Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and dire

Identify the coordinates of the vertex and focus, the...

Key Concepts

Directrix

The directrix is a fixed line used in the definition of a parabola and is located opposite the focus relative to the vertex. For any point on the parabola, the distance to the focus is equal to its distance to the directrix. This concept is essential in understanding the reflective and geometric properties of parabolic curves.

Axis of Symmetry

The axis of symmetry is a straight line that passes through the vertex of the parabola and divides it into two mirror-image halves. Its orientation (vertical or horizontal) is determined by the form of the equation, and it provides a line about which the parabola is symmetric.

Direction of Opening

The opening direction of a parabola is determined by the coefficient of the squared term in its standard form. A positive coefficient generally indicates the parabola opens in one direction (upward for vertical, rightward for horizontal), while a negative coefficient indicates it opens in the opposite direction (downward for vertical, leftward for horizontal).

Vertex

The vertex of a parabola is the point where the curve changes direction and is the minimum or maximum point of the parabola. In the standard form, the vertex (h, k) serves as the central reference point for determining other aspects of the parabola, such as the focus, axis of symmetry, and directrix.

Parabola Standard Form

Parabolas can be expressed in a standard form which makes it easy to identify their key features. For a parabola that opens horizontally, the equation is typically written as x = a(y - k)^2 + h, where (h, k) represents the vertex. This form helps in extracting properties like the vertex, focus, and directrix directly from the equation.

Focus

The focus is a fixed point located within the parabola that, together with the directrix, defines the curve. According to the parabola’s geometric definition, each point on the parabola is equidistant from the focus and the directrix. The distance from the vertex to the focus is crucial in determining the parabola’s shape.

Latus Rectum

The latus rectum is a line segment that passes through the focus and is perpendicular to the axis of symmetry. It provides a measure of the width of the parabola at the level of the focus, and its length is directly related to the distance between the vertex and the focus. This property is important in the precise geometrical description of the parabola.

Transcript

00:01 For the equation of the parabola, x equals negative 3y squared plus 1, we want to find the coordinates of the vertex and the focus, the equations of the directrix, and the axis of symmetry, and the direction of the opening.

00:18 Then we're going to find the length of the lattice rectum, and we're going to graph the parabola.

00:22 So to begin with, the coordinates of the vertex, well, h and k are the coordinate are the vertex.

00:33 So h for this one, since it's x equals, the h is 1.

00:38 So our vertex would be the point 1, and then y minus what squared, 0.

00:48 There we go.

00:49 So that's the vertex.

00:51 Then to find the focus, we take h plus 1 over 4a, okay.

00:56 Our a is negative 3, so our focus is going to be 1 plus 1 over 4 times negative 3.

01:10 So negative 1 12th, so 1 minus 1 12th, our focus is going to be 1112th and 0.

01:24 The equation of the directrix.

01:27 Well, the directrix is just going to be.

01:30 Instead of 1 plus 1 over 4a, it's 1 minus.

01:38 It's going to be x equals 1 plus 112th.

01:46 So x equals 1 and 112th.

01:50 That's the equation of the directrix.

01:54 Then the axis of symmetry.

01:57 For this equation, it's just the equation, y equals k well our k is zero so y equals zero that's the axis of symmetry so we've got our vertex we've got our focus we've got our directrix and our axis of symmetry and the direction of the opening since a is less than zero then this opening is going to be to the left so it opens to the left...

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