Question

Скласти рівняння параболи, якщо її вершина знаходиться в початку координат, а вона симетрична заданій осі та проходить через задану точку. OX та А(-8;4)

Answer 1

Воспользуемся стандартной формой записи квадратного уравнения y=ax²+bx+c для определения уравнения параболы, проходящей через три данные точки. Построим систему уравнений, заменяя значения x и y каждой точки в стандартной формуле квадратного уравнения, формируя систему из трех уравнений.

Если вершина находится в начале координат, то она имеет точку, симметричную заданной, это В(8; 4).

Поэтому имеем 3 точки на параболе: О(0; 0), А(-8; 4) и В(8; 4).

0 = a*0² + b*0 + c, отсюда с = 0.

4 = a*(-8)² + b*(-8) + 0,

4 = a*8² + b*8 + 0.

Сложим 2 последние уравнения:

8 = а*2*64, отсюда а = 8/(2*64) = 1/16.

Тогда b = (4 - a*8²)/8 = (4-(64/16))/8 = 0.

Ответ: y=(1/16)x².

Answer 2

Equation of a Parabola with Vertex at the Origin and Symmetric to the x-axis

To find the equation of a parabola with its vertex at the origin (0,0) and symmetric to the x-axis, we can use the general form of the equation of a parabola:

y = a(x - h)^2 + k

where (h, k) represents the coordinates of the vertex.

In this case, since the vertex is at the origin (0,0), the equation simplifies to:

y = ax^2

Now, we need to find the value of 'a' in order to complete the equation. To do this, we can use the fact that the parabola passes through a given point, A(-8,4).

Substituting the coordinates of point A into the equation, we get:

4 = a(-8)^2

Simplifying further, we have:

4 = 64a

Dividing both sides of the equation by 64, we find:

a = 4/64 = 1/16

Therefore, the equation of the parabola with its vertex at the origin, symmetric to the x-axis, and passing through the point A(-8,4) is:

y = (1/16)x^2

Please let me know if you have any#### Equation of a Parabola with Vertex at the Origin and Symmetric to a Given Axis

To find the equation of a parabola with its vertex at the origin, symmetric to a given axis, and passing through a given point, we can use the standard form of the equation of a parabola:

y = a(x - h)^2 + k

where (h, k) represents the vertex of the parabola.

In this case, since the vertex is at the origin (0, 0), the equation becomes:

y = ax^2

To determine the value of 'a' in the equation, we need to use the fact that the parabola passes through the given point A(-8, 4).

Substituting the coordinates of point A into the equation, we get:

4 = a(-8)^2

Simplifying the equation, we have:

4 = 64a

Solving for 'a', we find:

a = 4/64 = 1/16

Therefore, the equation of the parabola with its vertex at the origin, symmetric to the x-axis, and passing through the point A(-8, 4) is:

y = (1/16)x^2

Please let me know if you need any further assistance!