Real-world applications of quadratic functions examples

📚 Quick Study Guide

• 🎯 A quadratic function is a polynomial function of the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.
• 📈 The graph of a quadratic function is a parabola.
• Vertex Form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• 🧮 The roots (or zeros) of a quadratic function are the values of $x$ for which $f(x) = 0$. These can be found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• ✨ Real-world applications often involve maximizing or minimizing quantities, such as area, profit, or height.

🧪 Practice Quiz

1. What is the primary shape represented by a quadratic function when graphed?
   1. A. Line
   2. B. Circle
   3. C. Parabola
   4. D. Hyperbola
2. A ball is thrown upwards. Its height $h(t)$ at time $t$ is given by $h(t) = -5t^2 + 30t$. At what time does the ball reach its maximum height?
   1. A. 2 seconds
   2. B. 3 seconds
   3. C. 4 seconds
   4. D. 5 seconds
3. A farmer wants to fence a rectangular area next to a river. He has 100 meters of fencing. What is the maximum area he can enclose?
   1. A. 1250 $m^2$
   2. B. 1000 $m^2$
   3. C. 625 $m^2$
   4. D. 2500 $m^2$
4. The profit $P(x)$ from selling $x$ items is given by $P(x) = -x^2 + 100x - 1000$. How many items should be sold to maximize profit?
   1. A. 25
   2. B. 50
   3. C. 75
   4. D. 100
5. What is the vertex form of a quadratic equation useful for identifying directly?
   1. A. x-intercepts
   2. B. y-intercepts
   3. C. Vertex
   4. D. Roots
6. The path of a projectile is modeled by $y = -0.5x^2 + 5x$. At what horizontal distance $x$ does the projectile hit the ground (y=0)?
   1. A. 2.5
   2. B. 5
   3. C. 10
   4. D. 20
7. A company's cost $C(x)$ to produce $x$ units is given by $C(x) = 0.1x^2 - x + 5$. What is the minimum cost?
   1. A. 2.5
   2. B. 3
   3. C. 3.5
   4. D. 4