What is the equation, in standard form, of a parabola that models the values in the table
x (-2,0,4)
F(x) (2.5,1.5,-60.5)
To find the equation of a parabola given three points, we need to use the standard form equation of a parabola, which is:
y = ax^2 + bx + c
where a, b, and c represent the coefficients we need to determine.
Given the table:
x (-2, 0, 4)
F(x) (2.5, 1.5, -60.5)
We have three points: (-2, 2.5), (0, 1.5), and (4, -60.5).
Step 1: Substitute the points into the equation y = ax^2 + bx + c.
We will have three equations:
2.5 = a(-2)^2 + b(-2) + c --(1)
1.5 = a(0)^2 + b(0) + c --(2)
-60.5 = a(4)^2 + b(4) + c --(3)
Step 2: Simplify the equations.
Equation (1):
2.5 = 4a - 2b + c
Equation (2):
1.5 = c
Equation (3):
-60.5 = 16a + 4b + c
Step 3: Solve the system of equations.
Using the values from equations (2) and (3), we can substitute them into equation (1).
2.5 = 4a - 2b + 1.5
Simplify the equation:
1 = 4a - 2b
Now we have a system of two equations:
1 = 4a - 2b
-60.5 = 16a + 4b + 1.5
Step 4: Solve the system of equations.
Solve equation (1) for a in terms of b:
1 = 4a - 2b
4a = 2b + 1
a = (2b + 1) / 4
Substitute this value of a into equation (2):
-60.5 = 16((2b + 1) / 4) + 4b + 1.5
Simplify the equation further:
-60.5 = 4(2b + 1) + 4b + 1.5
Expand and simplify:
-60.5 = 8b + 4 + 4b + 1.5
Combine like terms:
-60.5 = 12b + 5.5
Subtract 5.5 from both sides:
-66 = 12b
Divide by 12:
-5.5 = b
Now substitute this value of b back into equation (1) to find a:
1 = 4a - 2(-5.5)
1 = 4a + 11
4a = -10
a = -2.5
Step 5: Substitute the values of a, b, and c into the standard form equation of a parabola.
The equation of the parabola in standard form is:
y = (-2.5)x^2 - 5.5x + 1.5
So, the equation, in standard form, of the parabola that models the values in the given table is y = (-2.5)x^2 - 5.5x + 1.5.
Why did the parabola start a band? Because it had a F(x)pression for music! With the given table values, we can use the standard form of a parabolic equation, which is y = ax^2 + bx + c.
To find the values of a, b, and c, we can substitute the table values into the equation. Let's start with the first set of values (-2, 2.5):
2.5 = a(-2)^2 + b(-2) + c
Simplifying this equation gives us:
2.5 = 4a - 2b + c -----(1)
Similarly, we can substitute the second set of values (0, 1.5) into the equation:
1.5 = a(0)^2 + b(0) + c
Simplifying further gives us:
1.5 = c -----(2)
Lastly, we can substitute the third set of values (4, -60.5):
-60.5 = a(4)^2 + b(4) + c
Simplifying this equation gives us:
-60.5 = 16a + 4b + c -----(3)
Now we have a system of three equations (equations 1, 2, and 3) with three variables (a, b, and c), we can solve the system using algebraic techniques.
I'm sorry if my humor isn't making the process any easier, but solving the system of equations will give us the values of a, b, and c, allowing us to obtain the standard form equation of the parabola that models the given values.
To find the equation of a parabola in standard form, we need to use the general equation for a parabola:
y = ax^2 + bx + c
To determine the values of a, b, and c, we can substitute the given values from the table into the equation.
Using the first point (-2, 2.5):
2.5 = a*(-2)^2 + b*(-2) + c
Simplifying the equation:
2.5 = 4a - 2b + c
Using the second point (0, 1.5):
1.5 = a*0^2 + b*0 + c
Simplifying the equation:
1.5 = c
Using the third point (4, -60.5):
-60.5 = a*4^2 + b*4 + c
Simplifying the equation:
-60.5 = 16a + 4b + c
Now we have a system of three equations:
2.5 = 4a - 2b + c (Equation 1)
1.5 = c (Equation 2)
-60.5 = 16a + 4b + c (Equation 3)
Substituting equation (2) into equations (1) and (3), we get:
2.5 = 4a - 2b + 1.5 (Equation 1')
-60.5 = 16a + 4b + 1.5 (Equation 3')
Simplifying equations (1') and (3'):
2.5 = 4a - 2b + 1.5
=> 2.5 - 1.5 = 4a - 2b
=> 1 = 4a - 2b (Equation 4)
-60.5 = 16a + 4b + 1.5
=> -60.5 - 1.5 = 16a + 4b
=> -62 = 16a + 4b (Equation 5)
Now, solving equations (4) and (5) simultaneously, we can find the values of a and b.
Multiplying equation (4) by 2:
2 = 8a - 4b (Equation 6)
Adding equations (5) and (6) together:
-62 + 2 = 16a + 8a
-60 = 24a
a = -60/24
a = -5/2
Substituting the value of a into equation (4):
1 = 4*(-5/2) - 2b
1 = -10 - 2b
11 = -2b
b = -11/2
Substituting the values of a, b, and c into the general equation for a parabola:
y = (-5/2)x^2 - (11/2)x + 1.5
Therefore, the equation of the parabola in standard form, that models the values in the table, is:
y = (-5/2)x^2 - (11/2)x + 1.5
4a-2b+c = 5/2
c = 3/2
16a+4b+c = -121/2
F(x) = -5/2 x^2 - 11/2 x + 3/2