Problem 11

Question

For Problems $1-30$, find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ y^{2}=-8(x-3) $$

Step-by-Step Solution

Verified

Answer

Vertex: (3,0); Focus: (1,0); Directrix: x=5.

1Step 1: Recognize the Form of the Equation

The given equation is $ y^2 = -8(x-3) $. This is a horizontal parabola, as it can be rewritten in the standard form $ (y-k)^2 = 4p(x-h) $, where the vertex is $(h, k)$.

2Step 2: Determine the Vertex

The equation $ y^2 = -8(x-3) $ can be rewritten as $ (y - 0)^2 = -8(x - 3) $. Thus, the vertex $(h, k)$ of the parabola is $(3, 0)$.

3Step 3: Find the Value of $p$

In the equation $ (y-k)^2 = 4p(x-h) $, compare it with $ y^2 = -8(x-3) $ to identify $ 4p = -8 $. Solving for $ p $ gives $ p = \frac{-8}{4} = -2 $.

4Step 4: Find the Focus

Since $ p = -2 $ and the vertex is $(3, 0)$, the focus, which is $ p $ units from the vertex in the horizontal direction, is at $ (3 + p, 0) = (1, 0) $.

5Step 5: Determine the Directrix

The directrix of a horizontal parabola is a vertical line. It is $ p $ units to the opposite side of the vertex from the focus. Thus, it is at $ x = 3 - (-2) = 5 $.

6Step 6: Sketch the Parabola

Plot the vertex at $(3, 0)$, the focus at $(1, 0)$, and draw the directrix as the vertical line $ x = 5 $. The parabola opens to the left, given that $ p < 0 $.

Key Concepts

VertexFocus of a ParabolaDirectrix

Vertex

In the world of parabolas, the vertex is an essential concept. The vertex is the point on the parabola that represents its highest or lowest point. This is where the parabola changes direction. For horizontal parabolas, like the one in the given equation, the vertex captures the central balance of the curve. It is the "turning point," located at the coordinates

Finding the Vertex
- Parabola equations can often be written in the form \[(y-k)^2 = 4p(x-h)\] This shows that the vertex follows the $(h, k)$ format.- For the equation $ y^2 = -8(x-3) $, we rewrite it as $(y-0)^2 = -8(x-3)$.- Comparing, the vertex $ (h, k) $ is $(3, 0)$.

Understanding the vertex helps in graphing, as it provides a reference to locate the rest of the parabola's components. It also tells us where the focus and directrix will relate spatially.

Focus of a Parabola

The focus of a parabola is a special point that lies inside the curve. It serves as a point to which all points on the parabola have certain reflective properties. This point, combined with the directrix, helps define the parabola's unique shape.

Calculating the Focus
- For horizontal parabolas, the focus is found by adding $ p $ units to the $ x $-coordinate of the vertex.- Given that $ y^2 = -8(x-3) $, we found $ p = -2 $.- Hence, the focus is located at $ (3 + (-2), 0) = (1, 0) $.

The focus affects the direction and "openness" of the parabola.Its positioning is crucial, as it'll always lie on the opposite side of the directrix relative to the vertex. Knowing how to find the focus helps in both graphing and understanding the parabola's properties.

Directrix

In the anatomy of a parabola, the directrix is a line that complements the focus. Together, they share a fundamental connection, defining the parabola. The directrix lies outside the curve.

Determining the Directrix
- The directrix of a horizontal parabola is always a vertical line.- It is placed $ |p| $ units from the vertex, but on the opposite side of the focus.- For our parabola, correcting the placement gives us the directrix at $ x = 3 + 2 = 5 $.

The directrix and the focus work in tandem. Every point on the parabola is equidistant from the focus and the directrix, resulting in its characteristic shape.Recognizing the directrix is fundamental to understanding how the parabola is created and positioned in graph form.

Problem 11