Pre-Calculus Examples: Finding Parabola Equations from Various Conditions

📚 Quick Study Guide

• 🔍 The standard form of a parabola equation is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex.
• 📈 If the parabola opens upwards, $a > 0$; if it opens downwards, $a < 0$.
• 📍 Given the vertex $(h, k)$ and another point $(x, y)$ on the parabola, substitute these values into the standard form to find $a$.
• ↔️ For a horizontal parabola, the equation is $x = a(y-k)^2 + h$.
• 🎯 When given three points, substitute each point into the general form $y = ax^2 + bx + c$ to create a system of three equations. Solve this system to find $a$, $b$, and $c$.
• 💡 The focus of a parabola is at $(h, k + \frac{1}{4a})$ for a vertical parabola and $(h + \frac{1}{4a}, k)$ for a horizontal parabola. The directrix is $y = k - \frac{1}{4a}$ and $x = h - \frac{1}{4a}$ respectively.
• 📝 Remember to square binomials correctly: $(x - h)^2 = x^2 - 2hx + h^2$.

Practice Quiz

1. What is the vertex form of a parabola?
   1. $y = ax + b$
   2. $y = a(x-h)^2 + k$
   3. $y = ax^2 + bx + c$
   4. $x = ay^2 + by + c$
2. A parabola has a vertex at (2, 3) and passes through the point (4, 5). What is the value of 'a' in the equation $y = a(x-2)^2 + 3$?
   1. $a = 0.5$
   2. $a = 1$
   3. $a = -0.5$
   4. $a = 2$
3. Which direction does the parabola $y = -2(x+1)^2 - 4$ open?
   1. Upwards
   2. Downwards
   3. Right
   4. Left
4. What is the equation of a parabola with vertex (0, 0) and focus (0, 2)?
   1. $y = \frac{1}{8}x^2$
   2. $y = 8x^2$
   3. $x = \frac{1}{8}y^2$
   4. $x = 8y^2$
5. A parabola passes through the points (0, 1), (1, 2), and (2, 5). What is the equation of the parabola in the form $y = ax^2 + bx + c$?
   1. $y = x^2 + 1$
   2. $y = x + 1$
   3. $y = x^2 + x + 1$
   4. $y = 2x^2 - x + 1$
6. What is the directrix of the parabola $y = \frac{1}{4}(x-1)^2 + 2$?
   1. $y = 1$
   2. $y = 2$
   3. $y = 3$
   4. $y = 0$
7. What is the vertex of the parabola defined by the equation $x = 2(y-3)^2 + 1$?
   1. (1, 3)
   2. (3, 1)
   3. (-1, -3)
   4. (-3, -1)