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