To find the equation of a parabola with a vertex at (1, 1), we can use the vertex form of a quadratic equation. The vertex form is given by:
y = a(x – h)² + k
In this equation, (h, k) represents the vertex of the parabola. Given that our vertex is (1, 1), we can substitute h = 1 and k = 1 into the vertex form:
y = a(x – 1)² + 1
The value of ‘a’ determines the direction and width of the parabola. If ‘a’ is positive, the parabola opens upwards, and if ‘a’ is negative, it opens downwards. Thus, we can express our parabola’s equation with ‘a’ as an arbitrary non-zero constant:
y = a(x – 1)² + 1, where a is not equal to 0.
For example, if we choose a = 1, the equation would be:
y = (x – 1)² + 1
This represents a parabola that opens upwards with its vertex at (1, 1). You can choose any non-zero value for ‘a’ to create different parabolas with the same vertex.