I'm currently working with parabolas, and I'm not sure if I am doing this specific problem correctly..
12. x= y^2-6y+6
y= -b/2a = 6/2(1)= 6/2= 3
x= (3)^2+6(3)+6
x= 33
y=3
(33,3) <--- what I graphed
Ya gotta pay attention to details.
That's -6y not +6y, so
x = 3^2 - 6(3)+3 = -3
note that x = (y-3)^2 - 3
So the vertex is at (-3,3)
Oh, I noticed that. I just thought it would come out to be -(-6y) that would then make it +6y.
To solve this problem correctly, let's go through the steps one by one.
Step 1: Start by rearranging the equation into the standard form of a parabola, which is y = ax^2 + bx + c. In this case, you have x = y^2 - 6y + 6. To rearrange it, subtract "x" from both sides:
y^2 - 6y + 6 - x = 0
Step 2: Now, let's determine the value of "y" using the quadratic formula. The quadratic formula is given by:
y = (-b ± √(b^2 - 4ac)) / (2a)
In this case, "a" is 1, "b" is -6, and "c" is (6 - x). Substituting these values into the quadratic formula, we get:
y = (-(-6) ± √((-6)^2 - 4(1)(6 - x))) / (2(1))
y = (6 ± √(36 - 24 + 4x)) / 2
y = (6 ± √(12 + 4x)) / 2
y = (6 ± 2√(3 + x)) / 2
y = 3 ± √(3 + x)
Step 3: Now, let's find the corresponding values of "x" for each value of "y". Plug in the value of "y" (3 ± √(3 + x)) into the original equation:
x = (3 ± √(3 + x))^2 - 6(3 ± √(3 + x)) + 6
x = (3 ± √(3 + x))^2 - 18 ± 6√(3 + x) + 6
Simplifying further:
x = 9 ± 6√(3 + x) + (3 + x) - 18 ± 6√(3 + x) + 6
x = 9 ± 6√(3 + x) + 3 + x - 18 ± 6√(3 + x) + 6
x = 18 ± 6√(3 + x) - 9 + x - 18 ± 6√(3 + x) + 6
x = 15 + 2x ± 12√(3 + x)
Simplifying further:
2x ± 12√(3 + x) = -15
From here, you can solve for "x" by considering the two cases separately, one with the plus sign and the other with the minus sign.
Case 1: 2x + 12√(3 + x) = -15
Solving this equation will give you one value of "x".
Case 2: 2x - 12√(3 + x) = -15
Solving this equation will give you the other value of "x".
Once you have the values of "x," substitute them back into the equation y = 3 ± √(3 + x) to find the corresponding values of "y."
Plotting these points (x, y) on a graph will help you visualize the parabola.