Mathematics Functions

NOTE

Mr. Gordon generated this concept summary and the related examples using ChatGPT (here are the prompts given and responses received).

He then reviewed and edited the results for clarity and accuracy.

This page is intended as a reference (please bookmark for future use).

1. Absolute Value

What it does:
abs() gives the size (or distance from zero) of a number, ignoring whether it’s positive or negative.

Example 1:

“Let’s say you track how far your stock price moves up or down each day. Even if it drops by $5 one day, the absolute value of the change is 5 because we only care about the size of the change, not the direction.”

let change = -5
let distanceFromZero = abs(change)
print(distanceFromZero)  // Output: 5

Example 2 (Grade 11: Exponential Functions):

“When you calculate the error between two predictions, you only care how far off you were—whether your prediction was too high or low doesn’t matter.”

let predictedValue = 50.0
let actualValue = 45.0
let error = abs(predictedValue - actualValue)
print(error)  // Output: 5.0

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2. Minimum and Maximum

What they do:
min() gives the smallest number, and max() gives the largest.

Example 1 (Grade 12: Optimization in Calculus):

“Suppose you need to figure out the maximum distance a drone can fly based on two battery types.”

let batteryA = 30.0  // minutes
let batteryB = 45.0  // minutes
print(max(batteryA, batteryB))  // Output: 45.0

Example 2 (Grade 11: Personal Finance):

“You bought some groceries with two discounts, and you want to know the bigger discount!“

let discount1 = 10.0  // dollars
let discount2 = 7.5   // dollars
print(max(discount1, discount2))  // Output: 10.0

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3. Exponentiation

What it does:
pow(base, exponent) raises a base number to the power of the exponent.

Example 1 (Grade 11: Exponential Growth):

“If you invest $1000 at 5% interest per year, how much will you have after 3 years (assuming compound interest)?”

let principal = 1000.0
let rate = 1.05  // 5% interest
let years = 3.0
let amount = principal * pow(rate, years)
print(amount)  // Output: 1157.625

Example 2 (Grade 12: Population Growth):

“How many bacteria will grow if they double every hour, starting from 1 bacterium, after 4 hours?“

let initialBacteria = 1.0
let doublingTime = 4.0
let totalBacteria = initialBacteria * pow(2, doublingTime)
print(totalBacteria)  // Output: 16.0

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4. Square Root

What it does:
sqrt() gives the square root of a number (a number that multiplies by itself to give the original value).

Example 1 (Grade 11: Geometry):

“What is the side length of a square field with an area of 64 square meters?”

let area = 64.0
let sideLength = sqrt(area)
print(sideLength)  // Output: 8.0

Example 2 (Grade 11: Distance Between Two Points):

“If two points on a coordinate plane are (1, 1) and (4, 5), what’s the distance between them?“

let dx = 4.0 - 1.0
let dy = 5.0 - 1.0
let distance = sqrt(pow(dx, 2) + pow(dy, 2))
print(distance)  // Output: 5.0

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5. Rounding Functions

What they do:

• round() rounds to the nearest integer.
• ceil() rounds up.
• floor() rounds down.

Example 1 (Grade 12: Personal Finance):

“If you buy a product for $12.49, how does it round off?”

let price = 12.49
print(round(price))  // Output: 12
print(ceil(price))   // Output: 13
print(floor(price))  // Output: 12

Example 2 (Everyday Math):

“Your pizza bill is $47.85. If each friend pays their share, how much will it be if you round up to avoid underpaying?“

let totalBill = 47.85
let numFriends = 4.0
let sharePerPerson = totalBill / numFriends
print(ceil(sharePerPerson))  // Output: 12

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6. Trigonometric Functions

What they do:
The sin(), cos(), and tan() functions give the sine, cosine, and tangent of an angle, measured in radians.

