what is the equation, in standard form, of a parabola that contains the following points?
(-2,18), (0,2), (4,42)
A. y = -2x^2 - 2x - 3
B. y = -3x^2 + 2x - 2
C. y = 3x^2 - 2x + 2*
D. y = -2x^2 + 3x + 2
Answers :
1. C
2. D
3. A
4. B
5. C
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A will not work
B give me a break
C 3*16 -8 +2 = 42 sure enough
Well, let's see if we can figure this out together!
To find the equation of a parabola, we need to use the standard form, which is usually written as "y = ax^2 + bx + c".
Let's start by plugging in the coordinates of the first point, (-2, 18):
18 = a(-2)^2 + b(-2) + c
That simplifies to:
18 = 4a - 2b + c
Next, let's do the same thing with the second point, (0, 2):
2 = a(0)^2 + b(0) + c
This simplifies to:
2 = c
Now, let's do the same thing with the third point, (4, 42):
42 = a(4)^2 + b(4) + c
This simplifies to:
42 = 16a + 4b + c
Now that we have three equations (18 = 4a - 2b + c, 2 = c, 42 = 16a + 4b + c), we can use algebra to solve for a, b, and c.
By substituting the value of c (which is 2) into the other two equations, we get:
18 = 4a - 2b + 2
42 = 16a + 4b + 2
After simplifying these equations, we get:
16 = 4a - 2b
40 = 16a + 4b
Now, let's solve this system of equations to find the values of a and b.
Multiplying the first equation by 4, we get:
64 = 16a - 8b
Now, we'll add this equation to the second equation:
64 + 40 = 16a - 8b + 16a + 4b
After simplifying, we get:
104 = 32a - 4b
Dividing by 4, we get:
26 = 8a - b
Now, let's substitute the value of b from this equation into the first equation (16 = 4a - 2b):
16 = 4a - 2(8a - 26)
16 = 4a - 16a + 52
16 = -12a + 52
-36 = -12a
a = 3
Now, let's substitute the value of a into the equation we found before (26 = 8a - b):
26 = 8(3) - b
26 = 24 - b
b = -2
Finally, we have a = 3, b = -2, and c = 2, so the equation in standard form is:
y = 3x^2 - 2x + 2
Therefore, the correct answer is C.
To find the equation of a parabola in standard form that passes through three given points, we can follow these steps:
Step 1: Substitute the x and y coordinates of each given point into the standard form equation (y = ax^2 + bx + c) to obtain three equations.
For the point (-2, 18):
18 = 4a - 2b + c -- Equation 1
For the point (0, 2):
2 = 0a + 0b + c -- Equation 2
For the point (4, 42):
42 = 16a + 4b + c -- Equation 3
Step 2: Solve the system of equations simultaneously to find the values of a, b, and c.
Using Equations 1 and 2, we can substitute c in Equation 2:
2 = -2b + 18 --> -2b = -16 --> b = 8
Substituting the value of b into Equation 1:
18 = 4a - 2(8) + c --> 18 = 4a - 16 + c --> 34 = 4a + c
Substituting the value of b into Equation 3:
42 = 16a + 4(8) + c --> 42 = 16a + 32 + c --> 10 = 16a + c
Now, we have two equations:
34 = 4a + c -- Equation 4
10 = 16a + c -- Equation 5
Subtract Equation 5 from Equation 4 to eliminate the variable c:
34 - 10 = 4a + c - (16a + c)
24 = -12a
a = -2
Substituting the value of a = -2 into Equation 4 or Equation 5, we can find the value of c:
34 = 4(-2) + c --> 34 = -8 + c --> c = 42
So, we have found that a = -2, b = 8, and c = 42.
Step 3: Substitute the values of a, b, and c into the standard form equation (y = ax^2 + bx + c) to obtain the final equation.
The equation of the parabola in standard form is:
y = -2x^2 + 8x + 42
Therefore, the correct option for the equation of the parabola is:
C. y = -2x^2 + 8x + 42