To find the equation of a parabola with a specific vertex, we can use the vertex form of a quadratic equation. The vertex form of a parabola is given by:
y = a(x – h)^2 + k
In this equation, (h, k) represents the vertex of the parabola, and ‘a’ determines the direction and width of the parabola. Here, you specified that the vertex is at the point (1, 1), which means:
– h = 1
– k = 1
Substituting these values into the vertex form, we get:
y = a(x – 1)^2 + 1
The value of ‘a’ can be any non-zero number. If:
- a > 0: the parabola opens upwards.
- a < 0: the parabola opens downwards.
For example, if we choose ‘a’ to be 1, the equation would be:
y = (x – 1)^2 + 1
This represents a parabola that opens upwards with its vertex at (1, 1). If we choose ‘a’ to be -1, the equation would be:
y = -(x – 1)^2 + 1
This represents a parabola that opens downwards and also has its vertex at (1, 1). In summary:
The general equation of a parabola with a vertex at (1, 1) is:
y = a(x – 1)^2 + 1, where ‘a’ is any non-zero real number.