find the equation of a parabola with vertex on the line y=2x, axis parallel to the x-axis and passing through (3/2, 1) and (3,4)
since the vertex is on the line y=2x, suppose it is at (h,2h). Then
x = a(y-2h)^2 + h
Now use the points to get
a(1-2h)^2 + h = 3/2
a(4-2h)^2 + h = 3
solve to get
x = -1/18 (y-7)^2 + 7/2
x = 1/2 (y-2)^2 + 1
Well, I can certainly help you with that, but let's take a detour first. Did you hear about the mathematician who got stuck on a slippery slope while skiing? He just couldn't find the right slope intercept!
Now, back to business. To find the equation of a parabola with the given conditions, we can start by finding the vertex. Since the vertex lies on the line y = 2x, we can substitute these values into the equation to get:
y = 2x
2x = 2x
This equation holds true for all values of x. Therefore, the vertex is at (x, 2x).
Now, let's use the points (3/2, 1) and (3, 4) to form two equations. Substituting the first point into our general equation of a parabola, we get:
1 = 2(3/2)
1 = 3
Huh? That doesn't quite work. Let's try with the second point instead:
4 = 2(3)
4 = 6
Hmm, that doesn't work either. It seems there might have been an error in the problem statement or the given points. Without accurate points, we won't be able to determine the equation of the parabola.
And to "put the nail in the coffin," it's one thing to find a parabola's equation and another to find its humor equation. You might end up with a "parahahaha-bola" or a "parasilly-bola." Keep your math and humor separate, my friend!
To find the equation of a parabola with a vertex on the line y = 2x, axis parallel to the x-axis, and passing through the points (3/2, 1) and (3, 4), follow these steps:
Step 1: Determine the vertex coordinates:
Since the vertex lies on the line y = 2x, it will have the form (a, 2a) for some value of a.
Since the parabola is parallel to the x-axis, the y-coordinate of the vertex does not change.
Step 2: Substitute the vertex coordinates into the general vertex form of a parabola:
The general vertex form of a parabola is given by:
y = a(x - h)^2 + k, where (h, k) represents the vertex coordinates.
In this case, the equation becomes:
y = a(x - a)^2 + 2a.
Step 3: Use the given points to solve for the value of a:
Substitute the coordinates of each point into the equation determined in Step 2.
For point (3/2, 1):
1 = a((3/2) - a)^2 + 2a.
For point (3, 4):
4 = a(3 - a)^2 + 2a.
Step 4: Solve the equations from Step 3 for the value of a:
Solve the system of equations to find the value of a.
Solving the first equation:
1 = a((3/2) - a)^2 + 2a.
Expand and rearrange:
1 = a(9/4 - 3a + a^2) + 2a.
1 = (9a/4) - (3a^2) + (a^3) + (2a).
1 = a^3 - (3a^2) + (11a/4).
Solving the second equation:
4 = a(3 - a)^2 + 2a.
Expand and rearrange:
4 = a(9 - 6a + a^2) + 2a.
4 = (9a) - (6a^2) + (a^3) + (2a).
4 = a^3 - (6a^2) + (11a).
Step 5: Set the two equations equal to each other:
Since the two equations from Step 4 are both equal to a cubed term, we can set them equal to each other:
a^3 - (3a^2) + (11a/4) = a^3 - (6a^2) + (11a).
Step 6: Simplify and solve for a:
Rearrange the equation from Step 5 by combining like terms:
(3a^2) = (6a^2) - (11a) - (11a/4).
Multiply through by 4 to eliminate the fractions:
12a^2 = 24a^2 - 44a - 11a.
Combine the terms on the right-hand side:
12a^2 - 24a^2 + 44a + 11a = 0.
Combine like terms on both sides:
-12a^2 + 55a = 0.
Factor out a common factor of a:
a(-12a + 55) = 0.
From this, we get two possible values for a:
a = 0,
-12a + 55 = 0.
Solving the second equation for a:
-12a = -55,
a = 55/12.
Step 7: Substitute the value of a back into the equation from Step 2:
Using a = 0:
y = 0(x - 0)^2 + 0,
y = 0.
Using a = 55/12:
y = (55/12)(x - 55/12)^2 + (2)(55/12).
Simplifying the equation:
y = (55/12)(x^2 - (110/12)x + (3025/144)) + (110/6).
y = (55/12)x^2 - (275/6)x + (3025/144) + (220/12).
y = (55/12)x^2 - (330/12)x + (3025/144) + (440/12).
y = (55/12)x^2 - (330/12)x + (3465/144).
Thus, the equation of the parabola is y = (55/12)x^2 - (330/12)x + (3465/144).
To find the equation of a parabola, we can start by determining the vertex form of the equation. The vertex form equation for a parabola with vertex (h, k) is:
(y - k) = a(x - h)^2
In this case, since the vertex of the parabola lies on the line y = 2x, we know that the y-coordinate of the vertex is equal to 2 times the x-coordinate of the vertex. Therefore, we have the vertex (h, k) = (h, 2h).
Let's find the x-coordinate of the vertex first. Since the axis of the parabola is parallel to the x-axis, the x-coordinate of the vertex will remain constant. The x-coordinate of the vertex (h) is halfway between the x-coordinates of the given points on the parabola, which are (3/2, 1) and (3, 4). Therefore:
h = (3/2 + 3) / 2
h = (3/2 + 6/2) / 2
h = 9/4
Now, let's find the y-coordinate of the vertex (k). Since the vertex lies on the line y = 2x, we substitute the x-coordinate of the vertex into the equation:
k = 2h = 2 * (9/4) = 18/4 = 9/2
So the vertex of the parabola is located at (9/4, 9/2).
Next, we can use one of the given points, (3, 4), to find the value of 'a' in the vertex form equation. Plugging in the values (x, y) = (3, 4):
(4 - 9/2) = a(3 - 9/4)^2
(8/2 - 9/2) = a(3(4/4) - 9/4)^2
(-1/2) = a * (12/4 - 9/4)^2
(-1/2) = a(3/4)^2
(-1/2) = a(9/16)
(-1/2) = (9a/16)
-1 * 16 = 9a/2
-16 = 9a/2
-32 = 9a
a = -32/9
Substituting the values of (h, k) = (9/4, 9/2) and a = -32/9 into the vertex form equation, we can write the equation of the parabola as:
(y - 9/2) = (-32/9)(x - 9/4)^2
Simplifying,
9(y - 9/2) = -32(x - 9/4)^2
Therefore, the equation of the parabola with the given conditions is 9(y - 9/2) = -32(x - 9/4)^2.