Questions: The focus of a parabola is at (0, 1/2), and the vertex is at the point (0,0). What is the standard form of the equation for this parabola? Select one: a. y=1/2 x^2 b. y=x^2 c. y=2 x^2 d. y=4 x^2

The focus of a parabola is at (0, 1/2), and the vertex is at the point (0,0). What is the standard form of the equation for this parabola?

Select one:
a. y=1/2 x^2
b. y=x^2
c. y=2 x^2
d. y=4 x^2

Transcript text: The focus of a parabola is at $\left(0, \frac{1}{2}\right)$, and the vertex is at the point $(0,0)$. What is the standard form of the equation for this parabola? Select one: a. $y=\frac{1}{2} x^{2}$ b. $y=x^{2}$ c. $y=2 x^{2}$ d. $y=4 x^{2}$

Solution

Solution Steps

To find the standard form of the equation of a parabola given its focus and vertex, we use the fact that the vertex form of a parabola with a vertical axis of symmetry is $ y = ax^2 $. The distance from the vertex to the focus is $ \frac{1}{4a} $. Given the focus at $ \left(0, \frac{1}{2}\right) $ and the vertex at $ (0,0) $, we can determine $ a $ and thus the equation of the parabola.

Step 1: Identify the Given Information

The focus of the parabola is at the point $ \left(0, \frac{1}{2}\right) $ and the vertex is at the point $ (0,0) $.

Step 2: Calculate the Distance from Vertex to Focus

The distance from the vertex to the focus is given by: \[ \text{distance} = \frac{1}{2} - 0 = \frac{1}{2} \]

Step 3: Relate Distance to Parameter $ a $

The relationship between the distance from the vertex to the focus and the parameter $ a $ is given by: \[ \text{distance} = \frac{1}{4a} \] Substituting the distance we found: \[ \frac{1}{2} = \frac{1}{4a} \]

Step 4: Solve for $ a $

Rearranging the equation to solve for $ a $: \[ 4a = 2 \implies a = \frac{1}{2} \]

Step 5: Write the Standard Form of the Parabola

The standard form of the equation of the parabola is: \[ y = ax^2 \] Substituting the value of $ a $: \[ y = \frac{1}{2}x^2 \]