Question

как нужно сдвинуть вершину параболы у=-х2-6х-6,чтобы получить параболу у=-х2+4х-5?

Answer 1

$y=- x^{2} -6x-6=- x^{2} -6x-9+3=-(x+3)^2+3$
Вершина параболы в точке  $(-3;3)$
$y=- x^{2} +4x-5=- x^{2} +4x-4-1=-(x-2)^2-1$
Вершина параболы в точке  $(2;-1)$
Нужно сдвинуть вершину параболы из точки $(-3;3)$ в точку  $(2;-1)$,
т е на 5 единиц вправо и на 4 единицы вниз

Answer 2

Adjusting the Parabola to Match the Given Equation

To shift the vertex of the parabola from the original equation y = -x^2 - 6x - 6 to the new equation y = -x^2 + 4x - 5, we need to complete the square to find the new vertex.

The general form of a quadratic equation is y = ax^2 + bx + c, and the vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Completing the Square

To transform the equation y = -x^2 - 6x - 6 into vertex form, we complete the square by following these steps:

1. Factor out the coefficient of x^2: y = -(x^2 + 6x) - 6 2. Complete the square by adding and subtracting the square of half the coefficient of x: y = -(x^2 + 6x + 9 - 9) - 6 y = -(x^2 + 6x + 9) + 9 - 6 y = -(x + 3)^2 + 3 Now, we can see that the vertex of the original parabola is at (-3, 3).

Shifting the Vertex

To shift the vertex to match the new equation y = -x^2 + 4x - 5, we need to move the vertex 7 units to the right and 2 units down.

The new vertex will be at the point (4, 1).

Therefore, to shift the vertex of the original parabola to match the new equation, we need to move it to the point (4, 1).

This can be achieved by replacing x with (x - 4) in the vertex form of the original equation: y = -(x - 4 + 3)^2 + 3 - 2 y = -(x - 1)^2 + 1

So, to shift the vertex of the parabola y = -x^2 - 6x - 6 to match the equation y = -x^2 + 4x - 5, we need to adjust the equation to y = -(x - 1)^2 + 1.

This adjustment shifts the vertex from (-3, 3) to (4, 1), aligning it with the new equation.