What is the equation, in standard form, of a parabola that contains the following points?
(–2, 18), (0, 2), (4, 42)
(1 point)
Responses
y = –2x2 – 2x – 3
y = –2 x 2 – 2 x – 3
y = –3x2 + 2x – 2
y = –3 x 2 + 2 x – 2
y = 3x2 – 2x + 2
y = 3 x 2 – 2 x + 2
y = –2x2 + 3x + 2
To find the equation of a parabola in standard form, we need to use the general equation of a parabola: y = ax^2 + bx + c.
To find the values of a, b, and c, we can substitute each of the given points into the equation and form a system of three equations.
The first point (–2, 18) gives us the equation:
18 = a(-2)^2 + b(-2) + c
18 = 4a - 2b + c
The second point (0, 2) gives us the equation:
2 = a(0)^2 + b(0) + c
2 = c
The third point (4, 42) gives us the equation:
42 = a(4)^2 + b(4) + c
42 = 16a + 4b + c
Now we can solve this system of equations to find the values of a, b, and c.
From the second equation, we know that c = 2. Substituting this into the first and third equations:
18 = 4a - 2b + 2
42 = 16a + 4b + 2
Simplifying these equations:
18 - 2 = 4a - 2b
42 - 2 = 16a + 4b
16 = 4a - 2b
40 = 16a + 4b
We can solve this system of equations using elimination or substitution method.
Multiplying the first equation by 2 and adding it to the second equation:
32 = 8a - 4b
40 = 16a + 4b
72 = 24a
a = 3
Substituting a = 3 into the first equation:
16 = 4(3) - 2b
16 = 12 - 2b
-2b = 16 - 12
-2b = 4
b = -2
Now we have the values of a = 3, b = -2, and c = 2.
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 that passes through three given points, we can use the standard form of the equation for a parabola: y = ax^2 + bx + c.
To determine the values of a, b, and c, we can substitute the coordinates of each given point into the equation and solve the resulting system of equations.
Let's go through the process using the given points (–2, 18), (0, 2), and (4, 42):
1. Substituting the coordinates of the first point (–2, 18) into the equation, we get:
18 = a(-2)^2 + b(-2) + c
18 = 4a - 2b + c
2. Substituting the coordinates of the second point (0, 2) into the equation, we get:
2 = a(0)^2 + b(0) + c
2 = c
3. Substituting the coordinates of the third point (4, 42) into the equation, we get:
42 = a(4)^2 + b(4) + c
42 = 16a + 4b + c
Since we found that c is equal to 2 from the second equation, we can substitute it into the other two equations:
18 = 4a - 2b + 2 (equation 1)
42 = 16a + 4b + 2 (equation 2)
Now we have a system of two equations with two variables (a and b). We can solve it by elimination or substitution.
Let's use the elimination method. Multiply equation 1 by 2 to eliminate the b variable:
36 = 8a - 4b + 4 (equation 3)
42 = 16a + 4b + 2 (equation 2)
Add equation 3 to equation 2:
36 + 42 = 8a + 16a + 4
78 = 24a + 4
Subtract 4 from both sides:
74 = 24a
Divide both sides by 24:
a = 74/24
a = 37/12
Now substitute the value of a back into equation 1:
18 = 4(37/12) - 2b + 2
18 = 37/3 - 2b + 2
Subtract 37/3 + 2 from both sides:
16 = -2b
b = -16/2
b = -8
Finally, substitute the values of a, b, and c into the equation y = ax^2 + bx + c:
y = (37/12)x^2 - 8x + 2
Therefore, the equation, in standard form, of the parabola that passes through the given points is:
y = (37/12)x^2 - 8x + 2
To find the equation of a parabola in standard form, we can use the general equation:
y = ax^2 + bx + c
To determine the values of a, b, and c, we can substitute the coordinates of the given points into the equation.
For the point (–2, 18):
18 = a(-2)^2 + b(-2) + c
18 = 4a - 2b + c ......(1)
For the point (0, 2):
2 = a(0)^2 + b(0) + c
2 = c ............(2)
For the point (4, 42):
42 = a(4)^2 + b(4) + c
42 = 16a + 4b + c ......(3)
Now, we have a system of three equations:
18 = 4a - 2b + c
2 = c
42 = 16a + 4b + c
We can substitute the value of c from equation (2) into equations (1) and (3):
18 = 4a - 2b + 2
42 = 16a + 4b + 2
Simplifying these equations, we get:
4a - 2b = 16 ......(4)
16a + 4b = 40 ......(5)
Next, we can multiply equation (4) by 2 to eliminate the coefficient of b:
8a - 4b = 32 ......(6)
Adding equations (5) and (6), we can eliminate b:
24a = 72
Dividing both sides by 24, we find:
a = 3
Substituting this value of a into equation (4), we can solve for b:
4(3) - 2b = 16
12 - 2b = 16
-2b = 4
b = -2
Finally, substituting the values of a = 3, b = -2, and c = 2 into the standard form equation:
y = 3x^2 - 2x + 2
Therefore, the equation of the parabola in standard form that passes through the given points is:
y = 3x^2 - 2x + 2.