Answer:
The answer is D because the value of a (2/3) is positive, and only D has it positive in both forms.
Explanation:
The vertex form of the equation for a parabola is a(x – h)² + k, and the vertex is (h,k).
The standard form of the parabola equation is ax² + bx + c.
From your given information, the vertex is (-3, -6) so by plugging that into the equation, we get a[x – (-3)]² + (-6) = a(x + 3)² - 6
so f(x) = a(x +3)² - 6 and we put in the point (0,0) we get
0 = a(0 +3)² - 6 and solve for a.
0 = a(3)² - 6
0 = 9a -6 (add 6 to both sides to get a by itself)
+6 +6
6 = 9a (divide both sides by 9 to get a by itself)
9 9
6/9 = a (we simplify 6/9 by reducing both numbers by 3 to get 2/3)
a = 2/3 therefore our vertex form is now f(x) = (2/3)(x +3)² - 6.
We have to expand our equation to find the standard form.
(x +3)² = (x+3)(x+3) = x² + 3x + 3x +9 = x² +6x+ 9 (plug it back in)
(2/3)(x² +6x+ 9) - 6
2/3x² + (2/3)6x+ (2/3)9 - 6
2/3x² +4x+ 6 - 6
f(x) = 2/3x² + 4x is standard form; vertex form is now f(x) = (2/3)(x +3)² - 6