what is the equation in standard form of a parabola that models the value in the table? x -2,0,4 fx 1,5,-59
so you have the points:
(-2,1), (0,5), and (4,-59)
let the function be
y = ax^2 + bx + c
for (0,5)
5 = 0 + 0 + c -----> c = 5
so we could reduce our equation to
y = ax^2 + bx + 5
for (-2,1)
1 = 4a - 2b + 5
4a - 2b = -4
2a - b = -2 ***
for (4, -59)
-59 = 16a + 4b + 5
16a + 4b = -64
4a + b = -16 ****
add *** and ****
6a = -18
a = -3
into ****
-12 + b = -16
b = -4
y = -3x^2 - 4x + 5
change to whatever version of the parabola you need
A quick mental check shows that all 3 points lie on the curve
To find the equation of a parabola in standard form based on the given table, you need to consider the three points (x, fx).
Step 1: Plug in the values from the table into the general form of a parabola, y = ax^2 + bx + c.
Using the first point (-2, 1):
1 = a(-2)^2 + b(-2) + c
1 = 4a - 2b + c
Using the second point (0, 5):
5 = a(0)^2 + b(0) + c
5 = c
Using the third point (4, -59):
-59 = a(4)^2 + b(4) + c
-59 = 16a + 4b + c
Step 2: Now you have a system of three equations:
1 = 4a - 2b + c
5 = c
-59 = 16a + 4b + c
Step 3: Solve the system of equations. Start by using the second equation to substitute 'c' into the other two equations:
1 = 4a - 2b + 5
-59 = 16a + 4b + 5
Simplifying the equations:
4a - 2b - 4 = 0
16a + 4b + 54 = 0
Step 4: Rewrite the equations in standard form:
4a - 2b = 4
16a + 4b = -54
Step 5: Multiply the first equation by 2 and the second equation by -1 to eliminate 'b':
8a - 4b = 8
-16a - 4b = 54
Step 6: Add the two equations together:
8a - 4b + (-16a - 4b) = 8 + 54
-8a - 8b = 62
Step 7: Divide the equation by -8 to find the simplified form:
a + b = -7.75
The equation in standard form of the parabola that models the table's values is:
y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
However, note that the given table does not have any symmetry, which leads to unusual results for the parabola. Therefore, it is possible there was an error in the table values or equation.