What is the focus of the parabola 
y
=
4
x
2
y=4x 
2
 ?

To find the focus of the parabola given by the equation \( y = 4x^2 \), we start by recalling the standard form of a parabola's equation that opens upwards or downwards:

\[
y = \frac{1}{4p}x^2
\]

Here, \(p\) is the distance from the vertex to the focus. 

Comparing \( y = 4x^2 \) with the standard form, we have:

\[
\frac{1}{4p} = 4
\]

Now, solve for \(p\):

\[
4p = \frac{1}{4}
\]

\[
p = \frac{1}{16}
\]

Since the parabola opens upwards, the focus is located at a distance of \( p \) units above the vertex. Therefore, the coordinates of the focus are:

\[
\left(0, \frac{1}{16}\right)
\]

So, the focus of the parabola \( y = 4x^2 \) is at \(\left(0, \frac{1}{16}\right)\).

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How do you find the directrix of the parabola \(y = 4x^2\)?
2. What is the equation of a parabola with a focus at \((0, -\frac{1}{16})\)?
3. How does the value of \(p\) affect the shape of the parabola?
4. What would be the focus if the equation were \(y = \frac{1}{2}x^2\)?
5. How would the equation change if the parabola opened to the right?

**Tip:** The focus of a parabola is always located inside the curve, and it lies along the axis of symmetry.

Learn how to find the focus of a parabola given by the equation y = 4x^2. This problem covers the vertex form of a parabola, the calculation of p, and the location of the focus relative to the vertex. Suitable for high school students studying parabolas.

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What is the focus of the parabola y

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