottom of a fraction. It names the equal-piece size and must be nonzero. Like: How many equal slices make a whole pizza.

Common denominator KOM-un dih-NOM-ih-nay-ter

A matching nonzero denominator used to rewrite fractions before adding or subtracting them. Like: Cut both pizzas into the same size slices.

Reduce rih-DOOS

Rename a fraction by dividing numerator and denominator by the same nonzero shared factor. The value stays the same. Like: Trade two small slices for one larger slice.

Reciprocal rih-SIP-ruh-kul

The multiplicative inverse of a nonzero number. A nonzero fraction’s reciprocal swaps its numerator and denominator. Like: The operation that undoes multiplying by an amount.

Factor FAK-ter

One of the quantities being multiplied in a product. A factor can cancel only when it multiplies an entire numerator or denominator. Like: One repeated group in a packed box.

Factoring FAK-ter-ing

Rewriting a sum or difference as a product. It reverses distribution and makes shared factors visible. Like: Repack separate repeated items into groups.

Factor pair FAK-ter pair

Two integers whose product is a target number. Integers include whole numbers and their negatives. A negative target needs one positive and one negative factor. Like: Two multipliers that produce a chosen signed product.

Factored form FAK-terd form

An expression written as a product of factors. Keeping the product visible helps show what can cancel and what makes a denominator zero. Like: Leave the packed groups visible.

Expand ik-SPAND

Multiply out a product or power and combine like terms. Every factor must reach every term it multiplies. Like: Open packed boxes and list their contents.

Distributive property dih-STRIB-yoo-tiv PROP-er-tee

A multiplier outside parentheses multiplies each term inside. It works with positive and negative multipliers. Like: Give the same item to every group.

Term turm

A piece of an expression separated from other pieces by plus or minus signs. Its sign belongs with it. Like: One item in a list of amounts.

Like terms lyk turmz

Terms with the same variable factors and powers. Their numerical multipliers can combine; different variable patterns cannot. Like: Add counts of the same kind of item.

Difference of squares DIF-er-ens of skwairz

One square subtracted from another. It factors as a matching minus and plus pair because the middle products cancel. Like: Equal middle gains and losses cancel.

Square root skwair root

The nonnegative number whose square is the number inside ...$\sqrt{...}$. A real square root requires a nonnegative inside. Like: A square floor’s nonnegative side length from its area.

Cube root kyoob root

The real number whose cube is the input. Negative inputs have negative cube roots. Like: An edge recovered from a signed cube calculation.

Power POW-er

A repeated multiplication of the same base. The exponent records how many factors are used for a positive whole-number power. Like: Repeat the same multiplication step.

Exponent ik-SPOH-nunt

The raised number that tells the power. Positive whole-number exponents count repeated factors. Like: A repetition count above an instruction.

Undefined un-dih-FYND

Having no value under the rule being used. Division by zero is undefined. Like: A machine setting for which the recipe gives no result.

Vertical asymptote VUR-tih-kul AS-im-toht

A vertical line approached as a graph’s outputs grow without bound near that input. It is not part of the function graph. Like: A boundary the two graph branches approach.

Branch branch

One separate connected piece of a graph. A break can separate pieces that must be read individually. Like: A road divided by an impassable gap.

Identity function eye-DEN-tih-tee FUNK-shun

The toolkit function f(x) = x. It returns the input unchanged and increases on all real inputs. Like: A machine that hands back what you fed in.

Absolute value AB-suh-loot VAL-yoo

A number’s distance from 0, always nonnegative. The bars |x| measure distance and do not mean an absolute maximum or minimum. Like: Count spaces from zero without counting direction.

Quadratic function kwah-DRAT-ik FUNK-shun

A polynomial function of degree 2. Its graph is a parabola; x²$x^2$ is the basic toolkit example. Like: A bowl-shaped graph.

Cubic function KYOO-bik FUNK-shun

A polynomial function of degree 3. The basic toolkit example x³$x^3$ rises on all real inputs. Like: A smooth bend that continues lower left to upper right.

Reciprocal function rih-SIP-ruh-kul FUNK-shun

The toolkit function f(x) = 1x$\frac{1}{x}$, defined for x ≠ 0. It decreases on each separate side of 0. Like: Smaller shares when more nonzero groups divide one amount.

Reciprocal squared function rih-SIP-ruh-kul skwaird FUNK-shun

The toolkit function f(x) = 1x²$\frac{1}{x^2}$, defined for x ≠ 0. Outputs are positive, approaching 0 without reaching it. Like: Tiny positive shares as the squared divisor grows.

Linear equation LIN-ee-er ih-KWAY-zhun

An equation whose variable appears only to the first power. Equal operations on both sides isolate its unknown. Like: A balanced scale with equal groups.

