Which equation matches the graph shown below? The parabola passes through the points left parenthesis negative 6 comma 19 right parenthesis, left parenthesis negative 4 comma negative 5 right parenthesis, left parenthesis negative 2 comma 13 right parenthesis, left parenthesis 0 comma 5 right parenthesis, and left parenthesis 2 comma 19 right parenthesis. (1 point) Responses y = 8x² + 2x – 5 y = 8 x ² + 2 x – 5 y = 8x² + 2x + 5 y = 8 x ² + 2 x + 5 y = 2x² + 8x + 5 y = 2 x ² + 8 x + 5 y = 2x² + 8x – 5
To find the equation of the parabola, we need to determine the values of the coefficients a, b, and c in the general form of a quadratic equation y = ax² + bx + c.
Using the given points, we can substitute their x and y values into the equation and solve for a, b, and c.
For the point (-6, 19):
19 = a(-6)² + b(-6) + c
19 = 36a - 6b + c ----(1)
For the point (-4, -5):
-5 = a(-4)² + b(-4) + c
-5 = 16a - 4b + c ----(2)
For the point (-2, 13):
13 = a(-2)² + b(-2) + c
13 = 4a - 2b + c ----(3)
For the point (0, 5):
5 = a(0)² + b(0) + c
5 = c ----(4)
For the point (2, 19):
19 = a(2)² + b(2) + c
19 = 4a + 2b + c ----(5)
Now, let's solve the system of equations (1)-(5) to determine the values of a, b, and c.
From equation (4), we know that c = 5.
Substituting c = 5 in equations (1), (2), and (3):
19 = 36a - 6b + 5
-5 = 16a - 4b + 5
13 = 4a - 2b + 5
Making it simpler:
36a - 6b = 14 ----(6)
16a - 4b = -10 ----(7)
4a - 2b = 8 ----(8)
Now, we will multiply equation (8) by 2 to eliminate the b term:
8a - 4b = 16 ----(9)
We will eliminate the b term by subtracting equation (7) from equation (9):
(8a - 4b) - (16a - 4b) = 16 - (-10)
-8a = 26
a = -26/8 = -13/4
Now, substituting a = -13/4 in equation (7):
16*(-13/4) - 4b = -10
-52 - 4b = -10
-4b = -10 + 52
-4b = 42
b = 42/(-4) = -21/2
The equation of the parabola is:
y = ax² + bx + c
y = (-13/4)x² + (-21/2)x + 5
Simplifying the above equation, we get:
y = -13/4x² - 21/2x + 5
Therefore, the equation that matches the graph is y = -13/4x² - 21/2x + 5.