Sinusoidal functions

Lessons

1. A sinusoidal function is a repeating waveI can explain this
2. Amplitude measures height, not directionI can explain this
3. Period measures the width of a repeatI can explain this
4. Phase shift: why minus inside moves rightI can explain this
5. One cycle: solve the inside-input inequalityI can explain this
6. Midline, maximum, minimum and rangeI can explain this
7. Read all four features, including unusual signsI can explain this
8. Build an equation from a graphI can explain this
9. Build and check a daylight modelI can explain this

Vocabulary

Sinusoidal function sih-nuh-SOY-duhl FUNK-shuhn

A repeating wave made by stretching and shifting sine or cosine. This section studies waves that have changing heights. Like: A smooth rise and fall that repeats like a Ferris wheel seat's height.

Sine sine

The vertical coordinate of a point on a circle of radius 1 after turning through the input angle. Its abbreviation is sin. Like: How high or low the wheel's marker sits.

Cosine KOH-sine

The horizontal coordinate of a point on a circle of radius 1 after turning through the input angle. Its abbreviation is cos. Like: How far right or left the wheel's marker sits.

Cycle SY-kuhl

One complete run through a repeating pattern, from one position in the pattern to the next identical position with the same direction of motion. Like: One full lap around a track.

Periodic function peer-ee-AH-dik FUNK-shuhn

A function whose outputs repeat after a fixed positive input distance: f(x + T) = f(x) for every input in its domain. Like: A weekly schedule that repeats after seven days.

Amplitude AM-plih-tood

The vertical distance from a wave's middle line to a highest or lowest point. It equals |A| and is always positive for a changing wave. Like: How far a swing seat rises above or falls below its middle height.

A ay

The signed multiplier outside sine or cosine. Its absolute value gives the amplitude; a negative A turns the basic wave upside down. Like: A height setting with a separate switch for flipping the ride.

Absolute value AB-suh-loot VAL-yoo

A number's distance from 0 on the number line. Distance keeps size and removes direction, so absolute value is never negative. Like: The distance from home, whether you walked east or west.

Maximum MAK-sih-muhm

The greatest output a wave reaches. For y = A sin(ωx − φ) + B, it is B + |A|. Like: The highest height reached during a Ferris wheel ride.

Minimum MIN-ih-muhm

The least output a wave reaches. For y = A sin(ωx − φ) + B, it is B − |A|. Like: The lowest height reached during a Ferris wheel ride.

Peak peek

A highest point on a wave. Its vertical coordinate is the maximum output, while its horizontal coordinate tells where that height occurs. Like: The crest at the top of a rolling hill.

Trough troff

A lowest point on a wave. Its vertical coordinate is the minimum output, while its horizontal coordinate tells where that height occurs. Like: The bottom of a valley between two hills.

Period PEER-ee-uhd

The smallest positive horizontal distance after which the entire changing wave repeats. For positive ω, the period is 2πω$\frac{2\pi }{\omega }$. Like: The time between one bell ring and the next in a steady repeating schedule.

T tee

The letter used for a wave's period. It measures one complete repeat along the horizontal axis. Like: The length marked on a ruler beneath one repeating pattern.

ω (omega) oh-MAY-guh

The positive multiplier of the horizontal input in standard form. It controls how quickly the inside input changes and satisfies ω = 2πT$\frac{2\pi }{T}$. Like: A speed setting for the wheel that generates the wave.

φ (phi) fie

The signed number subtracted inside sin(ωx − φ) or cos(ωx − φ). Divide it by positive ω to get the phase shift. Like: A timing adjustment before the wheel begins its usual cycle.

Phase shift fayz shift

The signed horizontal shift φω$\frac{\phi }{\omega }$ in standard form with ω > 0. Positive means right; negative means left. Like: Moving the start mark of a repeating ride along its timetable.

Horizontal shift hor-ih-ZAHN-tuhl shift

Moving a graph left or right by adding the same horizontal distance to every point's position. Like: Sliding a drawing sideways across a desk.

Midline MID-line

The horizontal line halfway between a wave's greatest and least heights. Its equation is y = B, where B is the average of those heights. Like: The center height around which a swing moves up and down.

B bee

The number added outside sine or cosine. It is the middle height and gives the midline equation y = B. Like: The base floor from which the ride's up and down motion is measured.

Vertical shift VER-tih-kuhl shift

Moving every point of a graph up or down by the same amount. Adding B outside the function shifts it up if B is positive and down if negative. Like: Raising or lowering a whole shelf without tilting it.

Range raynj

All the output heights a function actually reaches. A sinusoidal wave's range includes every height from B − |A| through B + |A|. Like: All the shelf heights a moving lift can reach.

Domain doh-MAYN

All the inputs a function accepts. A sinusoidal formula made from sine or cosine accepts every real horizontal input. Like: All the button choices a machine accepts.

Radian RAY-dee-uhn

An angle unit measured by arc length divided by radius. A full turn is 2π radians, so a quarter turn is π2$\frac{\pi }{2}$ radians. Like: Measuring a wheel's turn by the distance its marker travels.