Newton NOO-tun

A unit that measures force, a push or pull. The force example treats newtons as output units, so no physics calculation is needed. Like: A label telling how much push is measured.

Average speed AV-rij speed

Distance traveled divided by elapsed time. If distance from home increases throughout a trip without reversals, its rate gives the same average speed. Like: Spread the trip’s traveled miles evenly across its hours.

Integer IN-tih-jer

A whole number or the negative of a whole number. Fractions between whole numbers are not integers. Like: Numbered steps in either direction from zero.

Charged particles charjd PAR-tih-kulz

Tiny pieces of matter with electric charge. In the example, their separation is an input and the push between them is an output. Like: Two tiny specks whose spacing affects a push.

Force fors

A push or pull, measured in newtons in the section’s distance example. Like: The push your hand gives a door.

Per pur

For each one unit of another quantity. It names the division used in a rate. Like: Divide a total into equal one-unit shares.

Quarter KWOR-ter

In a time-rate problem, one fourth of a year, equal to three months. In money, the same word can mean a 25-cent coin. Like: One of four equal calendar pieces.

Secant line SEE-kant lyn

A straight line through two distinct points on a graph. Its slope is the average rate between the two inputs. Like: Lay a ruler through two marked places on a trail.

Nonnegative non-NEG-uh-tiv

Zero or positive. It includes 0 and excludes every negative real number. Like: A balance with no debt.

Nonpositive non-POZ-ih-tiv

Zero or negative. It includes 0 and excludes every positive real number. Like: A balance with no credit above zero.

Bound bownd

A height or number that a set of values never passes. A bound need not be attained. Like: A ceiling the graph may approach without touching.

Increasing function in-KREE-sing FUNK-shun

A function whose outputs strictly rise whenever its inputs rise within the interval being discussed. It can increase while its outputs are negative. Like: Walk uphill while still below sea level.

Decreasing function dih-KREE-sing FUNK-shun

A function whose outputs strictly fall whenever its inputs rise within the interval being discussed. Read each separated branch on its own. Like: Walk downhill as you move forward.

Constant function KON-stunt FUNK-shun

A function returning one fixed output c for every allowed input. c names one chosen number. Like: A machine that returns the same item every time.

Absolute value function AB-suh-loot VAL-yoo FUNK-shun

The toolkit function f(x) = |x|. It returns the distance of x from 0, falls toward 0, and then rises. Like: A V-shaped valley centered at zero.

Square root function skwair root FUNK-shun

The toolkit function f(x) = x$\sqrt{x}$, with domain [0, ∞). It rises from its included endpoint and has absolute minimum 0. Like: A road beginning at ground level and rising.

Cube root function kyoob root FUNK-shun

The toolkit function f(x) = ³x$\sqrt[3]{x}$. It accepts every real input and increases throughout its domain. Like: A rising road that passes smoothly through zero.

Absolute extrema AB-suh-loot ek-STREE-muh

The greatest and least attained output values on the entire domain, if they exist. One value can occur at several inputs. Like: The highest and lowest places on the entire trip.

Absolute maxima AB-suh-loot MAK-sih-muh

The plural of Absolute maximum. An absolute maximum value can be reached at more than one input. Like: Several summit locations tied at the same highest height.

Absolute minima AB-suh-loot MIN-ih-muh

The plural of Absolute minimum. An absolute minimum value can be reached at more than one input. Like: Several valley locations tied at the same lowest height.

Polynomial pol-ee-NOH-mee-ul

An expression formed by adding constants and numerical multiples of positive whole-number powers of its variable. Its expanded form has no variable in a denominator or root. Like: A recipe assembled from allowed power pieces.

Coefficient koh-uh-FISH-unt

The numerical multiplier of a term, including its sign. Like: How many copies of a recipe piece to use.

Degree dih-GREE

The greatest variable power with a nonzero coefficient in a simplified polynomial. Like: The highest rung appearing in a recipe.

Variable VAIR-ee-uh-bul

A letter standing for a number, which can vary or be unknown. Like: An empty input slot in a recipe.

Expression ik-SPRESH-un

Numbers, variables, and operations that describe a value. An equation sets two expressions equal. Like: One side of a mathematical recipe or balance.

Linear term LIN-ee-er turm

A numerical multiple of the input to the first power. Like: A fixed number of copies of the input.

Constant term KON-stunt turm

A term with no changing variable. It stays fixed as the input changes. Like: One fixed ingredient amount in the recipe.

Quick checks

Before you start

• Explain it like I am five

  Picture a road trip. You pass one mile marker and later pass another. The two signs tell you how far your position changed. The clock tells you how long that change took. Sharing the change over the time tells you your average pace.

