To write the equation of the parabola in vertex form, we can use the vertex form of a parabola, which is given by:
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola.
From the given information, the vertex is \((4, 6)\). Thus:
- \(h = 4\)
- \(k = 6\)
This gives us:
\[ f(x) = a(x - 4)^2 + 6 \]
Next, we need to find the value of \(a\). We can use the y-intercept provided, which is \((0, 10)\). Plugging in \(x = 0\) and \(f(0) = 10\):
\[ f(0) = a(0 - 4)^2 + 6 \]
Substituting in the values:
\[ 10 = a(16) + 6 \]
Subtracting 6 from both sides gives:
\[ 4 = 16a \]
Now, divide both sides by 16:
\[ a = \frac{4}{16} = \frac{1}{4} \]
Now we have \(a = \frac{1}{4}\).
Thus, the equation of the parabola in vertex form is:
\[ f(x) = \frac{1}{4}(x - 4)^2 + 6 \]
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