To write the equation of a parabola in vertex form, we use the formula:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((1, 1)\), so we have:
\[ y = a(x - 1)^2 + 1 \]
Next, we need to find the value of \(a\). We can use the given y-intercept \((0, 2)\). Plugging in these coordinates into the equation, we have \(x = 0\) and \(y = 2\):
\[ 2 = a(0 - 1)^2 + 1 \]
This simplifies to:
\[ 2 = a(1) + 1 \]
\[ 2 = a + 1 \]
Now, solve for \(a\):
\[ a = 2 - 1 \] \[ a = 1 \]
Therefore, we can substitute \(a\) back into the vertex form equation:
\[ y = 1(x - 1)^2 + 1 \]
This can be simplified to:
\[ y = (x - 1)^2 + 1 \]
Thus, the equation of the parabola in vertex form is:
\[ \boxed{y = (x - 1)^2 + 1} \]
1 answer