What is the equation, in standard form, of a parabola that models the values in the table
x -2 0 4
f(x) 16 -6 46
Responses
−4x2−3x−6
3x2+4x−6
4x2+3x−6
4x2−3x−6
To find the equation of a parabola in standard form, we need to know either the coordinates of the vertex or the coordinates of at least two other points on the parabola.
Let's use the given points (-2, 16) and (0, -6) to determine the equation.
We start by using the vertex form of a parabola:
f(x) = a(x-h)^2 + k
where (h, k) represents the coordinates of the vertex.
Substituting the coordinates of the given points:
16 = a(-2-h)^2 + k (Equation 1)
-6 = a(0-h)^2 + k (Equation 2)
Let's solve this system of equations to find the values of a, h, and k.
From Equation 2, we have:
-6 = ah^2 + k
Rearranging Equation 2, we get:
k = -6 - ah^2
Substituting k into Equation 1, we get:
16 = a(-2-h)^2 + (-6 - ah^2)
Expanding and simplifying, we have:
16 = 4ah^2 + 4ah + 16 + 6 - ah^2
Combine like terms:
16 = 3ah^2 + 4ah + 22
Rearranging the equation to standard form, we have:
3ah^2 + 4ah + (22 - 16) = 0
3ah^2 + 4ah + 6 = 0
Now, let's factor this quadratic equation:
(3h + 2)(ah + 3) = 0
Setting each factor equal to zero:
3h + 2 = 0 or ah + 3 = 0
For the first factor:
3h = -2
h = -2/3
For the second factor:
ah = -3
a = -3/h
Therefore, we have determined the values of h and a.
Now, let's substitute these values back into either Equation 1 or Equation 2 to find the value of k.
Using Equation 1:
16 = a(-2-h)^2 + k
Substituting the values of a and h:
16 = (-3/(-2/3))^2 + k
16 = (-9/2)^2 + k
16 = 81/4 + k
Multiplying both sides by 4 to clear the fraction:
64 = 81 + 4k
4k = -17
k = -17/4
So the equation of the parabola in standard form is:
f(x) = a(x-h)^2 + k
f(x) = -3(x-(-2/3))^2 + (-17/4)
To simplify the equation:
f(x) = -3(x+2/3)^2 - 17/4
Therefore, the correct equation, in standard form, of the parabola that models the values in the table is:
4x^2 + 3x - 6. Answer: (D)