Example 1 (Grade 11: Right Triangles):

“What’s the sine of a 45-degree angle?”

let angle = Double.pi / 4  // 45 degrees in radians
let sineValue = sin(angle)
print(sineValue)  // Output: ~0.707

Example 2 (Grade 12: Wave Functions):

“Calculate the height of a wave at a given angle if the amplitude is 3 meters.”

let angle = Double.pi / 6  // 30 degrees
let amplitude = 3.0
let height = amplitude * sin(angle)
print(height)  // Output: 1.5

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7. Arccosine

What it does:
acos() (inverse cosine) returns the angle (in radians) whose cosine is the given value. Use it when you know the ratio of the adjacent side to the hypotenuse in a right triangle and need to find the angle.

Example 1 (Grade 11: Right-Angle Trigonometry):

“You’re standing at the base of a hill, and you measure that the slope makes a ratio of 0.5 between the adjacent side (horizontal ground) and the hypotenuse (slope length). What’s the angle of elevation of the hill?”

let ratio = 0.5
let angleInRadians = acos(ratio)  // Output: ~1.05 radians
let angleInDegrees = angleInRadians * (180 / Double.pi)
print(angleInDegrees)  // Output: ~60.0 degrees

Example 2 (Grade 12: Vectors and Dot Product):

“If the cosine of the angle between two vectors is 0.866, what is the angle between them?“

let cosTheta = 0.866
let angleInRadians = acos(cosTheta)
let angleInDegrees = angleInRadians * (180 / Double.pi)
print(angleInDegrees)  // Output: ~30.0 degrees

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8. Arcsine

What it does:
asin() (inverse sine) returns the angle (in radians) whose sine is the given value. Use it when you know the ratio of the opposite side to the hypotenuse and want to find the angle.

Example 1 (Grade 11: Height of a Kite):

“A kite string is 20 meters long, and the string makes a 30% (0.3) angle with the ground. How high is the kite?”

let ratio = 0.3
let angleInRadians = asin(ratio)  // Output: ~0.305 radians
let angleInDegrees = angleInRadians * (180 / Double.pi)
print(angleInDegrees)  // Output: ~17.5 degrees

Example 2 (Grade 12: Waves and Harmonic Motion):

“If a sine wave reaches a peak at 0.707 units, what angle corresponds to that point?“

let ratio = 0.707
let angleInRadians = asin(ratio)
let angleInDegrees = angleInRadians * (180 / Double.pi)
print(angleInDegrees)  // Output: ~45.0 degrees

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9. Arctangent

What it does:
atan() (inverse tangent) returns the angle (in radians) whose tangent is the given value. Use it when you know the ratio of the opposite side to the adjacent side and want to find the angle.

Example 1 (Grade 11: Inclined Plane Problem):

“A ramp rises 2 meters for every 5 meters along the ground. What is the angle of the ramp?”

let ratio = 2.0 / 5.0
let angleInRadians = atan(ratio)  // Output: ~0.38 radians
let angleInDegrees = angleInRadians * (180 / Double.pi)
print(angleInDegrees)  // Output: ~21.8 degrees

Example 2 (Grade 12: Physics – Projectile Motion):

“An object is launched at an angle, and the ratio of its vertical to horizontal velocity is 1.2. What angle was it launched at?“

let velocityRatio = 1.2
let angleInRadians = atan(velocityRatio)
let angleInDegrees = angleInRadians * (180 / Double.pi)
print(angleInDegrees)  // Output: ~50.2 degrees

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10. Logarithms

What they do:
log() returns the natural logarithm, and log10() returns the base-10 logarithm.

Example 1 (Grade 12: Logarithmic Functions):

“How long will it take for an investment to double if it grows continuously at 7% per year?”

let rate = 0.07
let time = log(2) / rate
print(time)  // Output: ~9.9 years

Example 2 (Grade 12: Sound Intensity):

“If the intensity of a sound wave is 1000 units, what is the decibel level?”

let intensity = 1000.0
let decibels = 10 * log10(intensity)
print(decibels)  // Output: 30.0