📚 Topic Summary
Converting a parabola from general form to standard form allows us to easily identify the vertex and axis of symmetry. The general form of a parabola is given by $y = ax^2 + bx + c$, while the standard form (also known as vertex form) is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The key to converting from general to standard form is completing the square. This involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored into a squared term. Remember to balance the equation by adding or subtracting the same value on both sides.
By converting to standard form, you can quickly graph the parabola and understand its key features. Understanding the vertex and the direction the parabola opens is crucial for many applications in mathematics and physics.
🧮 Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Vertex |
A. The line that divides the parabola into two symmetrical halves. |
| 2. Standard Form |
B. The point where the parabola changes direction. |
| 3. General Form |
C. $y = a(x - h)^2 + k$ |
| 4. Axis of Symmetry |
D. The process of adding a constant to a quadratic expression to form a perfect square trinomial. |
| 5. Completing the Square |
E. $y = ax^2 + bx + c$ |
✍️ Part B: Fill in the Blanks
Fill in the missing words in the following paragraph:
The general form of a parabola is $y = ax^2 + bx + c$. To convert this to _______ form, we need to complete the _______. The standard form is given by $y = a(x - h)^2 + k$, where $(h, k)$ represents the _______ of the parabola. The value of 'a' determines whether the parabola opens _______ or _______.
🤔 Part C: Critical Thinking
Explain why converting a quadratic equation to standard form is useful. Give at least two reasons.