Which of the following is the equation, in standard form, of a parabola that contains the following points?
(–2, 18), (0, 2), (4, 42)
(1 point)
Responses
y = –2x^2 – 2x – 3
y = –3x^2 + 2x – 2
y = 3x^2 – 2x + 2
y = –2x^2 + 3x + 2
To find the equation of a parabola that contains the given points, you can use the standard form of the parabolic equation, which is y = ax^2 + bx + c.
To determine the values of a, b, and c, substitute each point into the equation and solve the resulting system of equations.
Using the point (–2, 18):
18 = a(-2)^2 + b(-2) + c
18 = 4a - 2b + c
Using the point (0, 2):
2 = a(0)^2 + b(0) + c
2 = c
Using the point (4, 42):
42 = a(4)^2 + b(4) + c
42 = 16a + 4b + c
Now, substitute the value of c from the second equation into the other two equations:
18 = 4a - 2b + 2
42 = 16a + 4b + 2
Simplifying these equations, we get:
4a - 2b + 16 = 0
16a + 4b + 40 = 0
Solving this system of equations, we find that a = 3 and b = -2.
Thus, the equation of the parabola in standard form is:
y = 3x^2 - 2x + 2.
Therefore, the correct answer is: y = 3x^2 - 2x + 2.
To find the equation of a parabola given three points, we can use the standard form equation y = ax^2 + bx + c.
We can substitute the values of the points into the equation to form a system of three equations:
18 = a(-2)^2 + b(-2) + c
2 = a(0)^2 + b(0) + c
42 = a(4)^2 + b(4) + c
Simplifying these equations, we have:
18 = 4a - 2b + c
2 = c
42 = 16a + 4b + c
From the second equation, we can determine that c = 2. Substituting this value into the other two equations, we get:
18 = 4a - 2b + 2
42 = 16a + 4b + 2
Simplifying these equations further, we have:
4a - 2b = 16
16a + 4b = 40
We can solve this system of equations using any method of choice. For simplicity, let's solve this system by elimination:
Multiplying the first equation by 4, we get:
16a - 8b = 64
Subtracting this new equation from the second equation, we have:
(16a + 4b) - (16a - 8b) = 40 - 64
12b = -24
b = -2
Substituting this value of b into the first equation, we have:
4a - 2(-2) = 16
4a + 4 = 16
4a = 12
a = 3
Now that we have determined the values of a and b, we can substitute them, along with c = 2, into the standard form equation to form the final equation:
y = 3x^2 - 2x + 2
Therefore, the correct choice is:
y = 3x^2 - 2x + 2
To determine the equation of a parabola in standard form, we need to use the given points to solve for the coefficients.
Let's substitute the coordinates of the given points into the equation y = ax^2 + bx + c.
For the point (–2, 18):
18 = a(-2)^2 + b(-2) + c
18 = 4a - 2b + c -- (Equation 1)
For the point (0, 2):
2 = a(0)^2 + b(0) + c
2 = c -- (Equation 2)
For the point (4, 42):
42 = a(4)^2 + b(4) + c
42 = 16a + 4b + c -- (Equation 3)
Now, substitute c = 2 from Equation 2 into Equation 1 and Equation 3:
18 = 4a - 2b + 2 -- (Equation 4)
42 = 16a + 4b + 2 -- (Equation 5)
Now, let's solve this system of equations (Equation 4 and Equation 5) to find the values of a and b:
Multiply Equation 4 by 2:
36 = 8a - 4b + 4 -- (Equation 6)
Then, subtract Equation 6 from Equation 5:
42 - 36 = 16a + 4b - (8a - 4b + 4)
6 = 16a + 4b - 8a + 4b - 4
6 = 8a + 8b - 4
10 = 8a + 8b
Divide through by 2:
5 = 4a + 4b -- (Equation 7)
Now, multiply Equation 7 by 2:
10 = 8a + 8b -- (Equation 8)
Subtract Equation 7 from Equation 8:
10 - 5 = 8a + 8b - (4a + 4b)
5 = 8a + 8b - 4a - 4b
5 = 4a + 4b
Divide through by 4:
5/4 = a + b -- (Equation 9)
Now that we have found the values of a and b, we can substitute them back into Equation 4 to find c:
18 = 4a - 2b + 2
18 = 4(a + b) - 2b + 2 -- (Using Equation 9)
18 = 4(5/4) - 2b + 2
18 = 5 - 2b + 2
18 = 7 - 2b
2b = -11
b = -11/2
Now, substitute the values of a = 5/4 and b = -11/2 into the equation y = ax^2 + bx + c:
y = (5/4)x^2 + (-11/2)x + 2
Therefore, the equation, in standard form, of the parabola that contains the given points is:
y = (5/4)x^2 - (11/2)x + 2