Find the vertex, focus, and directrix of the parabola given by each equation. Sketch the graph.

Find the vertex, focus, and directrix of the parabola given...

Key Concepts

Parabola

A parabola is a type of conic section that can be 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). It has a symmetric shape and its orientation (opening upward, downward, left, or right) depends on the coefficients in its equation.

Vertex

The vertex of a parabola is the point where the curve changes direction and is exactly midway between the focus and the directrix. In the vertex form of a parabola's equation, the vertex serves as a reference point for transformations and defining the geometry of the parabola.

Focus

The focus is a fixed point used in the definition of a parabola. Every point on the parabola is equidistant from this focus and the directrix. The position of the focus relative to the vertex helps determine the 'width' or 'narrowness' of the parabola.

Directrix

The directrix is a fixed line that, together with the focus, defines a parabola. Each point on the parabola is equidistant from the focus and the directrix. Its position relative to the vertex provides additional geometric information about the parabola’s orientation and shape.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into its vertex form. This form makes it easier to identify the vertex of the parabola and, consequently, helps in finding the focus and directrix by reorganizing and simplifying the original equation.

Transcript

00:01 Y is equal to here x square plus 6x plus 10 and so here this can be written as here that is x square plus 6x and this becomes y minus 10 and this is written in the form that is x x x squared plus 2 multiplied x multiplied 3 and we add 9 on both sides so this equation becomes here and this can be written as here x plus 3 whole square equals to y and this is minus 1 and in a standard form we write here the equation parabola that is x x minus 3 whole square is equals to here 4 multiplied 1 by 4 multiplied here y minus 1 so we know that the equation of parabola in a standard form in a standard form that is x minus h whole a square equals to 4p multiplied y minus k so this is the standard form of equation of parabola we compare here that is p here 1 by 4 that is k here minus 1 and h is here minus 3 so here the given point h equals to minus 3 p equals to 1 4 and k equals to here 1 so vertex becomes here vertex that is h k k equals to minus 3 1 and here focus f equals to h k plus p and the focus here that is minus 3 and this becomes here that is minus 3 1 plus 1 by 4 this becomes here 5 by 4 so focus is here and equation of direct place that is y equals to here that is k minus p so this becomes 1 minus 1 by 4 that is becomes here that is y equals to that is 1 minus 1 bar 4 this becomes here 3 by 4 this is equation of directics so we draw the graph here of the given equations here this is x xx6 this is negative x -tax this is y and this is a negative direction of y this is taken to be origin here and we do know that y equals to here 3 by 4 that is nearly to 0 .75 0 .y equals 2 so we just like this point here.

02:35 So this 0 point something here.

02:38 So this is the equation of directrics here.

02:42 That is here.

02:43 Y is equal to 3 by 4.

02:47 This is the equation of directrics here.

02:50 And this is origin.

02:51 So we take the point here that is minus 3 minus 4.

02:55 So this is 1.

02:57 This is 2 and this is 3.

02:59 This is minus 1.

03:00 This is here minus 2, this is minus 3, this is 1, 2, 3, 4 and this is minus 1, this is minus 2, minus 3 and minus 4.

03:12 Take the point.