Alright, to find the equation of a parabola given its vertex and another point it passes through, we need to follow a few steps systematically. Here we go:
### Step-by-Step Solution
1. Understanding Vertex Form:
The vertex form of a parabola is given by:
\(\[
y = a(x - h)^2 + k
\]\)
where \(\((h, k)\)\) is the vertex of the parabola.
2. Identify Given Information:
- The vertex of the parabola is given as \(\((0, 2)\)\).
- The parabola passes through the point \(\((4, 6)\)\).
3. Substitute the Vertex:
Substitute \(\((h, k) = (0, 2)\)\) into the vertex form equation:
\(\[
y = a(x - 0)^2 + 2
\]\)
Simplifying, we get:
\(\[
y = ax^2 + 2
\]\)
4. Use the Given Point:
Substitute the point \(\((x, y) = (4, 6)\)\) into the simplified equation to find the value of \(\(a\)\):
\(\[
6 = a(4)^2 + 2
\]\)
Simplifying inside the parenthesis:
\(\[
6 = 16a + 2
\]\)
5. Solve for \(\(a\)\):
Isolate \(\(a\)\) by solving the equation:
\(\[
6 - 2 = 16a
\]\)
\(\[
4 = 16a
\]\)
\(\[
a = \frac{4}{16}
\]\)
Simplifying the fraction:
\(\[
a = \frac{1}{4}
\]\)
6. Write the Final Equation:
Substitute \(\(a = \frac{1}{4}\)\) back into the vertex form equation:
\(\[
y = \frac{1}{4}x^2 + 2
\]\)
### Conclusion
The equation of the parabola in vertex form, simplified, is:
\(\[
y = \frac{1}{4}x^2 + 2
\]\)
This completes our process of finding the equation of the parabola that has its vertex at \(\((0, 2)\)\) and passes through the point \(\((4, 6)\)\).
