how do i find the equaion of the following parabolas?
1)intercepts at (2,0),(8,0) and (0,48)
2)turning point at (1,-2) and passing through the origin.
can someone please teach me how to do this??thanks!
The first: Write you standard equation of a parabola (it has two x intercepts)
y= h(x-b)^2 + k
You have three points.
Put those in the equations, you then have three equations, solve for h, b, k.
Example (first point)
0=h(2-b)^2 +k
second point
0=h(8-b)^2 + k
Third point
48=hb^2 + k
solve for h, b k There will be a lot of algebra here. A trick is to note it is symetrically about x=5, so b=5
then from the first equation k=-9h then putting that in the third,
48= h25-9h or h= 4
solved.
You can also simplify matters by making use of the location of the zeroes. You know that they are at x = 2 and x = 8, so the function must be of the form:
y(x) = A (x-2)(x-8)
To find A you use that y(0) = 48 --->
A = 3
Suppose you are given three points (x1,y1), (x2,y2) and (x3,y3) and y1, y2 and y3 are not zero. Then this trick doesn't work. However, you can still make this trick work as follows. Instead of the function y(x) you consider the function z(x) defined s:
z(x) = y(x) - y1.
Then z(x1) = 0, z(x2) = y2-y1 and
z(x3)=y3-y1
Because z(x) is zero at x = x1, you know that it contains a factor (x-x1). If you divide z(x) by (x-x1) you'll get a linear function. So, we define:
h(x) = z(x)/(x-x1).
Then h(x2)= (y2-y1 )/(x2-x1)
and
h(x3)= (y3-y1 )/(x3-x1)
h(x) is linear, you know it's value at two points, so you can easily determine it.
good post.
To find the equation of a parabola with given information, we can use several methods.
For the first parabola with intercepts at (2,0), (8,0), and (0,48), we can use the standard form of a parabola equation, which is y = ax^2 + bx + c.
Substituting the x and y values of each point into the equation, we get:
(1) When x = 2, y = 0:
0 = 4a + 2b + c.
(2) When x = 8, y = 0:
0 = 64a + 8b + c.
(3) When x = 0, y = 48:
48 = c.
Now we have a system of three equations with three unknowns (a, b, and c), and we can solve the system to find the values of a, b, and c.
Solving equations (1) and (2) simultaneously, we can eliminate c and solve for a and b:
(4) 64a + 8b = -2(4a + 2b) [multiplied equation (1) by -2]
64a + 8b = -8a - 4b
72a + 12b = 0. [added 8b to both sides]
6a + b = 0. [divided both sides by 12]
Using equation (3), we have c = 48.
Substituting the values of a, b, and c back into the standard equation, we get:
y = ax^2 + bx + c
y = 6x^2 - x + 48.
Therefore, the equation of the first parabola is y = 6x^2 - x + 48.
For the second parabola with the turning point at (1,-2) and passing through the origin, we can use the vertex form of the parabola equation, which is y = a(x-h)^2 + k.
Since the turning point is (1,-2), we have h = 1 and k = -2.
Substituting these values into the vertex form equation, we get:
y = a(x-1)^2 - 2.
Since the parabola passes through the origin (0,0), we substitute x = 0 and y = 0 into the equation:
0 = a(0-1)^2 - 2,
0 = a - 2,
a = 2.
Thus, the equation of the second parabola is y = 2(x-1)^2 - 2.
Therefore, the equation of the second parabola is y = 2(x-1)^2 - 2.
To find the equation of the first parabola, we can use the vertex form of a parabola equation:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola and a is a constant representing the rate of curvature.
We are given the x-intercepts at (2, 0) and (8, 0) and another point (0, 48). Let's start by finding the vertex coordinates:
The x-coordinate of the vertex can be found by taking the average of the x-intercepts:
hx = (2 + 8) / 2 = 5
Now let's substitute this value into one of the x-intercepts equations to find the corresponding y-coordinate:
0 = a(2 - 5)^2 + k
0 = 9a + k
Since we also have the point (0, 48), we can substitute these values into the equation above:
48 = 9a + k
Now we have two equations:
0 = 9a + k
48 = 9a + k
We can solve this system of equations by subtracting the first equation from the second:
48 - 0 = 9a + k - (9a + k)
48 = 0
This is not possible, so there must have been a mistake in the information provided for the first parabola. Please double-check the given points and retry finding the equation.
Moving on to the second parabola:
We are given the turning point at (1, -2) and that it passes through the origin (0, 0). Let's find the equation of this parabola.
Since the parabola passes through the origin (0, 0), we can substitute these values into the vertex form equation and solve for a:
0 = a(0 - 1)^2 + (-2)
0 = a + (-2)
a = 2
Now we can substitute the vertex coordinates (1, -2) and the value of a into the vertex form equation:
y = 2(x - 1)^2 - 2
So, the equation of the second parabola is y = 2(x - 1)^2 - 2.