What is the equation, in standard form, of a parabola that models the values in the table?
x –2 0 4
ƒ(x) 1 5 -59
To determine the equation of a parabola given a set of points, we can use the standard form of the equation of a parabola: y = ax^2 + bx + c. The standard form allows us to solve for the coefficients a, b, and c.
Let's use the points (-2, 1), (0, 5), and (4, -59) to find the equation.
Step 1: Plug in the x and ƒ(x) values into the equation y = ax^2 + bx + c.
For the point (-2, 1):
1 = a(-2)^2 + b(-2) + c
For the point (0, 5):
5 = a(0)^2 + b(0) + c
For the point (4, -59):
-59 = a(4)^2 + b(4) + c
Step 2: Simplify and create a system of equations.
From the first equation: 1 = 4a - 2b + c
From the second equation: 5 = c
From the third equation: -59 = 16a + 4b + c
Step 3: Solve the system of equations.
Since we already found that c = 5 from the second equation, we can substitute it into the other two equations.
1 = 4a - 2b + 5
-59 = 16a + 4b + 5
Simplifying these equations further, we have:
4a - 2b = -4 (Equation 1)
16a + 4b = -69 (Equation 2)
Now, we can solve this system of equations by using any method, such as substitution or elimination.
I will use the elimination method to solve this system:
Multiply Equation 1 by 2 and Equation 2 by -1 to eliminate the b term:
8a - 4b = -8 (Equation 3)
-16a - 4b = 69 (Equation 4)
Adding Equation 3 and Equation 4 gives us:
-8a = 61
Dividing both sides by -8:
a = -61/8
Now, substitute this value back into Equation 1:
4(-61/8) - 2b = -4
-244/8 - 2b = -4
Simplifying further:
-31 - 2b = -4
-2b = 27
b = -27/2
So, we now have the values of both a and b. We can substitute these values, along with c = 5, into the standard form of the equation:
y = ax^2 + bx + c
y = (-61/8)x^2 - (27/2)x + 5
Hence, the equation in standard form that models the values in the table is:
y = (-61/8)x^2 - (27/2)x + 5
ax^2 + bx + c = 1
plug in first value
4a -2b + c = 1
do the same for the others:
0 + 0 + c = 5
c = 5
16a + 4b + c = -59
so we plug in c into the equations with and b:
16a + 4b + 5 = -59
4a -2b + 5 = 1
4a = 2b -4
2a = b - 2
16a + 4b = -64
8(b-2) + 4b = -64
2(b-2) + b = -16
2b - 4 + b = -16
3b = -12
b = -4
Plug in for a
2a = -4 - 2
2a = -6
a = -3
So the equation is:
-3x^2 - 4x + 5