find the equation of a parabola with vertex on line y = 2x, axis parallel to the y axis and passing through (3/2, 1) and (3, 4).
To find the equation of a parabola, we need to determine the values of the coefficients in the general form of a parabola equation, which is y = ax^2 + bx + c.
Let's start by finding the vertex of the parabola. Since the vertex is on the line y = 2x, we can equate the y-values and solve for x.
2x = ax^2 + bx + c
Next, we'll use the given points to form two equations. The point (3/2, 1) gives us:
1 = a(3/2)^2 + b(3/2) + c
Simplifying further:
1 = (9/4)a + (3/2)b + c
Similarly, using the point (3, 4):
4 = a(3)^2 + b(3) + c
Simplifying:
4 = 9a + 3b + c
Now we have a system of equations with three variables (a, b, and c):
2a - b = 0 (from equating the y-values)
(9/4)a + (3/2)b + c = 1
9a + 3b + c = 4
To solve this system, we can eliminate the variable c. By subtracting the third equation from the second, we get:
-1/4 a - 3/2 b = -3
Now we have two equations to work with:
2a - b = 0
-(1/4)a - (3/2)b = -3
Multiplying the second equation by 4 to eliminate fractions:
-a - 6b = -12
Now we have a system of two equations with two variables:
2a - b = 0
-a - 6b = -12
We can solve this system to find the values of a and b. Multiplying the first equation by 6 and then adding both equations, we have:
12a - 6b = 0
-a - 6b = -12
11a = -12
a = -12/11
Substituting the value of a into the first equation, we can solve for b:
2(-12/11) - b = 0
-24/11 - b = 0
b = -24/11
Now that we have the values of a and b, we can substitute them back into one of the original equations to solve for c. Let's use the second equation:
(9/4)(-12/11) + (3/2)(-24/11) + c = 1
-27/11 + (-36/11) + c = 1
-63/11 + c = 1
c = 1 + 63/11
c = 11/11 + 63/11
c = 74/11
Now we have the values of a (-12/11), b (-24/11), and c (74/11).
Therefore, the equation of the parabola is:
y = (-12/11)x^2 - (24/11)x + (74/11)
To find the equation of a parabola, we need to determine the coefficients that define it. The equation of a general parabola in vertex form is given by:
y = a(x-h)^2 + k
where (h, k) represents the vertex of the parabola. In this case, the vertex lies on the line y = 2x. We can substitute this equation into our general form to express the vertex as (h, 2h).
Now, let's consider the given points (3/2, 1) and (3, 4) that the parabola passes through. Substituting these points into the equation y = a(x-h)^2 + k, we get the following two equations:
1 = a((3/2)-h)^2 + k Equation 1
4 = a(3-h)^2 + k Equation 2
We have two equations with two unknowns (a and h). We can solve this system of equations to find the values of a and h.
First, let's solve for k. We can substitute the vertex (h, 2h) into Equation 1:
1 = a((3/2)-h)^2 + 2h
Expanding, we get:
1 = a((9/4) - 3h + h^2) + 2h
1 = (9a/4) - 3ah + ah^2 + 2h
Rearranging, we have:
ah^2 + (-3a + 2)h + (9a/4 - 1) = 0
Now, we can solve for h using the quadratic formula:
h = [-(-3a + 2) ± √((-3a + 2)^2 - 4a(9a/4 - 1))]/(2a)
Simplifying:
h = [3a - 2 ± √((9a^2 - 12a + 4) - (9a^2 - 4a))/2a
h = [3a - 2 ± √(4a)])/2a
h = (3a - 2 ± 2√a)/2a
h = (3a - 2)/2a ± sqrt(a)/a
Now, let's substitute this value of h back into Equation 1 to solve for a:
1 = a((3/2)-[(3a-2)/2a])^2 + (2a - 2)/a [Plug in the value of h]
Expanding:
1 = a((3 - (3a - 2))/(2a))^2 + (2a - 2)/a
1 = a((6 - 3a + 2)/(2a))^2 + 2(a - 1)/a
1 = a((8 - 3a)/(2a))^2 + 2(a - 1)/a
Now, let's simplify this equation further.
1 = a((8 - 3a)/(2a))^2 + 2(a - 1)/a
1 = a(8 - 3a)^2/(4a^2) + 2(a - 1)/a
1 = ((8 - 3a)^2 + 2(a - 1)(4a^2))/(4a^2)
1 = (64 - 48a + 9a^2 + 8a^3 - 8a^2 + 8a^3 - 8a^2)/(4a^2)
Now, we have a polynomial equation of degree 3. To calculate its roots and determine the values of a, we can use numerical methods or graphing technology. Once we find the value(s) of a, we can substitute it back into Equation 1 or Equation 2 to solve for h and k.
Unfortunately, the calculations involved in finding the equation of the parabola using this method are quite complex and lengthy. It may be more efficient to use graphing software or online tools to find the equation of the parabola based on the given points and conditions.