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Parabolas – Vertex, Focus, Directrix (Horizontal and Vertical)
A parabola is a quadratic curve that is U-shaped and symmetric about its axis. It is defined by a quadratic equation in two variables, typically $x$ and $y$. The vertex, focus, and directrix are essential components of a parabola that help in understanding its shape and properties.
Parabolas appear in various real-world applications, including: - Optics: Parabolic mirrors and lenses are used in telescopes, microscopes, and solar concentrators to focus light.- Physics: The trajectory of projectiles, such as thrown balls or rockets, follows a parabolic path under the influence of gravity.- Engineering: Parabolic curves are used in the design of bridges, arches, and other structures to distribute loads efficiently.
The vertex of a parabola is the point where the curve changes direction, typically represented by the coordinates $(h, k)$. The vertex form of a parabola is given by:
$$y = a(x - h)^2 + k$$
where $(h, k)$ is the vertex.
The focus of a parabola is a fixed point that lies on the axis of symmetry, typically represented by the coordinates $(f, 0)$. The focus is the point around which the parabola is symmetric.
The directrix of a parabola is a line that is perpendicular to the axis of symmetry, typically represented by the equation $y = d$. The directrix is the line that the parabola approaches but never touches.
The eccentricity of a parabola is a measure of how far the focus is from the vertex, typically represented by the formula:
$$e = \frac{c}{a}$$
where $c$ is the distance from the vertex to the focus.
To solve problems involving parabolas, follow these steps:
Find the vertex, focus, and directrix of the parabola $y = 2(x - 1)^2 - 3$.
To find the vertex, we can rewrite the equation in vertex form:
$$y = 2(x - 1)^2 - 3$$
Comparing this with the vertex form $y = a(x - h)^2 + k$, we can see that the vertex is $(h, k) = (1, -3)$.
To find the focus, we need to calculate the distance from the vertex to the focus, $c$, using the formula $c = \frac{1}{4a}$:
$$c = \frac{1}{4(2)} = \frac{1}{8}$$
The focus is $(f, 0) = (1 + \frac{1}{8}, 0) = (\frac{9}{8}, 0)$.
To find the directrix, we need to calculate the equation of the directrix, $y = d$, using the formula $d = k - c$:
$$d = -3 - \frac{1}{8} = -\frac{25}{8}$$
The directrix is $y = -\frac{25}{8}$.
Find the vertex, focus, and directrix of the parabola $y = -\frac{1}{2}x^2 + 2x - 3$.
$$y = -\frac{1}{2}x^2 + 2x - 3$$
Completing the square, we get:
$$y = -\frac{1}{2}(x - 1)^2 + 1 - 3$$
$$y = -\frac{1}{2}(x - 1)^2 - 2$$
Comparing this with the vertex form $y = a(x - h)^2 + k$, we can see that the vertex is $(h, k) = (1, -2)$.
$$c = \frac{1}{4(-\frac{1}{2})} = -\frac{1}{2}$$
The focus is $(f, 0) = (1 - \frac{1}{2}, 0) = (\frac{1}{2}, 0)$.
$$d = -2 - (-\frac{1}{2}) = -\frac{3}{2}$$
The directrix is $y = -\frac{3}{2}$.
Find the vertex, focus, and directrix of the parabola $y = x^2 + 2x + 1$.
$$y = x^2 + 2x + 1$$
$$y = (x + 1)^2$$
Comparing this with the vertex form $y = a(x - h)^2 + k$, we can see that the vertex is $(h, k) = (-1, 0)$.
$$c = \frac{1}{4(1)} = \frac{1}{4}$$
The focus is $(f, 0) = (-1 + \frac{1}{4}, 0) = (-\frac{3}{4}, 0)$.
$$d = 0 - \frac{1}{4} = -\frac{1}{4}$$
The directrix is $y = -\frac{1}{4}$.
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