Test questions on converting parabola equations using completing the square

📚 Quick Study Guide

• 🔍 The standard form of a parabola equation is $y = ax^2 + bx + c$.
• 💡 Completing the square helps convert this to vertex form: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.
• 📝 To complete the square, take half of the coefficient of $x$ (which is $b$), square it, and add/subtract it inside the equation.
• ➗ The general steps are: 1. Factor out 'a' from the $x^2$ and $x$ terms. 2. Complete the square inside the parentheses. 3. Distribute 'a' back. 4. Simplify to get the vertex form.
• ➕ Remember that $h = -b/(2a)$ and $k$ can be found by plugging $h$ back into the original equation.
• 🤓 Vertex form directly reveals the vertex of the parabola, making it easier to graph and analyze.

Practice Quiz

1. What is the first step in converting $y = x^2 + 6x + 5$ to vertex form by completing the square?
   1. Add and subtract 9 inside the equation.
   2. Factor out the coefficient of $x^2$.
   3. Move the constant term to the other side of the equation.
   4. Divide all terms by 2.


2. Convert $y = x^2 - 4x + 7$ 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. What value should be added and subtracted to complete the square for $y = x^2 + 8x - 3$?
   1. 4
   2. 8
   3. 16
   4. 64


4. Convert $y = 2x^2 + 8x + 5$ to vertex form.
   1. $y = 2(x + 2)^2 - 3$
   2. $y = 2(x - 2)^2 - 3$
   3. $y = 2(x + 2)^2 + 3$
   4. $y = 2(x - 2)^2 + 3$


5. Which of the following is the vertex of the parabola $y = (x - 3)^2 + 4$?
   1. (3, 4)
   2. (-3, 4)
   3. (3, -4)
   4. (-3, -4)


6. Convert $y = -x^2 + 2x + 3$ to vertex form.
   1. $y = -(x - 1)^2 + 4$
   2. $y = -(x + 1)^2 + 4$
   3. $y = -(x - 1)^2 - 4$
   4. $y = -(x + 1)^2 - 4$


7. What is the vertex of the parabola $y = -2(x + 1)^2 - 5$?
   1. (-1, -5)
   2. (1, -5)
   3. (-1, 5)
   4. (1, 5)