Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Vertex \(V(-1,0)\) focus \(F(-4,0)\)

Step-by-Step Solution

Answer

The parabola equation is (y^2 = -12(x + 1)).

Step 1: Determine the Parabola's Orientation

Since both the vertex and the focus have the same y-coordinate (0), the parabola is horizontally oriented. This means it opens either to the left or the right.

Step 2: Identify the Direction of Opening

The focus, (-4,0), is to the left of the vertex, (-1,0). Therefore, the parabola opens to the left.

Step 3: Calculate the Distance Between Vertex and Focus

The distance p, between the vertex V(-1,0) and the focus F(-4,0), is | -1 - (-4) | = 3. The value of p is negative because the parabola opens to the left, so p = -3.

Step 4: Use the Standard Form of a Parabola

For a horizontally oriented parabola, the standard form is (x - h)^2 = 4p(y - k). In this case, since it opens to the left, we have (y - k)^2 = 4p(x - h). The vertex is (h, k) = (-1, 0) and p = -3, so we substitute these values into the equation.

Step 5: Substitute the Values into the Equation

Substitute (p = -3) and (h, k) = (-1, 0) into the equation (y - k)^2 = 4p(x - h), to get (y - 0)^2 = 4(-3)(x + 1).

Step 6: Simplify the Equation

Simplify the equation to get the final form: (y^2 = -12(x + 1)).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point and serves as its peak or the lowest point, depending on its orientation. In mathematical terms, the vertex is the point \(h, k\) in the equation form of the parabola. For our problem, the vertex is given as \(V(-1, 0)\).

This point is the most important when setting up the equation of the parabola. The vertex helps in determining the overall orientation of the parabola, whether it opens vertically or horizontally. It also anchors the entire graph, serving as a central point around which the parabola is shaped. In practice, when the vertex \(h\) is combined with the parameter \(p\), fitting it into the equation helps plot and understand the shape and direction of the parabola.

For any parabola-related problems, identifying the vertex is a first substantial step. It tells us both the direction and the symmetry line of the parabola.

Focus of a Parabola

The focus of a parabola is another essential component. This is a point from which the parabolic curve is equidistant from a line called the directrix. For our given equation, the focus is \(F(-4, 0)\). This specific location affects the shape and the hyperbola's spread.

Importantly, the location of the focus relative to the vertex helps determine the parabola's orientation. It tells us whether the parabola opens left, right, up, or down. Here, since the x-coordinate of the focus is smaller than that of the vertex \(x = -4 \; is \; less \; than \; -1\), the parabola opens to the left.

The distance between the vertex and the focus is called \(p\), and it's significant in defining the parabola's equation. This distance tracks how far the parabola "spreads" around its vertex.

Orientation of a Parabola

Understanding the orientation of a parabola allows us to know how the parabola is situated on a plane. This is typically determined by the position of the vertex and the focus.

• For our problem, both vertex and focus lie on the x-axis (y-coordinate is similar).
• The x-coordinate of the focus \((-4, 0)\) is to the left of the vertex \((-1, 0)\).

This setup indicates the parabola opens horizontally. More specifically, it opens to the left, as the focus is to the left of the vertex.

Orientation not only affects how we draw or visualize a parabola but guides us in using the right equation format. Horizontally-opening parabolas use the equation \((y - k)^2 = 4p(x - h)\). This distinction between different orientations is crucial; it ensures that when we plug in specific variables like \(h, k,\) and \(p\), they fit correctly in the equation. This understanding is key to solving parabola-related equations correctly.