Quadratic Trajectories Worksheet
Algebra I & II Applied Modeling
Quadratic Trajectories Worksheet
Solving Real-World Problems: Maximum Values & X-Intercepts
Name:
Date: Period:
Finding the Maximum (Vertex): For \( y = ax^2 + bx + c \) (\(a < 0\)), input is \( x = -\frac{b}{2a} \); maximum output is \( y = f\left(-\frac{b}{2a}\right) \).
Finding X-Intercepts (Zeros): Set \( y = 0 \) (ground/break-even) and solve via factoring or \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
1
Model Rocket Altitude
Physics Context
A high school physics team launches a model rocket from ground level. Its altitude \( h(t) \) in feet after \( t \) seconds is modeled by the function: \( h(t) = -16t^2 + 96t \)
(a) Find the time it takes to reach maximum altitude and calculate that maximum altitude.
Time: Max Altitude:
(b) Calculate the x-intercepts. Interpret what each value means in the context of the rocket's flight.
Intercepts: Airtime:
2
Charity Concert Merch Revenue
Economics Context
The student council sells commemorative concert shirts. Market research shows total revenue \( R(p) \) in dollars depends on the selling price \( p \) in dollars according to the function: \( R(p) = -10p^2 + 300p \)
(a) What shirt price \( p \) generates maximum revenue? What is the maximum revenue?
Best Price: Max Revenue:
(b) Find the x-intercepts of \( R(p) \). Why does setting the price at either intercept produce \$0 in revenue?
Intercepts: Meaning:
Page 1 of 4 • Real-World Quadratic Applications
Quadratic Trajectories Worksheet Part I: Trajectory & Flow Dynamics
3
Marine Biology: Dolphin's Leap
\( h(t) = -4t^2 + 16t \)
A dolphin leaps out of the water. Its height \( h(t) \) in feet above the ocean surface \( t \) seconds after breaching is given by \( h(t) = -4t^2 + 16t \).
(a) Find the vertex of this parabola. What was the dolphin's maximum height, and at what time was it reached?
Time: Max Height:
(b) Solve for the x-intercepts where \( h(t) = 0 \). How many total seconds was the dolphin in the air before re-entering the water?
Roots: Airtime:
4
Architectural Design: Plaza Fountain Stream
\( y = -0.5x^2 + 4x \)
An arc of water from a park fountain nozzle is modeled by \( y = -0.5x^2 + 4x \), where \( x \) is horizontal distance from nozzle (ft) and \( y \) is vertical height above pool surface (ft).
(a) How far horizontally from the nozzle does the water stream attain its maximum height? What is the peak height?
Distance: Peak:
(b) Calculate the x-intercepts. How wide across the pool basin does this water stream span from nozzle to splashdown?
Intercepts: Total Span:
5
Soccer Goal Kick Trajectory
\( h(t) = -5t^2 + 25t \)
A goalkeeper punts a soccer ball from pitch level. Its height \( h(t) \) in meters after \( t \) seconds is given by \( h(t) = -5t^2 + 25t \).
(a) Find the vertex coordinates \((t, h)\). State the maximum height attained by the ball and when it occurs.
Vertex: Max Height:
(b) Determine the x-intercepts. How long is the soccer ball airborne before hitting the ground?
Zeros: Airborne Time:
Page 2 of 4 • Real-World Quadratic Applications
Quadratic Trajectories Worksheet Part II: Multi-Phase Trajectories & Business Models
6
Fireworks Display Over Harbor
\( h(t) = -16t^2 + 64t + 80 \)
A shell is launched from a barge platform 80 ft above harbor water. Its height in feet after \( t \) seconds is \( h(t) = -16t^2 + 64t + 80 \).
(a) What is the shell's maximum height, and at what time after launch does it reach this apex?
Apex Time: Max Height:
(b) Find both mathematical x-intercepts. Which root represents when the dud shell lands in the water, and why is the other root non-viable?
Roots: Splash Time:
7
Skateboard Manufacturing: Profit Optimization
\( P(x) = -x^2 + 16x - 28 \)
A workshop calculates monthly profit \( P(x) \) (in hundreds of dollars) based on producing \( x \) custom boards: \( P(x) = -x^2 + 16x - 28 \).
(a) How many boards must be produced to achieve maximum profit? Calculate the maximum profit in dollars.
Boards: Max Profit:
(b) Solve for the break-even points (x-intercepts where \( P(x) = 0 \)). Over what production interval is the business profitable?
Break-Even: Profitable Range:
8
Stunt Motorcycle Ramp Jump
\( h(x) = -0.04x^2 + 3.2x \)
A stunt rider launches off a ramp. The motorcycle's elevation \( h(x) \) in feet at horizontal distance \( x \) feet from the ramp is \( h(x) = -0.04x^2 + 3.2x \).
(a) What is the motorcycle's peak height above the arena floor, and at what horizontal distance is it reached?
Horiz. Dist: Peak Height:
(b) Calculate the x-intercepts. How long of a landing ramp gap can the stunt rider clear from takeoff to touchdown?
Zeros: Total Clearance:
Page 3 of 4 • Real-World Quadratic Applications
Quadratic Trajectories Worksheet Part III: Elevated Starts & Synthesis
9
Acapulco Cliff Diver
\( h(t) = -16t^2 + 16t + 32 \)
A cliff diver leaps upward from a 32-foot seaside precipice. The diver's height in feet after \( t \) seconds is given by \( h(t) = -16t^2 + 16t + 32 \).
(a) When does the diver reach maximum height, and what is that maximum height above the ocean?
Time: Max Height:
(b) Solve for the x-intercepts. How many seconds after diving does the diver splash into the ocean?
Zeros: Ocean Entry:
10
Botanical Tunnel Greenhouse Arch
\( y = -0.25x^2 + 3x \)
An agricultural greenhouse cross-section is shaped as a parabolic arch. The height \( y \) in meters at distance \( x \) meters across the floor is \( y = -0.25x^2 + 3x \).
(a) Find the vertex of the arch. What is the peak ceiling clearance and where does it occur?
Center Dist: Peak Clearance:
(b) Calculate the x-intercepts. What is the total span (width) across the greenhouse foundation?
X-Intercepts: Total Width:
Mathematical Reasoning & Strategy Reflection
When reading an unfamiliar real-world word problem, how do you decide whether the question is asking you to calculate the vertex (maximum) or an x-intercept? Complete the rule:
Look for Vertex keywords when:
e.g., "highest", "peak", "maximum profit", "apex"
Look for X-Intercept keywords when:
e.g., "hits ground", "break-even", "touchdown", "total flight time"
Page 4 of 4 • Real-World Quadratic Applications
