Use quadratic regression to find the equation of the parabola going through these 3 points.
(-3,88), (2,-27), (4,-17)
y= 4x2-19x+?
y = 4x
2 + 19x − 5
You can factor the right-hand side by using the "splitting the middle term" method.
Multiply the coefficient of the rst term by the constant.
4 × − 5 = − 20
Find two numbers that when added equal −19, which is the coefcient of the middle term, and that multiply to −20. The numbers 20 and −1 meet the
requirements.
Split 19x as the sum of 20x and −x.
y = 4x
2 + 20x − x − 5
Factor out the common terms in the rst two terms and the second two terms.
y = 4x(x + 5) − (x + 5)
It is understood that −(x + 5) is −1(x + 5).
Factor out the common term x + 5.
y = (x + 5)(4x − 1)
using your three points, you get
88 = 36+57+?
? = -4
-27 = 8-38+?
? = -3
-17 = 64-76+?
? = -5
so, what do you think?
To find the equation of the parabola using quadratic regression, follow these steps:
Step 1: Write down the three points given.
(-3, 88), (2, -27), (4, -17)
Step 2: Use the general equation of a parabola, y = ax^2 + bx + c, and substitute the x and y values of each point into the equation.
For (-3, 88):
88 = a(-3)^2 + b(-3) + c
For (2, -27):
-27 = a(2)^2 + b(2) + c
For (4, -17):
-17 = a(4)^2 + b(4) + c
Step 3: Simplify the equations obtained in Step 2.
For (-3, 88):
9a - 3b + c = 88
For (2, -27):
4a + 2b + c = -27
For (4, -17):
16a + 4b + c = -17
Step 4: Solve the system of equations formed in Step 3. This can be done by substituting and eliminating variables, or by using matrix methods. Here, we will use the elimination method.
Multiply the equation for (-3, 88) by 4:
36a - 12b + 4c = 352
Multiply the equation for (2, -27) by 16:
64a + 32b + 16c = -432
Subtract the equation for (4, -17) from the sum of the other two equations:
(36a - 12b + 4c) + (64a + 32b + 16c) - (16a + 4b + c) = 352 - 432 - (-17)
Simplifying, we get:
84a + 20c = -197
Step 5: Continue solving the system of equations.
Multiply the equation for (4, -17) by 25:
400a + 100b + 25c = -425
Subtract the equation for (2, -27) from the previous equation:
(400a + 100b + 25c) - (64a + 32b + 16c) = -425 - (-27)
Simplifying, we get:
336a + 68b + 9c = -398
Step 6: Solve the obtained system of equations.
Using the system:
84a + 20c = -197 (Equation 1)
336a + 68b + 9c = -398 (Equation 2)
We can solve this system by using any standard method (substitution, elimination, matrices, etc.). Solving for 'c' in Equation 1, we get:
c = (-197 - 84a)/20
Substituting this value of 'c' into Equation 2, we get:
336a + 68b + 9[(-197 - 84a)/20] = -398
Simplifying this equation, we get:
6720a + 1360b - 3543 - 1512a = -7960
Combining like terms, we have:
5208a + 1360b = -4423 (Equation 3)
Step 7: Solve Equation 3 for 'b'.
Rearranging Equation 3, we get:
b = (-4423 - 5208a)/1360
Step 8: Solve for 'c' using the value of 'b' obtained in Step 7.
Substituting the value of 'b' in Equation 1, we have:
c = (-197 - 84a)/20
Step 9: Substitute the values of 'a', 'b', and 'c' obtained in Steps 7 and 8 into the general equation of a parabola, y = ax^2 + bx + c.
Thus, the equation of the parabola is:
y = ax^2 + bx + c
y = a x^2 + (-4423 - 5208a)x + (-197 - 84a)/20
Simplifying further, we get:
y = ax^2 - 4423x - 5208ax - 197/20 - 84a/20
y = ax^2 - 5208ax - 4423x - 197/20 - 84a/20
Hence, the equation of the parabola passing through the given points (-3, 88), (2, -27), and (4, -17) can be expressed as:
y = ax^2 - 5208ax - 4423x - 197/20 - 84a/20
To find the equation of the parabola that passes through the given three points using quadratic regression, we need to follow these steps:
Step 1: Organize the data
Let's arrange the given points into two lists: one for the x-values and another for the corresponding y-values.
x = [-3, 2, 4]
y = [88, -27, -17]
Step 2: Determine the quadratic regression equation
We want to find the equation in the form y = ax^2 + bx + c. To do this, we'll use a statistical calculator or software that performs quadratic regression.
Using a graphing calculator like Desmos or an online regression calculator, input the data points and perform quadratic regression. The calculator will provide the values of a, b, and c.
After performing the quadratic regression, the calculator will provide the values of a, b, and c. In this case, the equation of the parabola is y = 4x^2 - 19x + c.
Step 3: Solve for c
To determine the value of c, substitute the x and y values from any of the given points into the equation y = 4x^2 - 19x + c. Let's use the first point (-3, 88).
88 = 4(-3)^2 - 19(-3) + c
88 = 36 + 57 + c
88 = 93 + c
By rearranging the equation to solve for c, we get:
c = 88 - 93
c = -5
So, the equation of the parabola that passes through the given three points is:
y = 4x^2 - 19x - 5