To find the equation of the quadratic function represented by the given table, you can use the general form of a quadratic function:
f(x) = ax^2 + bx + c
First, let's find the values of the coefficients a, b, and c.
Plug in the values from the table into the equation:
For (x, f(x)):
(5, -4)
(6, 5)
(7, 8)
Use these data points to set up a system of three equations:
-4 = 25a + 5b + c
5 = 36a + 6b + c
8 = 49a + 7b + c
You can use a variety of methods to solve this system of equations, such as substitution or matrices. Let's use substitution to solve it step by step.
From the first equation, isolate c:
c = -4 - 25a - 5b
Now, substitute this expression for c in the second equation:
5 = 36a + 6b + (-4 - 25a - 5b)
Now, simplify and solve for b:
5 = 36a + 6b - 4 - 25a - 5b
Combine like terms:
5 = 11a + b - 4
Add 4 to both sides:
9 = 11a + b
Now, substitute this expression for b in the third equation:
8 = 49a + 7(-4 - 25a - 5b)
Simplify and solve for a:
8 = 49a - 28 - 175a - 35b
Combine like terms:
8 = -126 - 126a - 35b
Now, add 126 to both sides:
8 + 126 = -126a - 35b + 126
134 = -126a - 35b
Now, you have a system of two equations:
9 = 11a + b
134 = -126a - 35b
Solve this system of equations to find the values of a and b.
Let's subtract the first equation from the second equation to eliminate b:
(134 - 9) = (-126a - 35b) - (11a + b)
125 = -137a
Now, divide by -137 to find a:
a = -125/137
Now that you have the value of a, you can substitute it back into the first equation to find b:
9 = 11a + b
9 = 11(-125/137) + b
Now, solve for b:
b = 9 - 11(-125/137)
b ≈ 9 + 1125/137
b ≈ (1233/137)
With the values of a and b, you can now write the equation of the quadratic function:
f(x) = ax^2 + bx + c
f(x) = (-125/137)x^2 + (1233/137)x + c
Now, to find c, you can use any of the data points from the table. Let's use the point (5, -4):
-4 = (-125/137)(5^2) + (1233/137)(5) + c
Now, solve for c:
-4 = (-125/137)(25) + (1233/137)(5) + c
-4 = -2875/137 + 6165/137 + c
To combine the fractions on the right side:
-4 = (6165 - 2875)/137 + c
-4 = 3290/137 + c
Now, to isolate c, subtract 3290/137 from both sides:
-4 - 3290/137 = c
Now, find a common denominator:
-548/137 - 3290/137 = c
Combine the fractions:
c = (-548 - 3290)/137
c = -3838/137
c = -28
Now, you have the values of a, b, and c:
a = -125/137
b = 1233/137
c = -28
So, the equation of the quadratic function represented by the table is:
f(x) = (-125/137)x^2 + (1233/137)x - 28