  Now picture the road from the side. Read it from left to right. An uphill stretch gets higher, a downhill stretch gets lower, and a level stretch stays at one height. A drawing of inputs and outputs works the same way.

  A hill can be tallest near your car while a taller hill lies farther down the road. That is the difference between a local high and the highest place on the whole trip. The end of the road can also be the highest or lowest place you reach.

• Read the names and symbols before using them

  Think of a function as a named vending machine. You choose an input, it follows one rule, and it returns one output. A letter such as t is a place where you can put a number. The notation f(t) is read f of t: f is the machine’s name and t is its input. Parentheses hold that input together. An ordered pair records the same information as an address, with the input across first and the output height second.

• What to know cold, rebuild, and put on a cheat sheet

  Pack your study tools like a small travel bag. A few items must be within reach every time. Other items can be rebuilt from a short method. A third group belongs on a reference page when reference pages are allowed. For a closed-book exam, practice rebuilding those items before you leave the page. You do not need to memorize every multiplied-out expression or every worked answer.

• Subtracting signed numbers

  Subtraction answers a walking question on the number line: how far, and in which direction, do you walk from the second number to reach the first? Walking right counts as positive and walking left as negative. So 2 − (−1) asks how to get from −1 to 2: three steps right, which is +3. And 1 − 4 asks how to get from 4 to 1: three steps left, which is −3. A useful shortcut: subtracting a negative number is the same as adding the positive one, so 2 − (−1) = 2 + 1. Nonnegative means 0 or positive. Nonpositive means 0 or negative.

• Fraction names, reducing, multiplying, and dividing

  A fraction describes equal pieces, like slices of one pizza. The denominator is the bottom number: it tells you how many equal pieces make one whole. The numerator is the top number: it counts the pieces. Multiplying a fraction means taking a part of a part. Dividing asks how many of the other amount fit. Reducing renames the same amount with fewer, larger pieces, like replacing two small slices by one larger slice.

• Square roots, radical products, and real inputs

  Imagine a square floor. If its area is 16 square feet, its side is 4 feet because 4 times 4 is 16. A square root asks for that nonnegative side length. The braces in 16$\sqrt{16}$ hold everything under the root. You square a number by multiplying it by itself. The root goes backward from the area to the nonnegative side. A negative number has no real square root because a real square is never negative.

• Interval notation

  Interval notation is shorthand for a stretch of the number line. You write the left end, a comma, and the right end, wrapped in parentheses or square brackets. A parenthesis, ( or ), means that endpoint is left out; a square bracket, [ or ], means it is included. Infinity (∞) always gets a parenthesis, because it is not a number you can land on. The symbol ∪, read 'union', glues two separate stretches together. Set-builder notation names a collection by a condition. In {x | x > 2}, the vertical bar is read such that, so this says all x such that x is greater than 2.

• Domain, range, endpoints, and graph windows

  A map has a border, but the real road can continue beyond the map. A graph window has the same problem: the edge of the picture need not be the end of the function. The domain is every allowed input, and the range is every output actually reached. An endpoint is a boundary of an allowed stretch. A filled endpoint is included. A hollow endpoint is left out. An arrow says the graph continues; it does not mark a last point.

• Reading an output from a graph

  Read a graph like a street map with two directions. The horizontal axis runs across and locates the input. The vertical axis runs up and down and measures the output. An ordered pair (x, y) is one address: across first, height second. To find f(−1), start at input −1, move straight up or down to the graph, and then read its height. First inspect the scale because one grid space can mean more than one unit.

• Slope: rise over run

  Slope measures how steep a straight line is, the way a road sign warns that a hill climbs 6 feet for every 100 feet forward. You pick two points on the line and ask two questions: how far up did I go (the rise), and how far right did I go (the run)? Slope is rise divided by run: the number of units the line climbs for each single unit you move to the right. A negative slope means the line goes down as you move right, and a slope of 0 means it is flat.

• Evaluating a function at a number

  A function is a machine: you drop in an input and it hands back an output. The notation g(−2) means 'feed −2 into g'. You do it by erasing every x in the formula and writing the input in its place, inside parentheses. The parentheses matter most for negative numbers and for whole expressions such as x + h, because they keep the input in one piece while you square it or multiply it.

• Subtracting fractions and dividing a fraction by a whole number

  A fraction counts pieces: the bottom says what size the pieces are (fourths, eighths), and the top says how many pieces you have. You can subtract only pieces of the same size, the way you can compare pizza slices only if both pizzas were cut the same way. So first rewrite the fractions over a common bottom, then subtract the tops. To divide a fraction by 2, cut every piece in half again: the bottom doubles and the top stays.

