Consider the parabola that passes through the points (2,6)(-1,12) and (0,6)
Find the equation and the vertex
To find the equation of the parabola, we need to use the vertex form of a parabola equation, which is given by:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
Given three points (2,6), (-1,12), and (0,6), we can use these points to form three equations and solve for the values of a, h, and k.
1. Using the point (2,6):
6 = a(2 - h)^2 + k -- Equation 1
2. Using the point (-1,12):
12 = a(-1 - h)^2 + k -- Equation 2
3. Using the point (0,6):
6 = a(0 - h)^2 + k -- Equation 3
To solve these equations, we substitute the given x and y values into each equation.
1. Using the point (2,6):
6 = a(2 - h)^2 + k -- Equation 1
6 = a(4 - 4h + h^2) + k
2. Using the point (-1,12):
12 = a(-1 - h)^2 + k -- Equation 2
12 = a(1 + 2h + h^2) + k
3. Using the point (0,6):
6 = a(0 - h)^2 + k -- Equation 3
6 = ah^2 + k
Now we have three equations with three unknowns (a, h, k). We can solve these equations simultaneously to find their values.
Subtract equation 3 from equation 1:
0 = 4a - ah^2 -- Equation 4
Subtract equation 3 from equation 2:
6 = 2ah^2 + 2k -- Equation 5
Multiply equation 4 by -2:
0 = -8a + 2ah^2 -- Equation 6
Add equation 5 and equation 6:
6 = 2ah^2 - 8a + 2k
Simplify the equation above:
6 = 2a(h^2 - 4) + 2k
Divide the equation by 2:
3 = ah^2 - 4a + k -- Equation 7
Now we have two equations left (Equation 4 and Equation 7) with two unknowns (a and h). We can try to solve for a and h.
From Equation 4:
0 = 4a - ah^2
Rearrange the equation:
ah^2 = 4a
Divide both sides by a:
h^2 = 4
Taking the square root of both sides:
h = ±2
Substitute h = 2:
2a = 4
a = 2/2
a = 1
Substitute h = -2:
2a = 4
a = 4/2
a = 2
So, we have two possible values for a: a = 1 and a = 2.
Now substitute the value of a and h in Equation 7 to solve for k.
For a = 1:
3 = (1)(2^2) - 4(1) + k
3 = 4 - 4 + k
3 = 0 + k
k = 3
For a = 2:
3 = (2)(2^2) - 4(2) + k
3 = 8 - 8 + k
3 = 0 + k
k = 3
Therefore, we found that both values of a (1 and 2) give the same value of k (3). So, the equation of the parabola is:
y = a(x - h)^2 + k
y = a(x - 2)^2 + 3
The vertex of the parabola is given by (h, k), so the vertex is (2, 3).
To find the equation and vertex of a parabola that passes through three given points, you can use the standard form equation of a parabola: y = ax^2 + bx + c.
Since the parabola passes through the points (2, 6), (-1, 12), and (0, 6), we can substitute these points into the equation to get a system of three equations.
1) At point (2, 6): 6 = a(2)^2 + b(2) + c
Simplifying, we get: 4a + 2b + c = 6
2) At point (-1, 12): 12 = a(-1)^2 + b(-1) + c
Simplifying, we get: a - b + c = 12
3) At point (0, 6): 6 = a(0)^2 + b(0) + c
Simplifying, we get: c = 6
Now we have a system of three equations with three unknowns (a, b, c):
4a + 2b + c = 6 [Equation 1]
a - b + c = 12 [Equation 2]
c = 6 [Equation 3]
Substituting Equation 3 into Equations 1 and 2, we can simplify the system:
4a + 2b + 6 = 6 [Equation 1]
a - b + 6 = 12 [Equation 2]
Simplifying Equation 1, we get: 4a + 2b = 0
Simplifying Equation 2, we get: a - b = 6
Now, we have a system of two equations with two unknowns (a, b):
4a + 2b = 0 [Equation 4]
a - b = 6 [Equation 5]
Solving equations 4 and 5 simultaneously, we can find the values of a and b.
From Equation 5, we can express a in terms of b:
a = b + 6
Substituting this into Equation 4, we get:
4(b + 6) + 2b = 0
4b + 24 + 2b = 0
6b = -24
b = -4
Now, substitute the value of b into Equation 5 to find the value of a:
a - (-4) = 6
a + 4 = 6
a = 2
We have found the values of a and b, which are a = 2 and b = -4.
Finally, substitute these values back into Equation 3 to find the value of c:
c = 6
Therefore, the equation of the parabola that passes through the given points is:
y = 2x^2 - 4x + 6
To find the vertex of the parabola, you can use the formula x = -b / (2a).
In this case, a = 2 and b = -4.
x = -(-4) / (2 * 2)
x = 4 / 4
x = 1
To find the corresponding y-value, substitute the value of x into the equation:
y = 2(1)^2 - 4(1) + 6
y = 2 - 4 + 6
y = 4
Therefore, the vertex of the parabola is (1, 4).