Question
What is the equation of a parabola with vertex at origin, opening upward and focal length 5 inches?
Answers
GPT-4o mini
The equation of a parabola that opens upward with its vertex at the origin (0, 0) can be expressed in the standard form:
\[
y = \frac{1}{4p} x^2
\]
where \( p \) is the distance from the vertex to the focus. Since the focal length is given as 5 inches, we have \( p = 5 \).
Substituting \( p = 5 \) into the equation gives:
\[
y = \frac{1}{4 \cdot 5} x^2
\]
This simplifies to:
\[
y = \frac{1}{20} x^2
\]
Thus, the equation of the parabola is:
\[
y = \frac{1}{20} x^2
\]
This parabola opens upward with its vertex at the origin and a focal length of 5 inches.
\[
y = \frac{1}{4p} x^2
\]
where \( p \) is the distance from the vertex to the focus. Since the focal length is given as 5 inches, we have \( p = 5 \).
Substituting \( p = 5 \) into the equation gives:
\[
y = \frac{1}{4 \cdot 5} x^2
\]
This simplifies to:
\[
y = \frac{1}{20} x^2
\]
Thus, the equation of the parabola is:
\[
y = \frac{1}{20} x^2
\]
This parabola opens upward with its vertex at the origin and a focal length of 5 inches.