Quadratics in Sports
This type of graph, shaped like a curve, is used to model the path of a basketball shot, soccer kick, or a football throw.
Parabola
This simple trigonometric ratio helps construction workers find the height of a building using just a ladder and a protractor.
What is tangent?
(Explanation: tan(θ) = opposite/adjacent — often used to find the height of a building using the angle and distance.)
You buy 3 bags of coffee and 2 muffins. Each bag of coffee costs c dollars, and each muffin costs $2. Your total bill is $20. Find the cost of one bag of coffee.
Each bag of coffee costs $5.33
Write equation:
3c+2(2)=20
Simplify:
3c+4=20
Subtract 4:
3c=16
Divide:
c=16/3≈5.33
You earn $20 per hour walking dogs.
Write the linear function that represents your earnings after x hours.
What is y = 20x?
You get $10 every week as allowance. After 5 weeks, how much total money will you have received?
Use arithmetic sum formula
First term: a1=10a1=10
Common difference: d=0d=0 (it's constant)
Number of terms: n=5n=5
Use sum formula:
S5=5/2(2⋅10+(5−1)⋅0)=5/2⋅20=50Answer:$50
A soccer ball is kicked and follows the equation
h(t)= -16t2 +40t
What is the maximum height the ball reaches?
25
A roof has a slant height of 10 meters and makes an angle of 30° with the horizontal. This is how you find the vertical height of the roof.
What is 10 × sin(30°)?
(Explanation: sin(θ) = opposite/hypotenuse → height = slant height × sin(θ).)
In a gift shop, 2 candles and 1 mug cost $18. Three candles and two mugs cost $28. How much is each candle and each mug?
Answer:
Candle = $8, Mug = $2
A car is worth $25,000 and depreciates $2,500 per year.
Write a function for the value V(t) of the car after t years.
What is V(t) = -2500t + 25000?
A population of bacteria triples every hour. If you start with 100 bacteria, how many will there be after 4 hours?
Use geometric nth term formula
a1=100, r=3r=3, n=4n=4
Use geometric term
a4=100⋅3 a la 3=100⋅27=2700Answer: 2,700 bacteria
A soccer ball follows the path
h(t) = −4.9t2 + 19.6t
What’s the ball’s maximum height?
19.6
A crane’s support cable stretches from the top of a 25-meter pole to the ground 40 meters away. Which law helps find the angle the cable makes with the ground?
What is the Law of Cosines?
(Use it when you know two sides and the included angle or all three sides of a triangle.)
A $60 pair of shoes is marked 25% off. What’s the sale price after 8% tax is added?
Answer: $48.60
Discount: 60×0.25=1560 \times 0.25 = 1560×0.25=15
Sale price before tax: 60−15=4560 - 15 = 4560−15=45
Tax: 45×0.08=3.6045 \times 0.08 = 3.6045×0.08=3.60
Final price: 45+3.60=48.6045 + 3.60 = 48.6045+3.60=48.60
A tank is being filled with water at a constant rate. After 4 minutes, it contains 60 liters. After 10 minutes, it contains 120 liters.
Find the rate of change and write a function for the volume V(t) in terms of time t (in minutes).
Rate of change: (120 - 60) / (10 - 4) = 10 L/min
Function: V(t) = 10t + 20
You climb a staircase where each step is 2 cm higher than the last. The first step is 5 cm high. How high is the 10th step?
Use arithmetic nth term formula
a1=5, d=2d=2, n=10
Use nth term:
a10=5+(10−1)⋅2=5+18=23
Answer: 23 cm
A golfer hits a ball modeled by
h(x) = -0.01x2+0.4x+1
where x is horizontal distance in meters. How far does the ball travel before hitting the ground?
41
A builder wants to measure the angle between two walls that meet at a corner. They know the lengths of all three sides of the triangle formed. Which law should they use?
What is the Law of Cosines?
(To find an angle when all three sides are known.)
At a store, 2 protein bars and 1 drink cost $7. One protein bar and 2 drinks cost $6. What is the cost of one bar and one drink?
Bar = $2.67, Drink = $1.67
You’re tracking the temperature in a freezer. At 2 p.m., it’s -4°C. At 6 p.m., it’s -12°C.
Assume the temperature changes linearly. Write a function for temperature T(t) where t is hours after 2 p.m.
Slope: (-12 - (-4)) / (6 - 2) = -2°C/hour
Function: T(t) = -2t - 4
A ball bounces to 80% of its previous height. If the first bounce is 2 meters, what is the total vertical distance traveled after 4 bounces (up + down)?
Use geometric sum formula, considering up and down movement
First down: 22
Then up & down distances for 3 more bounces:
Up #1: 2⋅0.8=1.62⋅0.8=1.6
Down #1: 1.61.6
Up #2: 1.6⋅0.8=1.281.6⋅0.8=1.28, and so on.
We sum:
Total=2+2⋅(2⋅0.8+2⋅0.8 a la 2 +2⋅0.8 a la 3)
=2+2 (1.6+1.28+1.024)=2+2(3.904)=2+7.808=9.808=
Answer: 9.81 meters (rounded)
A basketball is shot and follows
h(x) = −0.15x2 + 1.2x + 2
A hoop is at 10 feet high. Can the ball go in? What is the maximum height?
Yes, the ball reaches 5.6ft
A construction crew needs to place a diagonal beam between two points on a building. They know two angles and one side. Which law allows them to find the length of the beam?
What is the Law of Sines?
(Use Law of Sines when given ASA or AAS situations in non-right triangles.)
You’re buying video games for $40 each, plus a $15 membership fee that gets you 10% off each game. Write a function C(n) for the total cost of n games with the membership. Then find the cost for 5 games.
Discounted price per game: 40×0.9=3640
Function: C(n)=15+36n
For 5 games: C(5)=15+36×5=15+180=195
Function: C(n)=15+36n
Cost for 5 games: $195
A taxi service charges a $3 base fee plus $2.40 per mile. You paid $27.
How many miles did you travel, and what was the linear equation used?
Equation: y = 2.4x + 3
Set 27 = 2.4x + 3 → x = 10 miles
You’re paid $1 on the first day of work, and your pay doubles each day. How much will you be paid on day 15, and what is your total pay after 15 days?
a1=1, r=2,n=15
🔹 Day 15 Pay:
a15=1⋅2 a los 14 =16,384
🔹 Total Pay:
S15=1⋅1−2a la 15/1−2 =1−32,768/−1=32,767
Answer:
Day 15 pay: $16,384
Total pay after 15 days: $32,767