Parabola examples with solutions

📚 Quick Study Guide

• 📈 Definition: A parabola is a symmetrical open curve formed by the intersection of a cone with a plane parallel to its side. It's defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
• ✏️ Standard Equation: The standard form equation of a parabola opening upwards or downwards is $(x - h)^2 = 4p(y - k)$, where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus.
• 🧭 Vertex Form: The vertex form highlights the vertex coordinates: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• 🔥 Focus and Directrix: For a parabola $(x - h)^2 = 4p(y - k)$, the focus is at $(h, k + p)$ and the directrix is the line $y = k - p$.
• ↔️ Parabola Opening Sideways: The standard equation is $(y - k)^2 = 4p(x - h)$, where $(h, k)$ is the vertex. If $p > 0$, it opens to the right; if $p < 0$, it opens to the left.
• 📐 Axis of Symmetry: The axis of symmetry is a line that divides the parabola into two mirror-image halves. For a parabola opening upwards or downwards, the axis of symmetry is $x = h$. For a sideways parabola, it's $y = k$.
• 🔑 Key Steps to Solve:
  • Identify the vertex $(h, k)$.
  • Determine the value of $p$.
  • Use the appropriate standard equation based on the parabola's orientation.

Practice Quiz

1. What are the coordinates of the vertex of the parabola given by the equation $y = 2(x - 3)^2 + 1$?
   1. (3, 1)
   2. (-3, 1)
   3. (3, -1)
   4. (-3, -1)
2. The focus of a parabola is at (2, 5) and its directrix is the line $y = 1$. What is the equation of the parabola?
   1. $(x - 2)^2 = 8(y - 3)$
   2. $(x + 2)^2 = 8(y + 3)$
   3. $(x - 2)^2 = 4(y - 3)$
   4. $(x + 2)^2 = 4(y + 3)$
3. What is the equation of the axis of symmetry for the parabola $y = -3(x + 1)^2 - 2$?
   1. x = 1
   2. x = -1
   3. y = -2
   4. y = 2
4. Which of the following equations represents a parabola that opens to the left?
   1. $(x - 1)^2 = 4(y + 2)$
   2. $(y - 1)^2 = 4(x + 2)$
   3. $(y - 1)^2 = -4(x + 2)$
   4. $(x - 1)^2 = -4(y + 2)$
5. What is the value of $p$ in the equation $(x - 2)^2 = 12(y + 1)$?
   1. 3
   2. -3
   3. 12
   4. 6
6. The directrix of a parabola is $x = 4$, and its vertex is at (1, 2). What is the equation of the parabola?
   1. $(y - 2)^2 = -12(x - 1)$
   2. $(y + 2)^2 = -12(x + 1)$
   3. $(y - 2)^2 = 12(x - 1)$
   4. $(y + 2)^2 = 12(x + 1)$
7. What is the focus of the parabola given by the equation $(y + 3)^2 = 8(x - 2)$?
   1. (0, -3)
   2. (4, -3)
   3. (2, -3)
   4. (2, 1)