This worksheet explores the diverse applications of quadratic functions in real-world scenarios. Quadratic functions, represented by the equation f(x) = ax² + bx + c (where 'a', 'b', and 'c' are constants and 'a' ≠ 0), describe parabolic curves and are surprisingly prevalent in various fields. Understanding their applications is crucial for anyone studying algebra or related disciplines.
Section 1: Understanding the Basics
Before diving into applications, let's solidify our understanding of key quadratic function concepts.
1.1 Identifying Quadratic Functions:
Determine whether the following equations represent quadratic functions. Explain your reasoning.
- a)
y = 3x² - 2x + 1 - b)
y = x³ + 4x - 7 - c)
y = 5x + 2 - d)
y = -x² + 9 - e)
y = 2(x-1)(x+3)
1.2 Key Features of a Parabola:
Sketch a sample parabola and label the following key features:
- Vertex: The highest or lowest point on the parabola.
- Axis of Symmetry: The vertical line that divides the parabola into two symmetrical halves.
- x-intercepts (roots or zeros): The points where the parabola intersects the x-axis.
- y-intercept: The point where the parabola intersects the y-axis.
Section 2: Real-World Applications
Quadratic functions model numerous real-world phenomena. Let's explore some examples.
2.1 Projectile Motion:
The trajectory of a projectile (e.g., a ball thrown into the air) can often be modeled using a quadratic function. The height (h) of the projectile at time (t) is often represented as: h(t) = -16t² + vt + h₀, where 'v' is the initial vertical velocity and 'h₀' is the initial height.
Problem 1: A ball is thrown upwards with an initial velocity of 64 ft/s from a height of 4 feet.
- a) Write the quadratic function that models the ball's height.
- b) Find the maximum height the ball reaches.
- c) When does the ball hit the ground?
2.2 Area and Optimization:
Quadratic functions are essential for solving area optimization problems.
Problem 2: A farmer has 100 feet of fencing to enclose a rectangular garden. What dimensions will maximize the area of the garden?
2.3 Revenue and Profit:
In business, quadratic functions can model revenue and profit. The relationship between price, quantity, and profit often yields a quadratic equation.
Problem 3: A company sells widgets for dollars per widget. The number of widgets sold is given by q = 100 – p. The revenue is R = pq. Find the price that maximizes the revenue.
2.4 Engineering and Physics:
Quadratic functions appear frequently in engineering and physics, describing phenomena such as the shape of a cable hanging between two supports (catenary curve, approximated by a parabola) or the path of a satellite in orbit (under specific simplifications).
Problem 4 (Challenge): A suspension bridge cable hangs in the shape of a parabola. The lowest point of the cable is 10 meters above the roadway, and the towers supporting the cable are 100 meters apart and 20 meters high. Find a quadratic function that models the shape of the cable.
Section 3: Further Exploration
Research and describe one additional real-world application of quadratic functions not covered in this worksheet.
This worksheet provides a starting point for understanding the numerous applications of quadratic functions. Further exploration will reveal their importance across various disciplines. Remember to show your work and explain your reasoning thoroughly for each problem.