Inside input in-SIDE IN-put

The entire expression fed into sine or cosine, such as ωx − φ. Calculate that expression before finding the sine or cosine output. Like: The number handed from the first machine to the next machine.

Quarter period KWOR-ter PEER-ee-uhd

One fourth of a wave's period. Five key points separated by this distance mark the start, three interior points and the end of one cycle. Like: One of four equal stretches of a complete lap.

Key points kee points

Useful points that anchor a wave sketch. Across one cycle, choose the start and four more points spaced one quarter period apart. Like: Fence posts that guide the curve between them.

Even function EE-vuhn FUNK-shuhn

A function satisfying f(−u) = f(u): opposite inputs give the same output. Its graph has matching left and right halves. Like: A face whose left and right halves match in a mirror.

Odd function ahd FUNK-shuhn

A function satisfying f(−u) = −f(u): opposite inputs give opposite outputs. Its graph matches itself after a half turn about the origin. Like: The same motion reversed in both direction and height.

Sinusoidal model sih-nuh-SOY-duhl MAH-duhl

A sine or cosine formula chosen to approximate observations that rise and fall in a repeating pattern. Predictions may differ from the observed values. Like: A smooth sketch through measurements of a yearly rhythm.

Latitude LAT-uh-tood

The angle telling how far a place lies north or south of Earth's equator, the line around Earth's middle. Like: A north-or-south address on a globe.

Time variable t time VAIR-ee-uh-buhl tee

The input letter counting time in a model. For the daylight data, t counts months continuously, with January labeled 1 and the next January labeled 13. Like: A calendar counter that keeps counting into the next year.

Horizontal line hor-ih-ZAHN-tuhl line

A straight line that stays at one height while extending left and right. Its equation has the form y = a fixed number. Like: A level shelf whose height is the same at every position.

Data DAY-tuh

Recorded observations or measurements. A model uses these observations to estimate a pattern and predict other values. Like: Numbers written in a notebook after measuring something each month.

Daylight hours DAY-lite OW-erz

The length of time during a day when the Sun is above the horizon, measured in hours. Like: The part of a day's clock during which the sky has daylight.

Sine model sine MAH-duhl

A model using sine to describe a repeating rise and fall. With positive A and ω, its shifted cycle starts at an upward middle crossing. Like: Starting a ride's timetable as the seat crosses its middle height going up.

Cosine model KOH-sine MAH-duhl

A model using cosine to describe a repeating rise and fall. With positive A and ω, its shifted cycle starts at a peak. Like: Starting a ride's timetable when the seat is at its highest position.

Unit circle YOO-nit SUR-kuhl

The circle of radius 1 centered at (0, 0). Its point at angle u has coordinates (cos u, sin u). Like: A wheel with its rim exactly one unit from its center.

Square root skwair root

The nonnegative number whose square equals the quantity inside $\sqrt{}$. It gives a positive length when that quantity is positive. Like: Recover a square tile's side length from its area.

Pythagorean theorem pih-THAG-uh-REE-uhn THEE-uh-ruhm

In a right triangle, the squares of the two shorter sides add to the square of the longest side. Like: Two smaller square areas together equal the largest square area.

Coefficient koh-uh-FISH-uhnt

A number multiplying a variable or expression. Its location tells what quantity it scales. Like: The number of identical bags in a purchase.

Parameter puh-RAM-uh-ter

A fixed number in a family of formulas that controls a feature of the graph. Like: A dial setting you choose before a machine runs.

Interval notation IN-ter-vuhl noh-TAY-shuhn

A compact way to name all numbers between two endpoints. Square brackets include an endpoint; parentheses exclude it. Like: A label describing a permitted stretch on a ruler.

Three-part inequality three part in-ee-KWAHL-uh-tee

Two comparisons joined together so one quantity has a lower and upper bound at once. Like: A thermostat reading that must stay between two limits.

Substitution sub-stih-TOO-shuhn

Replacing each copy of an input letter with the chosen value before calculating. Like: Fill every matching blank on an instruction card.

Reflection ree-FLEK-shuhn

A flip across a line that preserves distances. Compared with the same positive multiplier and center, a negative outside multiplier exchanges heights above and below the midline. Like: Turn a drawing over in a mirror.

Real number REE-uhl NUM-ber

Any number represented on the ordinary number line, including whole numbers, fractions, negative numbers and nonrepeating decimals. Like: Any marked position along an unbroken ruler.

Equilateral triangle ee-kwee-LAT-er-uhl TRY-ang-guhl

A triangle with three equal sides and three equal 60° angles. Like: Three identical sticks joined into a triangle.

Midline crossing MID-line KRAH-sing

A point where the wave passes through its center height. Upward and downward crossings are different stages of the cycle. Like: A moving elevator passing its middle floor while rising or falling.

