Solved Examples: Finding Vertex and Standard Form of Parabolas

📚 Quick Study Guide

• 📈 The vertex form of a parabola is given by $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• 🔍 To convert a quadratic equation from standard form ($y = ax^2 + bx + c$) to vertex form, complete the square.
• 📝 The x-coordinate of the vertex can also be found using the formula: $h = \frac{-b}{2a}$. Then, substitute $h$ into the original equation to find the y-coordinate $k$.
• 💡 Completing the Square: Add and subtract $(\frac{b}{2a})^2$ inside the parenthesis to rewrite the equation in vertex form.
• 🧮 Remember that the value of 'a' in both standard and vertex forms determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
• 📌 Knowing the vertex helps determine the maximum or minimum value of the quadratic function.

Practice Quiz

1. Question 1: What is the vertex of the parabola $y = (x - 2)^2 + 3$?
   1. (2, -3)
   2. (-2, 3)
   3. (2, 3)
   4. (-2, -3)
2. Question 2: Convert the equation $y = x^2 + 4x + 1$ to vertex form.
   1. $y = (x + 2)^2 - 3$
   2. $y = (x - 2)^2 + 3$
   3. $y = (x + 2)^2 + 3$
   4. $y = (x - 2)^2 - 3$
3. Question 3: The vertex of a parabola is at (1, -2). If the parabola also passes through the point (2, 0), what is the equation of the parabola in vertex form?
   1. $y = 2(x - 1)^2 - 2$
   2. $y = (x - 1)^2 - 2$
   3. $y = -2(x - 1)^2 - 2$
   4. $y = -2(x + 1)^2 - 2$
4. Question 4: What is the value of 'a' in the standard form equation of a parabola that opens downwards?
   1. a > 0
   2. a = 0
   3. a < 0
   4. a = 1
5. Question 5: Find the vertex of the parabola $y = 2x^2 - 8x + 5$.
   1. (2, -3)
   2. (-2, 5)
   3. (2, 5)
   4. (-2, -3)
6. Question 6: Convert the equation $y = -x^2 + 6x - 4$ to vertex form.
   1. $y = -(x - 3)^2 + 5$
   2. $y = -(x + 3)^2 - 5$
   3. $y = (x - 3)^2 + 5$
   4. $y = -(x - 3)^2 - 5$
7. Question 7: A parabola has a vertex at (-1, 4) and passes through the point (0, 2). What is the equation of the parabola in vertex form?
   1. $y = -2(x + 1)^2 + 4$
   2. $y = -2(x - 1)^2 + 4$
   3. $y = 2(x + 1)^2 + 4$
   4. $y = 2(x - 1)^2 + 4$