To find the equation of a parabola with a vertex at the origin and passing through the point (1, 6), follow these steps:
1. Understand the Standard Form of a Parabola:
The standard form of a parabola with the vertex at the origin is given by:
[tex]\[
y = ax^2 + bx + c
\][/tex]
Since the vertex is at the origin (0, 0), this simplifies further:
[tex]\[
y = ax^2 + bx
\][/tex]
Because the vertex is at (0, 0), both [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are zero:
[tex]\[
y = ax^2
\][/tex]
2. Substitute the Given Point into the Equation:
You know the parabola passes through the point (1, 6). Substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 6\)[/tex] into the simplified equation [tex]\(y = ax^2\)[/tex]:
[tex]\[
6 = a(1)^2
\][/tex]
Simplifying this:
[tex]\[
6 = a
\][/tex]
3. Plug [tex]\(a\)[/tex] Back into the Standard Form:
Now that you have [tex]\(a = 6\)[/tex], substitute this value back into the equation [tex]\(y = ax^2\)[/tex]:
[tex]\[
y = 6x^2
\][/tex]
4. Determine the Coefficients:
From the equation above, we see that:
[tex]\[
a = 6, \quad b = 0, \quad c = 0
\][/tex]
Therefore, the equation of the parabola in standard form is:
[tex]\[
y = 6x^2 + 0x + 0
\][/tex]
So to write it neatly, the standard form of the equation is:
[tex]\[
y = 6x^2
\][/tex]
This is the equation of the parabola whose vertex is at the origin and which passes through the point (1, 6).