• Subtracting a whole expression

  When a minus sign sits in front of parentheses, it applies to every term inside, not only the first. A term is a piece separated by a plus or minus sign. Think of taking away a debt: removing a debt of 2 makes you 2 richer. The distributive property says a multiplier reaches every term in parentheses. A minus in front is the multiplier −1, so it flips every sign. Like terms have the same letters with the same powers, such as 5x and −2x; x²$x^2$ and x are different kinds and cannot combine.

• Polynomial, coefficient, and degree before the algebra checks

  Think of an expression as a recipe made from terms. A variable is a letter standing for a number. A polynomial recipe adds constants and numerical multiples of positive whole-number powers of its variable. A coefficient is the numerical multiplier of a term, including its sign. The degree is the greatest variable power with a nonzero coefficient. Degree 1 is linear, degree 2 is quadratic, and degree 3 is cubic. These names describe the whole simplified polynomial, not any one term you happen to see.

• Squaring a sum: (x + h)²${)}^2$

  (x + h)²${)}^2$ means (x + h) times (x + h). It is tempting to write x²$x^2$ + h²$h^2$, but that loses two pieces. Picture a square garden whose sides are x + h feet long. Its area splits into four patches: an x by x square, an h by h square, and two x by h strips. Adding the patches gives x²$x^2$ + 2xh + h²$h^2$.

• Cubing a sum: (x + h)³${)}^3$

  A cube is three copies of the same factor multiplied together, like three equal edges used to find a box’s volume. To cube a sum, square it first and multiply that result by one more copy of the sum. Each term in the square must multiply both terms in the last copy. Keeping the six products on separate lines gives each product a place, so none gets lost. A power counts repeated factors: h²$h^2$ · h contains three h factors and is h³$h^3$.

• Factoring out a common factor and canceling

  Canceling works only on things that are multiplied. If every term on the top of a fraction contains an h, you can write each term as h times something, like taking one identical item from each packed group. Then the top is 'h times a group', and that h cancels with an h on the bottom, because h divided by h is 1.

• Factoring x²$x^2$ + bx + c

  Some expressions, like a²$a^2$ + 2a − 35, can be rewritten as a product of two parentheses, (a + 7)(a − 5). Think of it as running a multiplication backward: you know the answer and need the two things that were multiplied. The method is a number puzzle: find two numbers that multiply to the last number and add to the middle number.

• Difference of squares

  An expression like b²$b^2$ − 4 is one perfect square minus another, since 4 = 2²$2^2$. It always factors the same way: b²$b^2$ − 4 = (b − 2)(b + 2). You will meet it when an interval ends at a letter, like [2, b], and the function has an x²$x^2$ term. After subtracting, a common number usually comes out first, and a difference of squares is left.

• Subtracting fractions that contain letters

  Letters in the bottom of a fraction do not change the rules. To subtract 1x + h$\frac{1}{x+h}$ − 1x$\frac{1}{x}$, the pieces must be the same size, so you need a common bottom. The product of the two bottoms, x(x + h), always works. Each fraction gets multiplied, top and bottom, by the piece of the common bottom that it is missing.

• Solving a linear equation and a fraction equation

  An equation is like a balanced scale: both sides have the same value. Solving means finding the input that makes the balance true. To keep it balanced, perform the same operation on both sides. Undo the operation that was performed last. For 6b + 5, undo the added 5 before the multiplication by 6. A fraction equation can be solved by multiplying both sides by its nonzero denominator. Write forbidden denominator values first, because a solution must work in the original equation.

• The signed step h and the distance between inputs

  Think of a walk along numbered mile markers. A change has a direction as well as a size. The letter h names the signed change from your starting input x to your ending input x + h. A positive h goes right, and a negative h goes left. The distance you walk is the nonnegative size |h|. Those are different jobs: direction belongs to h, while distance belongs to |h|. A difference quotient can use either direction as long as its two inputs are allowed and different.

• Decimal division, rounding, dollars, and cents

  Money gives decimal places a concrete meaning. One dollar is 100 cents, so $1.37 is 137 cents. Sharing that change over seven years gives a rate in dollars per year, or the same rate in cents per year. Rounding is like marking the nearest small tick on a ruler: you keep a chosen number of places and inspect the next digit. Keep the exact fraction during work and label the rounded decimal with ≈.

• Elapsed minutes from clock times

  A clock starts a new hour after 60 minutes, so its face is not a decimal number line. To measure a short trip across an hour mark, count from the start to the next hour and then count onward to the finish. Add the two pieces. Once you know the elapsed minutes, a rate uses change divided by those minutes. A quarter of a year is three months; it is a time unit and has a different meaning from a 25-cent coin.

• One-page cheat sheet for rates and graph behavior

  Use this compact reference like a route card: it reminds you what to compare and which method gets you there. It contains the formulas and decision cues, not the answers to your homework. Print this refresher on its own during study. For a closed-book exam, cover it and rebuild the calculation before checking the card.