Respuesta :
Answer:
A
Step-by-step explanation:
Using the standard form of the parabola , y = ax² + bx + c ( a ≠ 0 )
Substitute the given points into the equation and solve for a, b, c using simultaneous equations.
(- 5, 10 )
- 10 = a(- 5)² - 5b + c, that is
25a - 5b + c = - 10 → (1)
(- 3, 2 )
2 = a(- 3)² - 3b + c, that is
9a - 3b + c = 2 → (2)
(2, - 3 )
- 3 = a(2)² + 2b + c , that is
4a + 2b + c = - 3 → (3)
Eliminate c from the equations
Subtract (2) from (1) term by term
16a - 2b = - 12 → (4)
Subtract (3) from (2) term by term
5a - 5b = 5 → (5)
Multiply (4) by 5 and (5) by - 2 then add to eliminate b
80a - 10b = - 60 → (6)
- 10a + 10b = - 10 → (7)
Add (6) and (7) term by term
70a = - 70 ( divide both sides by 70 )
a = - 1
Substitute a = - 1 into (4) and evaluate for b
- 16 - 2b = - 12 ( add 16 to both sides )
- 2b = 4 ( divide both sides by - 2 )
b = - 2
Substitute a = - 1, b = - 2 into (3) and evaluate for c
- 4 - 4 + c = - 3
- 8 + c = - 3 ( add 8 to both sides )
c = 5
Thus a = - 1, b = - 2 and c = 5
Substitute these values into the standard parabola, that is
y = - x² - 2x + 5 → A
Here we want to find the equation of the parabola that passes through the points (-5, -10), (-3, 2), and (2, -3).
We will get, by brute force, that the correct option is A:
y = -x^2 - 2x + 5
To find the correct option we can use brute force.
We do know the options, so what we can do is just evaluate the given options in the x-values of the points, and see if we get the same y-value that the point.
We can start with option A:
A: y = -x^2 - 2x + 5
For the first point (-5, -10) we have x = -5, evaluating in this we get:
y = -(-5)^2 - 2*(-5) + 5 = -25 + 10 + 5 = -10
Now let's try with the next point (-3, 2), we have x = -3
y = -(-3)^2 - 2*(-3) + 5 = -9 + 6 + 5 = 2
Now we can try with the last point (2, -3), where x= 2
y = -(2)^2 - 2*2 + 5 = -4 - 4 + 5 = -3
So we got all the correct y-values (the second value for each point) thus we can conclude that this one is the correct option.
The correct option is A: y = -x^2 - 2x + 5
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