Vertex for equation
Opens up or down, and passes
through (−6, −7), (−11, −2), and (−8, 1)
using the three points,
36a-6b+c = -7
121a-11b+c = -2
64a-8b+c = 1
So, (a,b,c) = (-1,-18,-79)
y = -x^2-18x-79
= -(x^2+18x+81) + 2
= -(x+9)^2 + 2
opens down, vertex at (-9,2)
To find the vertex of the quadratic equation, we can use the formula:
x = -b/(2a)
Given that the equation opens up or down, we can determine the sign of 'a' in the standard form of the quadratic equation, which is:
y = ax^2 + bx + c
Now, we'll substitute the given points into the equation to find the values of 'a', 'b', and 'c'.
Using the point (-6, -7):
-7 = a(-6)^2 + b(-6) + c
Using the point (-11, -2):
-2 = a(-11)^2 + b(-11) + c
Using the point (-8, 1):
1 = a(-8)^2 + b(-8) + c
These three equations can be rewritten as:
36a - 6b + c = -7 ....(Equation 1)
121a - 11b + c = -2 ....(Equation 2)
64a - 8b + c = 1 ....(Equation 3)
Next, we'll simultaneously solve these equations to find the values of 'a', 'b', and 'c'.
Subtracting Equation 1 from Equation 2, we get:
85a - 5b = 5 ....(Equation 4)
Subtracting Equation 2 from Equation 3, we get:
57a + 3b = 3 ....(Equation 5)
Now, let's solve Equations 4 and 5:
Multiply Equation 4 by 3 and Equation 5 by 5 to eliminate 'b':
255a - 15b = 15 ....(Equation 6)
285a + 15b = 15 ....(Equation 7)
Adding Equations 6 and 7, we get:
540a = 30
Dividing by 540, we find:
a = 1/18
Substituting the value of 'a' back into Equation 4, we can solve for 'b':
85(1/18) - 5b = 5
5 - 5b = 5
-5b = 0
b = 0
Finally, substituting the values of 'a' and 'b' back into Equation 1, we find 'c':
36(1/18) - 6(0) + c = -7
2 + c = -7
c = -9
So, the equation is:
y = (1/18)x^2 - 9
Now, we can find the vertex using the formula x = -b/(2a):
x = -(0)/(2(1/18))
x = 0
To find the y-coordinate (or the value of 'y' at that x-coordinate), we can substitute x = 0 into the equation:
y = (1/18)(0)^2 - 9
y = -9
Therefore, the vertex of the given equation is (0, -9).
To find the vertex of a quadratic equation, you can use the formula:
x = -b / (2a)
Where a, b, and c are the coefficients of the quadratic equation in the form of ax^2 + bx + c.
In this case, you don't have the equation of the quadratic function. However, you are given three points that the quadratic function passes through. We can use these points to find the equation and then determine the vertex.
Step 1: Use the given points to form a system of equations:
(-6, -7):
-7 = a(-6)^2 + b(-6) + c -- Equation 1
(-11, -2):
-2 = a(-11)^2 + b(-11) + c -- Equation 2
(-8, 1):
1 = a(-8)^2 + b(-8) + c -- Equation 3
Step 2: Simplify the equations:
Equation 1: 36a - 6b + c = -7
Equation 2: 121a - 11b + c = -2
Equation 3: 64a - 8b + c = 1
Step 3: Solve the system of equations to find the values of a, b, and c.
You can use various methods to solve this system of equations, such as substitution, elimination, or matrix methods. Once you find the values of a, b, and c, you can proceed to the next step.
Step 4: Find the x-coordinate of the vertex using the formula x = -b / (2a).
Plug in the values of a and b into the formula to find the x-coordinate.
Step 5: Substitute the x-coordinate of the vertex into the quadratic equation to find the y-coordinate of the vertex.
Plug in the x-coordinate into the quadratic equation to find the y-coordinate.
The coordinates of the vertex will be (x, y), where x is the x-coordinate of the vertex, and y is the y-coordinate of the vertex.
Brute force, check my arithmetic
y = a x^2 + b x + c
first two points:
-7 = 36 a - 6 b+ c
-2 = 121 a -11 b + c
---------------------------------subtract
-5 = -85 a + 5 b
second and third points:
-2 = 121 a - 11 b + c
1 = 64 a - 8 b + c
--------------------------- subtract again
-3 = +57 a - 3 b multiply by 5
-5 = -85 a + 5 b from above, multiply by 3
so
-15 = +285 a - 15 b
-15 = -540 a + 15 b
-------------------------------add
-30 = - 255 a
a = 30/255 = 0.118
-3 = +57 a - 3 b
-3 = 57 (0.118) - 3 b
3 b = 9.7
b = 3.24
-7 = 36 a - 6 b+ c
-7 = 36(0.118) - 6(3.24) + c
c = -7 - 4.25 + 19.44
c = 8.19
so
y = 0.118 x^2 + 3.24 x + 8.19 opens up, holds water
now the vertex, complete square
0.118 x^2 + 3.24 x = y - 8.19 divide by 0.118
x^2 + 27.5 x = 8.47 y - 69.4
add (b/2)^2
x^2 + 27.5 x + 189 = 8.47 y - 69.4 + 189
(x+13.75)^2 = 8.47 ( y + 14.1)
(-13.75 , -14.1)