Problem 47 - Precalculus: Mathematics for Calculus

Question

Complete the square to determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an cllipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$y^{2}=4(x+2 y)$$

Step-by-Step Solution

Answer

The equation is a parabola with vertex (-4, 4), focus (-3, 4), and directrix x = -5.

Step 1: Rearrange the Equation

Start by rearranging the given equation into a familiar conic form.Original equation: \( y^2 = 4(x + 2y) \)Expand the terms and bring them to one side:\( y^2 - 8y = 4x \)

Step 2: Complete the Square for y

Now, complete the square on the left-hand side to make it easier to analyze the form of the equation.Left-hand side: \( y^2 - 8y \)To complete the square, take half of the coefficient of \( y \) (which is \( -8 \)), square it, and add inside the square:\( y^2 - 8y \rightarrow (y - 4)^2 - 16 \)So, the equation becomes: \( (y - 4)^2 - 16 = 4x \) Add \( 16 \) to both sides: \( (y - 4)^2 = 4x + 16 \)

Step 3: Identify the Conic Type

From the equation \( (y - 4)^2 = 4x + 16 \), notice that it can be rearranged into \((y - k)^2 = 4p(x - h)\) form, which is the standard form of a horizontal parabola.The given equation thus represents a **parabola**.

Step 4: Find the Vertex, Focus, and Directrix

For the standard form of a parabola \((y - k)^2 = 4p(x - h)\), the vertex is \((h, k)\). We rewrite the equation as:\( (y - 4)^2 = 4(x + 4) \)Thus, \( h = -4 \) and \( k = 4 \). The vertex is \((-4, 4)\).Since \( 4p = 4 \), we have \( p = 1 \).The focus, given by \((h + p, k)\), is \((-4 + 1, 4) = (-3, 4)\).The directrix is \(x = h - p = -4 - 1 = -5\).

Step 5: Sketch the Graph

Plot the vertex \((-4, 4)\), and locate the focus \((-3, 4)\). Draw the directrix line \(x = -5\).Since this is a horizontal parabola opening to the right, sketch the curve passing through the vertex and focusing in the positive x-direction, opening towards the focus.

Key Concepts

Parabola

A parabola is a type of conic section that you can recognize by its distinctive U-shaped curve. It is defined as the set of all points that are equidistant from a fixed point, called the focus, and a line, called the directrix. This unique property sets the parabola apart and gives it a symmetrical shape.

When working with parabolas, you'll often encounter their standard equation forms, such as \((y-k)^2 = 4p(x-h)\) for horizontal parabolas, and \((x-h)^2 = 4p(y-k)\) for vertical ones. These equations help identify key features of the parabola, like its vertex, focus, and directrix, which we'll discuss in more detail in the following sections.

The orientation of a parabola, whether it opens left-right or up-down, is determined by the squared term in its equation. The vertex form for the given parabola is particularly useful, as it allows easy identification of the vertex coordinates, merely by inspection.

Vertex

The vertex of a parabola is its highest or lowest point, also serving as the curve's point of symmetry. For a horizontal parabola, the vertex form \((y-k)^2 = 4p(x-h)\), lets you quickly find the vertex at the coordinates \((h, k)\).

For the equation \((y-4)^2 = 4(x+4)\), we pinpoint the vertex at \((-4, 4)\). This vertex acts as a pivot where the parabola changes direction, displaying perfect symmetry.*+ around this point.

Understanding the vertex is key because it gives us a lot of information about the graph of the parabola: the axis of symmetry runs through the vertex, and it signifies the parabola's minimum or maximum value in a coordinate direction. Getting comfortable with spotting it in standard forms will simplify sketching and analyzing parabolas significantly.

Focus

The focus of a parabola is a fixed point that helps define its shape. Every point on the parabola is equidistant from the focus and the directrix. For the standard parabola form \((y-k)^2 = 4p(x-h)\), we calculate the focus as \((h+p, k)\).

In our example, with \(p = 1\), and calculated vertex at \((-4, 4)\), the focus is located at \((-3, 4)\).

Locating the focus is crucial because it dictates how "open" or "narrow" the parabola is. In sketching the parabola, it represents the point towards which the parabola will extend indefinitely. Understanding the focus' role can aid in creating accurate and proportionate parabola sketches.

Directrix

The directrix is an essential component of a parabola, being a s1traight line that, along with the focus, helps define the locus of the parabola's points. For a parabola in the form \((y-k)^2 = 4p(x-h)\), the directrix is described by the equation \(x = h - p\).

In this exercise, with \(h = -4\) and \(p = 1\), the directrix is found at \(x = -5\).

Think of the directrix as a boundary for the parabola; the curve will never cross this line as it stretches in the direction of the focus. It helps in visualizing the parabola's symmetry and defining its curvature. A proper understanding of the interaction between the focus and the directrix will enable one to sketch more accurate graphs of parabolas.