Real world examples of quadratic functions and parabolas in action.

📚 Quick Study Guide

• 🔢 A quadratic function is a polynomial function of degree 2, generally written as $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$.
• 📈 The graph of a quadratic function is a parabola, a U-shaped curve.
• 顶点 The vertex of a parabola is the point where the parabola changes direction. Its x-coordinate is given by $x = -\frac{b}{2a}$.
• 🧮 The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is $x = -\frac{b}{2a}$.
• 💡 The roots (or zeros) of a quadratic function are the x-values where the parabola intersects the x-axis. They can be found by solving the quadratic equation $ax^2 + bx + c = 0$ using factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• 🎯 The discriminant, $b^2 - 4ac$, determines the number of real roots: if it's positive, there are two real roots; if it's zero, there is one real root; if it's negative, there are no real roots.
• 🎢 Parabolas can open upwards (if $a > 0$) or downwards (if $a < 0$).

🧪 Practice Quiz

1. What real-world scenario can be modeled by a quadratic function?
   1. A) The motion of a car moving at a constant speed.
   2. B) The height of a ball thrown into the air.
   3. C) The growth of bacteria with unlimited resources.
   4. D) The decay of a radioactive substance.
2. The path of a projectile follows what shape?
   1. A) A straight line.
   2. B) A circle.
   3. C) A parabola.
   4. D) A hyperbola.
3. A satellite dish is designed with a parabolic cross-section. What is the primary reason for this shape?
   1. A) To minimize wind resistance.
   2. B) To maximize the surface area for signal collection.
   3. C) To focus incoming signals to a single point.
   4. D) To evenly distribute the weight of the dish.
4. The cables of a suspension bridge often form a curve that closely resembles which shape?
   1. A) A sine wave.
   2. B) A parabola.
   3. C) An exponential curve.
   4. D) A logarithmic curve.
5. Which part of a parabola represents the maximum or minimum value of a quadratic function?
   1. A) The y-intercept.
   2. B) The x-intercept.
   3. C) The vertex.
   4. D) The axis of symmetry.
6. Suppose a football is kicked, and its height (in feet) is modeled by the equation $h(t) = -16t^2 + 64t$, where $t$ is the time in seconds. What is the maximum height reached by the football?
   1. A) 16 feet
   2. B) 32 feet
   3. C) 64 feet
   4. D) 128 feet
7. A company's profit, $P$, can be modeled by $P(x) = -x^2 + 10x - 9$, where $x$ is the number of units sold. What is the break-even point (where profit is zero)?
   1. A) x = 0 and x = 10
   2. B) x = 1 and x = 9
   3. C) x = -1 and x = -9
   4. D) x = 2 and x = 8