Interactive quiz: Identify standard form of parabolas after conversion

📚 Quick Study Guide

• 🔍 The standard form of a parabola is given by either $(y - k)^2 = 4p(x - h)$ or $(x - h)^2 = 4p(y - k)$, where $(h, k)$ is the vertex.
• ✏️ To convert a general quadratic equation to standard form, complete the square. This involves grouping the terms with the same variable together.
• 🧮 Identify whether the parabola opens horizontally or vertically based on which variable is squared. If $x$ is squared, it's vertical; if $y$ is squared, it's horizontal.
• 📐 Remember to factor out any leading coefficients before completing the square. This ensures you're adding the correct value to both sides of the equation.
• 💡 The value of $p$ determines the width and direction of the parabola. It's the distance from the vertex to the focus and from the vertex to the directrix.

Practice Quiz

1. What is the standard form of the parabola given by $y^2 + 4y - 8x + 20 = 0$?
1. (A) $(y + 2)^2 = 8(x - 2)$
2. (B) $(y - 2)^2 = 8(x + 2)$
3. (C) $(x - 2)^2 = 8(y + 2)$
4. (D) $(x + 2)^2 = 8(y - 2)$
2. Convert the equation $x^2 - 6x - 4y + 5 = 0$ to standard form.
1. (A) $(x - 3)^2 = 4(y + 1)$
2. (B) $(x + 3)^2 = 4(y - 1)$
3. (C) $(y - 3)^2 = 4(x + 1)$
4. (D) $(y + 3)^2 = 4(x - 1)$
3. The equation $y^2 - 2y + 12x - 35 = 0$ in standard form is:
1. (A) $(y - 1)^2 = -12(x - 3)$
2. (B) $(y + 1)^2 = -12(x + 3)$
3. (C) $(x - 1)^2 = -12(y - 3)$
4. (D) $(x + 1)^2 = -12(y + 3)$
4. What is the standard form of the parabola given by $x^2 + 8x + 8y = 0$?
1. (A) $(x + 4)^2 = -8(y - 2)$
2. (B) $(x - 4)^2 = -8(y + 2)$
3. (C) $(y + 4)^2 = -8(x - 2)$
4. (D) $(y - 4)^2 = -8(x + 2)$
5. Convert $2y^2 + 8y - 4x + 10 = 0$ to standard form.
1. (A) $(y + 2)^2 = 2(x - 1)$
2. (B) $(y - 2)^2 = 2(x + 1)$
3. (C) $(x + 2)^2 = 2(y - 1)$
4. (D) $(x - 2)^2 = 2(y + 1)$
6. The standard form of the parabola $3x^2 - 12x - 6y + 18 = 0$ is:
1. (A) $(x - 2)^2 = 2(y - 1)$
2. (B) $(x + 2)^2 = 2(y + 1)$
3. (C) $(y - 2)^2 = 2(x - 1)$
4. (D) $(y + 2)^2 = 2(x + 1)$
7. What is the standard form of the equation $x = \frac{1}{4}y^2 + y - 1$?
1. (A) $(y + 2)^2 = 4(x + 2)$
2. (B) $(y - 2)^2 = 4(x - 2)$
3. (C) $(x + 2)^2 = 4(y + 2)$
4. (D) $(x - 2)^2 = 4(y - 2)$