Question
Using vertex form, write the equation of the parabola with a vertex at (1,1)
and a y-intercept at (0,2)
.(1 point)
The equation in vertex form is
.
and a y-intercept at (0,2)
.(1 point)
The equation in vertex form is
.
Answers
GPT-4o mini
The vertex form of a parabola is given by the equation:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
Given that the vertex is at \((1, 1)\), we can substitute \(h = 1\) and \(k = 1\) into the equation:
\[
y = a(x - 1)^2 + 1
\]
Next, we need to determine the value of \(a\). We know that the parabola has a y-intercept at \((0, 2)\). We can substitute \(x = 0\) and \(y = 2\) into the equation to find \(a\):
\[
2 = a(0 - 1)^2 + 1
\]
\[
2 = a(1) + 1
\]
\[
2 = a + 1
\]
\[
a = 1
\]
Now we can substitute \(a\) back into the vertex form equation:
\[
y = 1(x - 1)^2 + 1
\]
Thus, the equation of the parabola in vertex form is:
\[
y = (x - 1)^2 + 1
\]
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
Given that the vertex is at \((1, 1)\), we can substitute \(h = 1\) and \(k = 1\) into the equation:
\[
y = a(x - 1)^2 + 1
\]
Next, we need to determine the value of \(a\). We know that the parabola has a y-intercept at \((0, 2)\). We can substitute \(x = 0\) and \(y = 2\) into the equation to find \(a\):
\[
2 = a(0 - 1)^2 + 1
\]
\[
2 = a(1) + 1
\]
\[
2 = a + 1
\]
\[
a = 1
\]
Now we can substitute \(a\) back into the vertex form equation:
\[
y = 1(x - 1)^2 + 1
\]
Thus, the equation of the parabola in vertex form is:
\[
y = (x - 1)^2 + 1
\]