Write the equation of the parabola with vertex at (2,1), axis the line y-1=0 and passes through (5,0).
parabola with vertex at (2,1) ... y = a(x-2)^2 + 1
axis the line y-1=0 ... oops, a horizontal parabola ... x = a(y-1)^2 + 2
passes through (5,0)
a(0-1)^2 + 2 = 5
a = 3
So, we are left with
x = 3(y-1)^2 + 2
Why did the parabola go to the comedy club? Because it wanted to find its vertex!
To find the equation of the parabola, we first need to determine the equation of its directrix, and the distance from the vertex to the directrix, which is also the distance from the vertex to the focus point.
1. The axis of the parabola is the line y - 1 = 0, which can be rewritten as y = 1. Since the vertex is at (2,1), this means the directrix is the line y = 0.
2. Determine the distance from the vertex to the directrix (and the focus point). This distance is denoted as "p." Since the focus point is above the vertex, p is positive.
The distance from the vertex (2,1) to the directrix (line y = 0) is 1 unit. Therefore, p = 1.
3. The general equation of a parabola with a vertical axis and vertex at (h,k) is given by:
(y - k) = 4p(x - h)
Plugging in the values for the vertex (h, k) and the distance p, we have:
(y - 1) = 4(1)(x - 2)
Simplifying the equation:
y - 1 = 4x - 8
4. Rearranging the equation to its standard form:
y = 4x - 7
Therefore, the equation of the parabola with vertex (2,1), axis the line y - 1 = 0 and passing through (5,0), is y = 4x - 7.
To determine the equation of a parabola, you need to know its vertex and a point that lies on the parabola. In this case, you are given the vertex at (2, 1) and a point that the parabola passes through, which is (5, 0).
Step 1: Determining the equation of the axis of symmetry.
The axis of symmetry is a vertical line that passes through the vertex of the parabola. Since the vertex is given as (2, 1), the equation of the axis of symmetry is x = 2.
Step 2: Determining the equation of the parabola.
The equation of a parabola can be written in the form: (x - h)^2 = 4p(y - k), where (h, k) represents the vertex and p represents the distance between the vertex and the focus of the parabola.
In this case, the vertex is (2, 1). Therefore, we can rewrite the equation as:
(x - 2)^2 = 4p(y - 1).
Step 3: Finding p.
To find the value of p, we need to use the fact that the parabola passes through the point (5, 0). By substituting this point into the equation, we can solve for p.
(5 - 2)^2 = 4p(0 - 1)
3^2 = -4p
9 = -4p
p = -9/4
Step 4: Writing the final equation of the parabola.
Now that we have the value of p, we can substitute it back into our equation:
(x - 2)^2 = 4(-9/4)(y - 1)
Simplifying further:
(x - 2)^2 = -9(y - 1)
Therefore, the equation of the parabola with a vertex at (2, 1), axis the line y - 1 = 0, and passes through (5, 0) is (x - 2)^2 = -9(y - 1).