To solve this problem, we need to understand the properties of a parabola given by the equation $ y^2 = 4px $.
Step-by-Step Solution:
1.
Identify the form of the equation:
The given equation is $ y^2 = 12x $. This matches the standard form $ y^2 = 4px $, where $ p $ is the distance from the vertex to the focus and also from the vertex to the directrix.
2.
Determine the value of \( p \):
From the equation $ y^2 = 12x $, we can see that $ 4p = 12 $. Solving for \( p \):
\[
p = \frac{12}{4} = 3
\]
3.
Find the vertex:
For the equation $ y^2 = 4px $, the vertex is always at the origin $(0, 0)$.
4.
Locate the focus:
Since $ p = 3 $, the focus is located at $(p, 0)$, which is $(3, 0)$.
5.
Determine the directrix:
The directrix is a vertical line located at $ x = -p $. Therefore, the directrix is at $ x = -3 $.
6.
Match with the graphs:
We need to find the graph that shows:
- A parabola opening to the right (since $ y^2 = 12x $).
- Vertex at $(0, 0)$.
- Focus at $(3, 0)$.
- Directrix at $ x = -3 $.
By examining the provided graphs, we look for one that fits all these criteria.
Final Answer:
The correct graph is the one that shows a parabola opening to the right with the vertex at $(0, 0)$, the focus at $(3, 0)$, and the directrix at $ x = -3 $.
Based on the description, the correct choice is the first option in both columns (assuming they are identical as per the image description).
Final Answer: First option in both columns.
Parent Tip: Review the logic above to help your child master the concept of analytic geometry worksheet.
