Find the equation of the parabola described. Find the two points that define the latus rectum, a

Find the equation of the parabola described. Find the two...

Key Concepts

Standard Form Equation

The standard form of a parabola’s equation is derived from its definition involving distances from the focus and directrix. Depending on whether the parabola opens vertically or horizontally, the equation is expressed in a form that clearly reveals the vertex, axis of symmetry, and the parameter that governs its width.

Latus Rectum

The latus rectum is a line segment that passes through the focus of the parabola and is perpendicular to the axis of symmetry. Its endpoints lie on the parabola, and its length is directly related to the distance between the vertex and the focus, providing insight into the parabola's width at the level of the focus.

Vertex

The vertex of a parabola is the point where it attains its minimum or maximum value, depending on its orientation. It is located exactly halfway between the focus and the directrix, serving as an essential reference point for the parabola’s symmetry and for writing its equation in standard form.

Parabola

A parabola is a conic section defined as the set of all points that are equidistant from a fixed point, called the focus, and a fixed line, known as the directrix. This unique geometric property leads to its characteristic shape and allows for the derivation of its algebraic equation.

Focus and Directrix

The focus is a specific point that, in conjunction with the directrix (a fixed line), determines the shape and position of the parabola. The distance from any point on the parabola to the focus is equal to its perpendicular distance to the directrix, a defining property used in formulating its equation.

Transcript

00:03 For this problem, we're given a parabola with a focus at the coordinates negative 4 ,4, and a directrix line at y equals negative 2.

00:12 From here, we need to figure out the vertex of our parabola, its equation, the points that define the lattice rectum, before finally sketching out the graph.

00:22 So to begin, i'm going to plot the points of our focus and our directrix line.

00:27 So our focus is at negative 4, 4, and our directrix line is at y equals negative.

00:35 2.

00:41 So we can figure out the vertex from here as we know that it will be halfway between our focus and our directrix line.

00:49 So our focus and our directrix go from negative 2 all the way up to 4.

00:55 This is a value of 6 units between them.

00:58 So we know that our vertex is going to be three units either direction.

01:03 So it's going to be also the same x distance as our focus.

01:09 So our vertex, is going to be right there at negative 4 .1.

01:19 From here, we can figure out the equation of our parabola.

01:23 And because our focus is above our vertex, we know that our parabola is going to open upwards.

01:32 So for our equation, we are going to use the format of x minus h squared is equal to 4a times y minus k.

01:50 We can plug in the values of h and k by using the coordinates of our vertex.

01:56 Our h value is going to be the x value of our vertex, and our k value will be the y value...

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