What is the equation, in standard form,, of a parabola that models the values in the table?
x| -2| 0| 4
f(x) 2.5| 1.5| -60.5|
To find the equation of a parabola, we need three points on the parabola. From the table given, we have three points (-2, 2.5), (0, 1.5), and (4, -60.5).
We can write the equation of a parabola in the form of y = ax^2 + bx + c.
Using the first point (-2, 2.5),
2.5 = a(-2)^2 + b(-2) + c
2.5 = 4a - 2b + c
4a - 2b + c = 2.5
Using the second point (0, 1.5),
1.5 = a(0)^2 + b(0) + c
1.5 = c
c = 1.5
Using the third point (4, -60.5),
-60.5 = a(4)^2 + b(4) + c
-60.5 = 16a + 4b + 1.5
16a + 4b = -60.5 - 1.5
16a + 4b = -62
So now we have the system of equations:
4a - 2b + c = 2.5
16a + 4b = -62
c = 1.5
Substituting c = 1.5 in the first equation:
4a - 2b + 1.5 = 2.5
4a - 2b = 1
Rearranging the second equation:
16a + 4b = -62
4a + b = -15.5
To eliminate b, multiply the first equation by 4 and subtract the result from the second equation:
4a + b = -15.5
-16a - 4b = -4
-16a - 4b - 4a - b = -4 - (-15.5)
-20a - 5b = -4 + 15.5
-20a - 5b = 11.5
Divide both sides of the equation by -5:
4a + b = -15.5
4a + b = -2.3
Since the coefficients of a and b are the same in both equations, but the constants are different, this system of equations has no solution.
Therefore, there is no equation, in standard form, of a parabola that models the given values in the table.