Question

The axis of a parabola is $x=0$. If the vertex is at a distance 3 from the origin above the x-axis, the vertex of the parabola is at ________.

• A. $(3, 0)$
• B. $(-3, 0)$
• C. $(3, -3)$
• D. $(3, 3)$
• E. $(0, 3)$

Hint:

Axis of symmetry always contains the vertex.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
This question asks for the coordinates of the vertex of a parabola based on a description of its location and orientation. We need to interpret the geometric description into Cartesian coordinates.
Step 2: Detailed Explanation
1. Interpret "The axis of a parabola is x = 0".
The line \(x=0\) is the y-axis. The axis of symmetry of the parabola is the y-axis. The vertex of a parabola always lies on its axis of symmetry. Therefore, the x-coordinate of the vertex must be 0.
2. Interpret "the vertex is at a distance 3 from the origin".
This means the vertex is on a circle of radius 3 centered at the origin. Its coordinates \((x, y)\) must satisfy \(x^2 + y^2 = 3^2 = 9\).
3. Interpret "above the x-axis".
"Above the x-axis" means the y-coordinate of the point must be positive (\(y>0\)).
4. Combine the information.
- From point 1, the vertex has the form \((0, y)\).
- From point 3, \(y\) must be positive.
- Substitute the form \((0, y)\) into the distance condition from point 2:
\[ 0^2 + y^2 = 9 \] \[ y^2 = 9 \] \[ y = \pm 3 \] - Since the vertex is above the x-axis, we must choose the positive value for y, so \(y=3\).
Therefore, the coordinates of the vertex are (0, 3).
Step 4: Final Answer
The vertex of the parabola is at (0, 3).