Question
Write a function in vertex form if the vertex is (2,4), and a non-vertex point is (0,16).
Answers
GPT-4o mini
To write a quadratic function in vertex form, we start with the general vertex form formula:
\[
f(x) = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the quadratic function. Given that the vertex is \((2, 4)\), we can substitute \(h = 2\) and \(k = 4\):
\[
f(x) = a(x - 2)^2 + 4
\]
Next, we need to find the value of \(a\). We can use the non-vertex point \((0, 16)\) to do this. Since the point \((0, 16)\) lies on the curve of the function, we can substitute \(x = 0\) and \(f(x) = 16\) into the equation:
\[
16 = a(0 - 2)^2 + 4
\]
Simplifying this gives:
\[
16 = a(4) + 4
\]
\[
16 - 4 = 4a
\]
\[
12 = 4a
\]
\[
a = 3
\]
Now that we have \(a\), we can substitute it back into the vertex form equation:
\[
f(x) = 3(x - 2)^2 + 4
\]
Thus, the function in vertex form is:
\[
f(x) = 3(x - 2)^2 + 4
\]
\[
f(x) = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the quadratic function. Given that the vertex is \((2, 4)\), we can substitute \(h = 2\) and \(k = 4\):
\[
f(x) = a(x - 2)^2 + 4
\]
Next, we need to find the value of \(a\). We can use the non-vertex point \((0, 16)\) to do this. Since the point \((0, 16)\) lies on the curve of the function, we can substitute \(x = 0\) and \(f(x) = 16\) into the equation:
\[
16 = a(0 - 2)^2 + 4
\]
Simplifying this gives:
\[
16 = a(4) + 4
\]
\[
16 - 4 = 4a
\]
\[
12 = 4a
\]
\[
a = 3
\]
Now that we have \(a\), we can substitute it back into the vertex form equation:
\[
f(x) = 3(x - 2)^2 + 4
\]
Thus, the function in vertex form is:
\[
f(x) = 3(x - 2)^2 + 4
\]