Quick checks

Before you start

• Function input and substitution

  Think of a vending machine that accepts a number instead of a coin. A function is its instruction card: it takes the input you give it and returns one output. In f(x) = 2x + 5, x marks the empty input slot. The request f(3) means: put 3 into that slot and find the output. It does not mean f times 3. Substitution means replacing each copy of the input letter with the chosen number. Parentheses keep that number together while you follow the instructions.

• Signed arithmetic and absolute value

  Picture yourself walking along a number line. Positive numbers point right and negative numbers point left. Adding a positive number moves you right; adding a negative number moves you left. Subtracting reverses the direction of the number being subtracted. Absolute value measures your distance from 0, without saying which side you are on. The marks |−6| ask for that distance, so |−6| = 6. A negative starting point and a positive distance can describe the same spot. Keep direction and distance separate when reading the numbers outside a wave formula.

• Read a graph, a table and an interval

  Think of a map with one direction across the page and another direction up the page. A function graph uses the across direction for an input and the up direction for its output. A point (2, 5) means input 2 and output 5, in that order. The horizontal axis is the input line; the vertical axis is the output line. A real number is any position on the ordinary number line. The domain lists allowed inputs. The range lists outputs the function reaches. A table pairs each input with its output in one column. An interval names a whole stretch of numbers, including the numbers between its endpoints.

• Fractions containing π and dividing by a fraction

  A fraction describes equal pieces. If a ribbon has length π, cutting it into four equal pieces gives each piece length π4$\frac{\pi }{4}$. The symbol π is the fixed ratio of a circle's distance around to its diameter, the distance across its center. It is about 3.14159. Since π is a number, fraction arithmetic treats it like any other length. Leave π written exactly rather than replacing it with a rounded decimal. Division asks how many of one amount fit into another. Dividing by 13$\frac{1}{3}$ has the same effect as tripling: three thirds fit into one whole. Keep the amount you started with, turn the divisor upside down, and multiply.

• Decimal sum, difference and average

  Think of decimal amounts as money written in dollars and cents. A value of 18.72 has 1,872 hundredths, and 5.88 has 588 hundredths. Keeping the decimal points aligned keeps dollars with dollars and hundredths with hundredths. The average of two values is their sum divided by 2. On a number line it is the point halfway between them. Half their difference tells you how far that midpoint is from either end. These are the two calculations you need to find a wave's middle height and the distance from that middle to an extreme.

• Radians and sine/cosine quarter-turn values

  Picture a wheel centered at (0, 0), with radius 1 and a marker starting at its rightmost point. A radius is the distance from the center to the rim, so here that distance is 1. A degree is one of 360 equal parts of a full turn; ° is its symbol. A radian measures a turn by the distance traveled around that wheel. One full lap has length 2π, so a full turn is 2π radians and a quarter turn is π2$\frac{\pi }{2}$ radians. The marker's horizontal position is cosine. Its vertical position is sine. At the four compass directions you can read these positions without a calculator: right, up, left, down. The two tables below record those positions separately. Use the column under the input you need. This wheel is a Unit circle. In the point (horizontal, vertical), the first number is across and the second is up. We measure a positive turn counterclockwise, opposite a clock hand; this is a convention that fixes which direction counts positive. We use u as an input-angle letter. P(u) names the point reached after that turn, so P(u) = (cos u, sin u).

• An exact special-angle value, from the triangle

  You will estimate daylight at an inside angle of π3$\frac{\pi }{3}$. That is 60° because π radians is a half turn of 180°. To get its sine without rounding, picture an Equilateral triangle, meaning all three sides are equal, each of length 2. Cut it from the top straight down to the middle of the bottom side. You get two right triangles, meaning triangles with a square 90° corner. Each has a short leg of 1, a slanted side of 2 and a 60° angle at the bottom. The other leg comes from the Pythagorean theorem.

• Factoring and solving a linear equation with substitution

  Factoring is packing a repeated multiplier outside parentheses. Two bags that each contain the same items can be written as one description of a bag, multiplied by 2. Solving an equation is a different job: find the input that makes the two sides equal. Think of a balanced scale. You can add the same amount to both pans or divide both amounts by the same nonzero number and keep the balance. For 2x − π = 0, you want the input that makes the expression 2x − π come out as 0. The balance holds for any Real number that makes both sides equal.

• Three-part inequalities

  A Three-part inequality gives a quantity a lower limit and an upper limit at the same time. It is like a temperature that must stay between two allowed readings. The chain 0 ≤ 2x − π ≤ 2π says the expression 2x − π must be at least 0 and at most 2π. To discover the allowed x values, undo the expression's operations. Apply each move to all three parts, so both limits move with the quantity between them. The ≤ signs allow the endpoints themselves.

• Even cosine and odd sine

  Picture a marker turning around a wheel. Turning the same amount clockwise instead of counterclockwise puts it at the mirror point across the horizontal line through the center. Its horizontal position stays the same, but its vertical position changes sign. Cosine reads horizontal position, so cos(−u) = cos u. Sine reads vertical position, so sin(−u) = −sin u. These properties are called even and odd. The names describe what happens when you reverse an input's sign; they do not say the input itself must be an even or odd whole number.