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
To find the equation in standard form of a parabola that models the values in the table, we need to use the formula for a parabola, which is given by:
f(x) = ax^2 + bx + c
To find the values of a, b, and c, we can substitute the coordinates (x, f(x)) from the table into the equation, and then solve the resulting system of equations.
Let's substitute the first set of coordinates (-2, 1) into the equation:
1 = a(-2)^2 + b(-2) + c
1 = 4a - 2b + c
Next, substitute the second set of coordinates (0, 5) into the equation:
5 = a(0)^2 + b(0) + c
5 = c
Since the vertex of the parabola lies at the coordinates (0, 5), we know that c = 5.
Now, substitute the third set of coordinates (4, -59) into the equation:
-59 = a(4)^2 + b(4) + 5
-59 = 16a + 4b + 5
-64 = 16a + 4b
Now we have a system of two equations:
1 = 4a - 2b + 5 (from the first coordinate)
-64 = 16a + 4b (from the third coordinate)
To solve this system, we can use the method of substitution, elimination, or any other method of solving a system of equations.
Let's solve this system using the substitution method:
First, isolate b in the first equation:
-2b = -4a + 4
b = 2a - 2
Now substitute this expression for b into the second equation:
-64 = 16a + 4(2a - 2)
-64 = 16a + 8a - 8
-64 = 24a - 8
-56 = 24a
a = -56/24
a = -7/3
Now substitute the value of a back into the expression for b:
b = 2(-7/3) - 2
b = -14/3 - 2
b = -14/3 - 6/3
b = -20/3
Substituting the values of a, b, and c back into the equation for a parabola:
f(x) = (-7/3)x^2 - (20/3)x + 5
Therefore, the equation in standard form of the parabola that models the values in the table is:
f(x) = (-7/3)x^2 - (20/3)x + 5
This is what I have so far if someone could help me finish.
a(-2)^2+b(-2)+c=1
a(0)^2+b(0)+c=5
a(4)^2+b(4)+c=-59
4a-2b+c=1
0a+0b+c=5 c=5
16a+4b+c=-59
4a-2b+5=1
16a+4b+5=-59
4a-2b+5-5=1-5
4a-2b=-4
16a+4b+5-5=-59-5
16a+4b=-64
Now I'm not sure if I'm doing this right.
4(4a-2b)=-4*4
16a+4b=-16
16a-8b=-16
16a+4b=-64
16a-8b-(16a+4b)=-36-(-64)
I'm lost!
Can someone please help?