The equation of a parabola can be expressed in several forms, but when the vertex is known, the vertex form is the most useful. The vertex form of a parabola is given by:
y = a(x - h)² + kIn this equation, (h, k) represents the vertex of the parabola. Given that the vertex in this case is (2, 0), we can substitute h = 2 and k = 0 into the equation. This results in:
y = a(x - 2)² + 0Which simplifies to:
y = a(x - 2)²At this point, you might be wondering about the value of ‘a’. The coefficient ‘a’ determines the direction and the width of the parabola:
- If ‘a’ is positive, the parabola opens upward.
- If ‘a’ is negative, the parabola opens downward.
- A larger absolute value of ‘a’ results in a narrower parabola, while a smaller absolute value of ‘a’ results in a wider parabola.
For example, if we set ‘a’ to 1, the equation becomes:
y = (x - 2)²This represents a standard upward-opening parabola with its vertex at the point (2, 0). On the other hand, if we choose ‘a’ to be -1, the equation would be:
y = -(x - 2)²This represents a downward-opening parabola also with its vertex at (2, 0).
In conclusion, the general equation of a parabola with a vertex at (2, 0) is y = a(x - 2)², where the specific value of ‘a’ will shape the parabola accordingly. You can choose different values of ‘a’ based on how you want the parabola to behave!