What is the equation in standard form of a parabola that models the values in the table?
x -2 0 4
f(x) 1 5 -59
Please help.I have somewhat of an idea
a(-2)^2+b(-2)+c c=1
a(0)^2+b(0)+c= c=5
a(4)^2+b(4)+c c=-59
4a+2b+c=1
0a+0b+c=5
16a+4b+c=-59
4a-2b+5=0
16a+4b+5=-59
4a-2b+5-5=0-5
4a-2b=-5
16a+4b+5-5=-59-5
16a+4b=-64
5(4a-2b)=5*5
20a-10b=25
20a-10b=25
16a+4b=-64
20a-10b-(16a+4b)=25-(64)
4a+14b=89
not sure what to do now
Not sure where I made my mistake.I lost myself after the 4a-2b+5=0, 16a+4b+5=-59. Could you please clarify what I need to do next.
To solve the system of equations, you can use the method of substitution or elimination. Let's use the method of elimination to solve for a and b.
Multiply the second equation by 10 to simplify:
20a - 10b = 25
160a + 40b = -640
Next, add the two equations together to eliminate b:
(20a - 10b) + (160a + 40b) = 25 - 640
180a + 30b = -615
Simplify the equation:
6a + b = -41
Now you have two equations:
4a + 14b = 89
6a + b = -41
To eliminate b, multiply the second equation by -14:
-84a - 14b = 574
4a + 14b = 89
Add the two equations together:
-80a = 663
Divide both sides of the equation by -80:
a = -8.29
Substitute the value of a back into one of the equations, let's use the second equation:
6(-8.29) + b = -41
-49.74 + b = -41
Add 49.74 to both sides:
b = 8.74
Thus, the values of a and b are approximately -8.29 and 8.74, respectively.
Now, substitute the values of a and b back into one of the original equations, let's use the first equation:
4a + 14b = 89
4(-8.29) + 14(8.74) = 89
-33.16 + 122.36 = 89
Simplify:
89 = 89
Since both sides of the equation are equal, we can consider this equation true. Therefore, the values of a = -8.29 and b = 8.74 satisfy the original system of equations.
Now, to find the value of c, substitute the values of a and b back into one of the original equations, let's use the second equation:
16a + 4b + c = -64
16(-8.29) + 4(8.74) + c = -64
-132.64 + 34.96 + c = -64
Simplify:
-97.68 + c = -64
Add 97.68 to both sides:
c = 33.68
Finally, the equation in standard form that models the values in the table is:
f(x) = -8.29x^2 + 8.74x + 33.68
To find the equation of a parabola in standard form, we need three points on the parabola. In this case, the table gives us three points: (-2, 1), (0, 5), and (4, -59). Given these points, we can set up a system of equations.
The standard form equation of a parabola is y = ax^2 + bx + c.
Using the first point, (-2, 1), we obtain the following equation:
1 = a(-2)^2 + b(-2) + c
1 = 4a - 2b + c
Using the second point, (0, 5), we obtain another equation:
5 = a(0)^2 + b(0) + c
5 = c
Using the third point, (4, -59), we obtain the third equation:
-59 = a(4)^2 + b(4) + c
-59 = 16a + 4b + c
Now we have a system of three equations with three variables (a, b, c):
1 = 4a - 2b + c
5 = c
-59 = 16a + 4b + c
To solve this system, we can use the method of substitution or elimination. Let's use the method of elimination:
First, let's eliminate the variable c. Subtract the second equation (5 = c) from the first and third equations:
1 - 5 = 4a - 2b + c - c
-4 = 4a - 2b
-59 - 5 = 16a + 4b + c - c
-64 = 16a + 4b
Now we have:
-4 = 4a - 2b
-64 = 16a + 4b
To eliminate the variable b, multiply the second equation by 2 and subtract it from the first equation:
-4 - 2(-64) = 4a - 2b - 2(16a + 4b)
-4 + 128 = 4a - 2b - 32a - 8b
124 = -28a - 10b
Simplifying further:
-4a - 10b = 124
Now you have a system of two equations with two variables:
-4 = 4a - 2b
-4a - 10b = 124
To solve this system, you can use the method of substitution or elimination. Let's use the method of substitution:
Rearrange the first equation to solve for a:
-4 = 4a - 2b
4a = 2b - 4
a = (2b - 4) / 4
a = (1/2)b - 1
Substitute this value for a in the second equation:
-4(1/2)b - 10b = 124
-2b - 10b = 124
-12b = 124
b = -124/12
b = -31/3
Now substitute the values of a and b into any of the original equations (e.g., 1 = 4a - 2b + c) to solve for c.
1 = 4a - 2b + c
1 = 4((1/2)b - 1) - 2b + c
1 = 2b - 4 - 2b + c
1 = -4 + c
c = 1 + 4
c = 5
Therefore, the equation in standard form of the parabola that models the values in the table is:
y = (1/2)x^2 - (31/3)x + 5
after subbing in c=5, you got here:
4a-2b+5=0
16a+4b+5=-59
You made a couple of errors here. You should have come up with:
4a - 2b = -4
16a + 4b = -64
or,
4a - 2b = -4
4a + b = -16
b = -4
a = -3
So, y = -3x^2 - 4x + 5
Take much care when proceeding from step to step, that you don't make transcription errors!