Use the information provided to write the standard form equation of each parabola.
1) Opens left or right, and passes though (18, -5) (18, -13) and (6, -11)
2) Opens left of right, vertex: (10, -10), passes through (-26, -7)
To find the standard form equation of a parabola, we need to determine the values of the coefficients a, b, and c in the equation y = ax^2 + bx + c. We can use the given information to determine these coefficients for each parabola.
1) Opens left or right, and passes through (18, -5), (18, -13), and (6, -11):
Since the parabola passes through these three points, we can substitute their x and y coordinates into the general equation y = ax^2 + bx + c to form a system of equations.
Using the point (18, -5):
-5 = a(18)^2 + b(18) + c -- (Equation 1)
Using the point (18, -13):
-13 = a(18)^2 + b(18) + c -- (Equation 2)
Using the point (6, -11):
-11 = a(6)^2 + b(6) + c -- (Equation 3)
We now have a system of three equations with three unknowns (a, b, and c). We can solve this system to find the coefficients.
Subtracting Equation 2 from Equation 1, we eliminate c:
-5 - (-13) = a(18^2 - 18^2) + b(18 - 18) + c - c
8 = 0a + 0b
8 = 0
The resulting equation 8 = 0 is not possible. It means that the given points do not form a parabola. There might be an error in the information provided. Please check the given points again.
2) Opens left or right, vertex: (10, -10), passes through (-26, -7):
Since we know the vertex and a point on the parabola, we can write the equation using the vertex form of a parabola: y = a(x - h)^2 + k, where (h, k) is the vertex.
Using the vertex (10, -10) and the point (-26, -7), we have:
-7 = a(-26 - 10)^2 - 10
Simplifying this equation:
-7 = a(-36)^2 - 10
-7 = a(1296) - 10
-7 + 10 = 1296a
3 = 1296a
a = 3/1296
The value of a is 3/1296. Now, substituting the value of a into the vertex form equation, we get the equation for the parabola:
y = (3/1296)(x - 10)^2 - 10
Simplifying this equation further gives the final standard form equation.
1) To find the standard form equation of a parabola that opens left or right, we need to use the formula:
(y - k)^2 = 4p(x - h), where (h, k) represents the coordinates of the vertex.
Step 1: Find the coordinates of the vertex:
Since the parabola passes through (18, -5) and (18, -13), we can conclude that the vertex has an x-coordinate of 18.
Therefore, the vertex is (18, k), and we still need to determine the value of k.
We also know that the parabola passes through (6, -11). To find the value of k, substitute the coordinates into the equation: (y - k)^2 = 4p(x - h).
(-11 - k)^2 = 4p(6 - 18)
(-11 - k)^2 = 4p(-12)
(-11 - k)^2 = -48p
Step 2: Using another point to eliminate p:
Let's use the point (18, -13) to eliminate p from the equation:
(-13 - k)^2 = 4p(18 - 18)
(-13 - k)^2 = 0
We can now solve the two equations simultaneously:
(-11 - k)^2 = -48p
(-13 - k)^2 = 0
Expanding and simplifying the first equation:
121 + 22k + k^2 = -48p
Since the second equation is equal to 0, we can solve it:
(-13 - k)^2 = 0
k = -13
Now we substitute k = -13 into the first equation:
121 + 22(-13) + (-13)^2 = -48p
121 - 286 + 169 = -48p
4 = -48p
p = -4/48
p = -1/12
Step 3: Write the equation of the parabola:
Using the collected information, we can write the standard form equation of the parabola:
(y + 13)^2 = -4/12(x - 18)
Simplifying further:
(y + 13)^2 = -(1/3)(x - 18)
Therefore, the standard form equation of the parabola is (y + 13)^2 = -(1/3)(x - 18).
2) The standard form equation of a parabola opening left or right is also given by: (y - k)^2 = 4p(x - h), where (h, k) represents the coordinates of the vertex.
Given the vertex is (10, -10), we need to determine the value of k.
To eliminate p, we will substitute the point (-26, -7) into the equation:
(-7 - k)^2 = 4p(-26 - 10)
(-7 - k)^2 = 4p(-36)
(-7 - k)^2 = -144p
Now, substitute the vertex coordinates into the equation we obtained:
(-10 - k)^2 = 4p(10 - 10)
(-10 - k)^2 = 0
Solving the two equations simultaneously:
(-7 - k)^2 = -144p
(-10 - k)^2 = 0
Expanding and simplifying the first equation:
49 + 14k + k^2 = -144p
As the second equation is equal to 0, we can solve it:
(-10 - k)^2 = 0
k = -10
Now substitute k = -10 into the first equation:
49 + 14(-10) + (-10)^2 = -144p
49 - 140 + 100 = -144p
9 = -144p
p = -1/16
Therefore, the equation of the parabola is:
(y + 10)^2 = -1/16(x - 10)