Find the focus and directrix for the parabola y=ax^2. What special meaning does the expression 1/(4a) have?
I write a parabola that is upright (holds water) as
(x-h)^2 = 4b (y -k)
where my 4 b is your 1/a and h and k are zero so vertex at (0,0)
vertex is at (0,0)
vertex to focus = b = 1/(4a) so focus at ( 0, 1/(4a) )
vertex to directrix = b but below so at y = -1/(4a)
Oh, we're talking about parabolas? Now that's a fun topic! The focus and directrix of the parabola y=ax^2 can be determined using some mathematical meanderings.
The focus of the parabola is given by the point (0, 1/(4a)), where "a" is the coefficient of x^2. So, you can imagine this focus sitting at the vertex, waving at all the points on the parabola.
Now, let's talk about the directrix. It's like a boundary line that the parabola never crosses. The directrix for this parabola is given by the equation y = -1/(4a). So, it's this imaginary line where the parabola says, "Sorry, directrix, I can't cross you. It's against my parabolic principles!"
As for what the expression 1/(4a) means, well, it holds a special place in the parabolic world. It gives us the distance from the vertex to the focus OR from the vertex to the directrix. So, it's like a secret code that tells us how far these points are from the vertex. It's quite parrific, isn't it?
So, there you have it, the focus, directrix, and a little parabolic humor to brighten up your day!
To find the focus and directrix of the parabola y = ax^2, where a is a constant, we can use the standard form of a parabola equation:
(y - k) = 4a(x - h)^2, where the vertex is at (h, k).
Comparing this with the given equation y = ax^2, we can see that the vertex is at (0, 0).
The focus of the parabola is given by (h, k + 1/(4a)) and the directrix is the vertical line y = k - 1/(4a).
Since the vertex is at (0, 0), the focus will be at (0, 1/(4a)) and the directrix will be the horizontal line y = -1/(4a).
Now, let's discuss the special meaning of the expression 1/(4a). In the standard form of a parabola equation, the coefficient of (x - h)^2 is equal to 4p, where p is the distance between the focus and the vertex. In this case, p = 1/(4a), which means that the distance between the vertex and the focus is 1/(4a). This value represents the focal length of the parabola.
To find the focus and directrix of the parabola y = ax^2, we can use the standard form equation of a parabola: (x-h)^2 = 4p(y-k), where (h,k) is the vertex and p is the distance from the vertex to the focus or directrix.
In this case, since the equation is y = ax^2, we can rewrite it as (x-0)^2 = 4a(1/a)y, which simplifies to x^2 = 4ay.
From this form, we can see that the vertex is located at the origin (h = 0, k = 0). Now, the coefficient of y is 4a, so we can determine that p = 1/(4a).
The sign of a determines the orientation of the parabola. If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards.
To find the focus and directrix:
1. Focus: The focus is located on the axis of symmetry and is at a distance p from the vertex. Since the vertex is at the origin, the focus is at the point (0, p). In this case, the coordinates of the focus are (0, 1/(4a)).
2. Directrix: The directrix is a horizontal line perpendicular to the axis of symmetry and is also at a distance p from the vertex in the opposite direction of the focus. For a parabola opening upwards, the directrix is given by the equation y = -p. The equation of the directrix in this case is y = -1/(4a).
Now, regarding the expression 1/(4a), it represents the distance between the vertex and the focus or directrix. It is significant because it determines the shape and position of the parabola. If the value of 1/(4a) is smaller, the parabola will be narrower, and if it is larger, the parabola will be wider. Additionally, the reciprocal, 4a, represents the coefficient of y in the standard form equation, which determines the steepness of the parabola.
Therefore, the expression 1/(4a) holds the key to understanding the position and characteristics of the parabola y = ax^2.