Answer:
The equation is:
f(x) = 4x² + 5x -6
Explanation:
A parabola is a second degree equation that has the general (standard) formula: f(x) = ax² + bx + c
Now, to get the equation of the parabola having points (-2,0) , (0,-6) and (4,78) we need to get the values of a, b and c.
1- Substitute with point (0, -6) in the general equation as follows:
f(x) = ax² + bx + c
-6 = a(0)² + b(0) + c
c = -6
Therefore, the equation of the parabola now becomes:
f(x) = ax² + bx - 6
2- Substitute with point (-2 , 0) in the equation we got from part 1:
f(x) = ax² + bx - 6
0 = a(-2)² + b(-2) - 6
0 = 4a - 2b - 6 ................> Divide all terms by 2
0 = 2a - b - 3
b = 2a - 3 ................> equation I
3- Substitute with point (4,78) in the equation we got from part 1:
f(x) = ax² + bx - 6
78 = a(4)² + b(4) - 6
78 = 16a + 4b - 6
78 + 6 = 16a + 4b
84 = 16a + 4b ................> equation II
Substitute with equation I in equation II and solve for a as follows:
84 = 16a + 4b
84 = 16a + 4(2a-3)
84 = 16a + 8a - 12
84 + 12 = 24a
96 = 24a
a = 4
Now, substitute with the value of a in equation I to get b as follows:
b = 2a - 3
b = 2(4) - 3 = 8 - 3 = 5
From the above calculations, we can conclude that:
The equation of the required parabola is:
f(x) = 4x² + 5x -6
Hope this helps :)
