To write the equation of a parabola given three points, we can use the general form of the equation of a parabola, which is \( y = ax^2 + bx + c \). Since the parabola passes through the points (-1, 3), (0.5, 4.5), and (0, 3), we can create a system of equations using these points.
1. Plug in the coordinates of the point (-1, 3) into the equation of the parabola:
\( 3 = a(-1)^2 + b(-1) + c \)
This simplifies to \( 3 = a - b + c \) ...(Equation 1)
2. Similarly, plug in the coordinates of the point (0.5, 4.5) into the equation:
\( 4.5 = a(0.5)^2 + b(0.5) + c \)
This simplifies to \( 4.5 = 0.25a + 0.5b + c \) ...(Equation 2)
3. Substitute the coordinates of the point (0, 3) into the equation:
\( 3 = a(0)^2 + b(0) + c \)
This simplifies to \( 3 = c \) ...(Equation 3)
Now, we have a system of three equations (Equation 1, Equation 2, Equation 3) that we can solve simultaneously to find the values of \( a \), \( b \), and \( c \) which represent the coefficients in the equation of the parabola.
By solving this system of equations, we can determine the specific equation of the parabola that passes through the given points (-1, 3), (0.5, 4.5), and (0, 3).
