Set to 0.
X^2 - 6X + 12 = 0
X^2 - 6X = -12
halve the linear term (-6), square it and add it to both sides
X^2 - 6X + 9 = -12 + 9
gather terms right side, factor terms left side
(X - 3)^2 = -3
(X - 3)^2 + 3 = 0
(3,3) are the vector coordinates
"y = 2x2 - 12x + 6" is a quadratic equation which describes a parabola whose vertex occurs at the point (3, -12) and which has a range of -12 → ∞. It intercepts the x-axis at the points (3 - √6, 0) and (3 + √6, 0).
4
The given equation (x^2 = 12y) represents a parabola that opens upwards. The length of the focal width (the distance between the two points on the parabola where it intersects a line parallel to the directrix) is equal to (4p), where (p) is the distance from the vertex to the focus. For this parabola, (4p = 12), so the focal width is (12). Therefore, (12) is the length of the focal width.
To find the equation of the parabola, we can use the vertex form, which is (y = a(x - h)^2 + k), where ((h, k)) is the vertex. Substituting the vertex ((-2, -20)), the equation becomes (y = a(x + 2)^2 - 20). Using the intercept ((0, -12)) to find (a), we substitute (x = 0) and (y = -12), resulting in (-12 = a(0 + 2)^2 - 20). Solving for (a) gives (a = 2), leading to the final equation (y = 2(x + 2)^2 - 20).
12 plus 9 equals 21.
"y = 2x2 - 12x + 6" is a quadratic equation which describes a parabola whose vertex occurs at the point (3, -12) and which has a range of -12 → ∞. It intercepts the x-axis at the points (3 - √6, 0) and (3 + √6, 0).
4
7
Y=3x^2 and this is in standard form. The vertex form of a prabola is y= a(x-h)2+k The vertex is at (0,0) so we have y=a(x)^2 it goes throug (2,12) so 12=a(2^2)=4a and a=3. Now the parabola is y=3x^2. Check this: It has vertex at (0,0) and the point (2,12) is on the parabola since 12=3x2^2
The given equation (x^2 = 12y) represents a parabola that opens upwards. The length of the focal width (the distance between the two points on the parabola where it intersects a line parallel to the directrix) is equal to (4p), where (p) is the distance from the vertex to the focus. For this parabola, (4p = 12), so the focal width is (12). Therefore, (12) is the length of the focal width.
it equals 72. just do 12 X 6
To find the equation of the parabola, we can use the vertex form, which is (y = a(x - h)^2 + k), where ((h, k)) is the vertex. Substituting the vertex ((-2, -20)), the equation becomes (y = a(x + 2)^2 - 20). Using the intercept ((0, -12)) to find (a), we substitute (x = 0) and (y = -12), resulting in (-12 = a(0 + 2)^2 - 20). Solving for (a) gives (a = 2), leading to the final equation (y = 2(x + 2)^2 - 20).
it equals 70
You can find the x-coordinate of it's vertex by taking it's derivative and solving for zero: y = -3x2 + 12x - 5 y' = -6x + 12 0 = -6x + 12 6x = 12 x = 2 Now that we have it's x coordinate, we can plug it back into the original equation to find it's y coordinate: y = -3x2 + 12x - 5 y = -3(2)2 + 12(2) + 5 y = -12 + 24 + 5 y = 17 So the vertex of the parabola y = -3x2 + 12x - 5 occurs at the point (2, 17).
12 plus 9 equals 21.
12+12
The curve turns at a minimum: (2.5, -12)