The question was:
Suppose a parabola has vertex (5, –3) and also passes through the point (6, 1). Write the equation of the parabola in vertex form.
The Bot's response was y = (x-5)^2 - 3
Clearly this is wrong, since (6-5)^2 - 3 = -2, not 1.
The correct solution starts by noting that y = a(x-5)^2 - 3
Then we can solve for a by proceeding with
a(6-5)^2 - 3 = 1
a-3 = 1
a = 4
That makes the real solution y = 4(x-5)^2 - 3
Apologies for the incorrect response earlier. You are correct, the correct equation in vertex form is y = 4(x-5)^2 - 3. Thank you for catching that mistake.
You are correct, my previous response was indeed incorrect. Thank you for pointing it out. The correct solution is as follows:
To derive the equation of the parabola in vertex form, we start with the general equation of a parabola in vertex form: y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.
Given that the vertex is (5, -3), we have h = 5 and k = -3. Thus, the equation becomes y = a(x-5)^2 - 3.
Since the parabola also passes through the point (6, 1), we can substitute these coordinates into the equation to find the value of a.
1 = a(6-5)^2 - 3
1 = a(1)^2 - 3
1 = a - 3
a = 1 + 3
a = 4
Therefore, the equation of the parabola in vertex form with the given vertex and point is:
y = 4(x-5)^2 - 3
I apologize for the incorrect response earlier. Let's go through the correct solution step by step.
To write the equation of a parabola in vertex form, we use the general equation y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola.
In this case, we are given that the vertex is (5, -3). So, we start with the equation y = a(x-5)^2 + k.
We also know that the parabola passes through the point (6, 1). We can substitute these coordinates into the equation to find the value of 'a'.
Substituting (6, 1) into the equation, we get 1 = a(6-5)^2 - 3.
Simplifying the equation, we have 1 = a(1)^2 - 3.
This further simplifies to 1 = a - 3.
To isolate 'a', we add 3 to both sides of the equation, giving us a = 4.
Now that we have the value of 'a', we can substitute it back into the equation y = a(x-5)^2 - 3.
Replacing 'a' with 4, we get y = 4(x-5)^2 - 3, which is the correct equation of the parabola in vertex form.
I apologize for the confusion caused by the earlier incorrect response. Thank you for bringing it to my attention.