Sketching a Parabola In Exercises 11-16, find the vertex, focus, and directrix of the parabola,

Sketching a Parabola In Exercises 11-16, find the vertex,...

Key Concepts

Graphing and Orientation

Graphing a parabola involves plotting its vertex, focus, and directrix, then sketching the curve as a smooth, symmetric arc. The orientation—whether it opens upwards, downwards, leftwards, or rightwards—is determined by the form of the equation and the sign of the parameter p. Recognizing the direction of opening helps in accurately drawing the parabola and understanding its behavior in the coordinate plane.

Focus and Directrix

The focus and directrix are defining features of a parabola, based on its geometric definition as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The focal distance, represented by p, determines how far the focus lies from the vertex along the axis of symmetry, while the directrix is found an equal distance on the opposite side. These elements are crucial for understanding the parabola’s reflective properties and its structural characteristics.

Completing the Square

Completing the square is an algebraic technique used to rewrite quadratic expressions in a way that reveals key features such as the vertex of a parabola. This method involves adding and subtracting a number that makes the quadratic term a perfect square trinomial, allowing the expression to be rewritten in a translated form. It is a critical step when converting a general conic section equation into its standard form, which eases the identification of geometric properties.

Vertex Form of a Parabola

The vertex form of a parabola is a rewritten equation that makes the vertex—the highest or lowest point—explicit. For a parabola that opens vertically, the equation is typically expressed as (x - h)² = 4p(y - k), where (h, k) is the vertex and p is a parameter related to the distance between the vertex and the focus. This form is especially useful for graphing as it immediately reveals the translation and stretching of the basic parabola.

Transcript

00:01 So we have this equation of a parabola x squared plus 4x plus 4y minus 4 equals 0.

00:08 And we need to find the vertex focus in directrix and then use this information to sketch the graph.

00:16 So the first thing we need to do is write this equation in standard form for a parabola.

00:21 And there's two possible equations for standard form.

00:25 First we could have x minus h all squared is equal to.

00:34 4p times y minus k or we could have y minus k all squared is equal to 4p times x minus h so if you look at our equation you see that we can see that we have an x squared term here but there's no y squared term so we're going to end up using this first standard form so we need to find a way to to write this equation in that.

01:19 So we start by isolating our x terms, so we get x squared plus 4x, and we move the other terms to the other side, equals negative 4y plus 4.

01:36 And then we need to complete the square to write this as a trinomial square that we can then factor in this form.

01:43 So we do that by taking half the coefficient on the x term, and we square that, and we add to the left side.

01:51 So we have x squared plus half of four is two, two squared is four.

01:58 So we're going to add four to the left side.

02:12 And then we have to add the four to the right side as well so that the value of the equation is the same.

02:20 It's still a true statement.

02:22 So then we factor this as a trinomial square.

02:25 Just get x plus 2, all squared equals negative 4y plus 8.

02:40 Which is simply negative 4 times y minus 2, factor out the negative 4.

02:51 So then if we compare that to our standard form, you see that our vertex would be at negative 2 because we're subtracting negative 2 is the same as adding 2, negative 2, 2, 2.

03:04 So, and for the focus, the x coordinate will be the same in this case, because we have the x minus 8 squared 4.

03:23 And the y coordinate will be the y coordinate of the vertex plus p and here we have 4 p times y minus k so here and here we have negative 4 so that's 4 times negative 1 so we got p is negative 1 so our focus will be at negative 2 and 2 minus 1 it's negative 2 and 1 it's negative 2 and 1 and our directrix will be a horizontal line...

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