Question
Write a function in vertex form if the vertex is (-2,-4), and a non-vertex point is (-3,1).
Answers
GPT-4o mini
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 \((-2, -4)\). Therefore, we can substitute \(h = -2\) and \(k = -4\):
\[
y = a(x + 2)^2 - 4
\]
Next, we need to determine the value of \(a\) using the non-vertex point \((-3, 1)\). We can substitute \(x = -3\) and \(y = 1\) into the equation to find \(a\):
\[
1 = a(-3 + 2)^2 - 4
\]
Now, simplify the equation:
\[
1 = a(-1)^2 - 4
\]
\[
1 = a(1) - 4
\]
\[
1 = a - 4
\]
\[
a = 1 + 4
\]
\[
a = 5
\]
Now that we have found \(a\), we can write the final equation in vertex form:
\[
y = 5(x + 2)^2 - 4
\]
So the equation of the parabola in vertex form is:
\[
y = 5(x + 2)^2 - 4
\]
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((-2, -4)\). Therefore, we can substitute \(h = -2\) and \(k = -4\):
\[
y = a(x + 2)^2 - 4
\]
Next, we need to determine the value of \(a\) using the non-vertex point \((-3, 1)\). We can substitute \(x = -3\) and \(y = 1\) into the equation to find \(a\):
\[
1 = a(-3 + 2)^2 - 4
\]
Now, simplify the equation:
\[
1 = a(-1)^2 - 4
\]
\[
1 = a(1) - 4
\]
\[
1 = a - 4
\]
\[
a = 1 + 4
\]
\[
a = 5
\]
Now that we have found \(a\), we can write the final equation in vertex form:
\[
y = 5(x + 2)^2 - 4
\]
So the equation of the parabola in vertex form is:
\[
y = 5(x + 2)^2 - 4
\]