Question
(6, -6) and also passes through the point (8, -14). Write the equation of the parabola in vertex form.(1 point) Responses y=3(x−6)2−6 y is equal to 3 times open paren x minus 6 close paren squared minus 6 y=−2(x−6)2−6 y is equal to negative 2 times open paren x minus 6 close paren squared minus 6 y=(x+6)2−6 y is equal to open paren x plus 6 close paren squared minus 6 y=−3(x+6)2−6
Answers
GPT 3.5
To find the equation of the parabola in vertex form, we will use the vertex form equation:
y = a(x - h)^2 + k
where the vertex of the parabola is (h, k).
We are given that the parabola passes through the point (6, -6), so we can substitute these values into the equation:
-6 = a(6 - h)^2 + k
We are also given that the parabola passes through the point (8, -14), so we can substitute these values into the equation:
-14 = a(8 - h)^2 + k
We now have a system of two equations with two variables (a and h). Let's solve this system of equations:
From the first equation:
-6 = a(6 - h)^2 + k
-6 = a(6^2 - 12h + h^2) + k
-6 = a(36 - 12h + h^2) + k
-6 = 36a - 12ah + ah^2 + k
From the second equation:
-14 = a(8 - h)^2 + k
-14 = a(8^2 - 16h + h^2) + k
-14 = a(64 - 16h + h^2) + k
-14 = 64a - 16ah + ah^2 + k
Now we can subtract the second equation from the first equation to eliminate k:
-6 - (-14) = (36a - 12ah + ah^2 + k) - (64a - 16ah + ah^2 + k)
-6 + 14 = 36a - 12ah + ah^2 - 64a + 16ah - ah^2
8 = -28a + 4ah
Simplifying further:
8 = -4a(7 - h)
2 = -a(7 - h)
Now we can substitute this expression for a in either equation to solve for h. Let's use the first equation:
-6 = a(6 - h)^2 + k
Substituting -a(7 - h) for a:
-6 = -a(7 - h)(6 - h)^2 + k
Using the expression we found for a earlier:
-6 = -2(7 - h)(6 - h)^2 + k
Since we are only interested in the equation of the parabola in vertex form, we do not need to solve for h and k explicitly. The equation can be written as:
y = -2(7 - h)(6 - h)^2 + k
y = a(x - h)^2 + k
where the vertex of the parabola is (h, k).
We are given that the parabola passes through the point (6, -6), so we can substitute these values into the equation:
-6 = a(6 - h)^2 + k
We are also given that the parabola passes through the point (8, -14), so we can substitute these values into the equation:
-14 = a(8 - h)^2 + k
We now have a system of two equations with two variables (a and h). Let's solve this system of equations:
From the first equation:
-6 = a(6 - h)^2 + k
-6 = a(6^2 - 12h + h^2) + k
-6 = a(36 - 12h + h^2) + k
-6 = 36a - 12ah + ah^2 + k
From the second equation:
-14 = a(8 - h)^2 + k
-14 = a(8^2 - 16h + h^2) + k
-14 = a(64 - 16h + h^2) + k
-14 = 64a - 16ah + ah^2 + k
Now we can subtract the second equation from the first equation to eliminate k:
-6 - (-14) = (36a - 12ah + ah^2 + k) - (64a - 16ah + ah^2 + k)
-6 + 14 = 36a - 12ah + ah^2 - 64a + 16ah - ah^2
8 = -28a + 4ah
Simplifying further:
8 = -4a(7 - h)
2 = -a(7 - h)
Now we can substitute this expression for a in either equation to solve for h. Let's use the first equation:
-6 = a(6 - h)^2 + k
Substituting -a(7 - h) for a:
-6 = -a(7 - h)(6 - h)^2 + k
Using the expression we found for a earlier:
-6 = -2(7 - h)(6 - h)^2 + k
Since we are only interested in the equation of the parabola in vertex form, we do not need to solve for h and k explicitly. The equation can be written as:
y = -2(7 - h)(6 - h)^2 + k
oobleck
what a load of rubbish!
If the vertex is at (6,-6) then the equation is
y = a(x-6)^2 - 6
Since the curve passes through (8,-14), we use that to get
a(8-6)^2 - 6 = -14
4a = -8
a = -2
so the equation is y = -2(x-6)^2 - 6
If the vertex is at (6,-6) then the equation is
y = a(x-6)^2 - 6
Since the curve passes through (8,-14), we use that to get
a(8-6)^2 - 6 = -14
4a = -8
a = -2
so the equation is y = -2(x-6)^2 - 6
GPT 3.5
Apologies for the error in my previous response. You are correct. The equation of the parabola in vertex form is y = -2(x-6)^2 - 6, given that the vertex is at (6, -6) and it passes through the point (8, -14). Thank you for pointing out the mistake and providing the correct equation.