Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ y^{2}+14 y+4 x+45=0 $$

Step-by-Step Solution

Answer

The vertex is (-1, -7), the focus is (0, -7), and the directrix is x = -2.

Step 1: Rearrange the Equation

Begin by rewriting the given parabola equation in a more useful form: \[ y^2 + 14y + 4x + 45 = 0 \]We want to complete the square for the \(y\)-terms. To do this, group the \(y\) terms together:\[ y^2 + 14y = -4x - 45 \]

Step 2: Complete the Square

To complete the square for the expression \(y^2 + 14y\), find a number that will complete the square. The number is \(\left(\frac{14}{2}\right)^2 = 49\).Add and subtract 49 inside the equation: \[ (y^2 + 14y + 49) = 49 - 4x - 45 \]This simplifies to:\[ (y + 7)^2 = 4x + 4 \]

Step 3: Rewrite the Equation in Standard Form

The equation becomes:\[ (y + 7)^2 = 4(x + 1) \]This is the standard form of a horizontal parabola: \[ (y - k)^2 = 4p(x - h) \]Here, \(h = -1\), \(k = -7\), and \(4p = 4\) which gives \(p = 1\).

Step 4: Identify the Vertex, Focus, and Directrix

For the standard form \((y - k)^2 = 4p(x - h)\), the vertex is \((h, k)\).In this parabola, the vertex is \((-1, -7)\).The focus lies at \((h + p, k)\), which is \((-1 + 1, -7) = (0, -7)\).The directrix is a vertical line at \(x = h - p\), which is \(x = -1 - 1 = -2\).

Step 5: Graph the Parabola with Key Features

To sketch the graph:1. Plot the vertex at \((-1, -7)\).2. Mark the focus at \((0, -7)\).3. Draw the directrix as the vertical line \(x = -2\).4. The parabola opens to the right because it is horizontal with a positive \(p\). It will be symmetric around the line \(y = -7\).

Key Concepts

Vertex of Parabola

In the case of parabolas, the vertex is a key feature that tells us the turning point of the curve. For the equation \[(y+7)^2 = 4(x+1)\], we can identify the vertex using the standard form of a horizontally oriented parabola: \[(y-k)^2 = 4p(x-h)\]. Here, \(h = -1\) and \(k = -7\), which means our vertex is at the point \((-1, -7)\). The vertex is located at the intersection of the axis of symmetry and the parabola. It is also the point from which the curve rises or falls when graphed. Understanding the vertex helps in orienting the parabola on a graph.

• For the equation \((y-k)^2 = 4p(x-h)\) when \(p\) is positive, the parabola opens to the right.
• If \(p\) were negative, it would open to the left.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a form that is easier to graph or analyze. For the quadratic expression \(y^2 + 14y\), we complete the square by adding and subtracting \( \left(\frac{14}{2}\right)^2 = 49\). This transforms the equation into \[(y+7)^2 = y^2 + 14y + 49\]. By manipulating the original equation, we achieve \[(y + 7)^2 = 4(x + 1)\]. This form reveals that the parabola's vertex can directly be read off and demonstrates symmetry. Completing the square is particularly useful in finding the vertex and simplifying the process of sketching the parabola.
By adjusting the equation into a perfect square, we are able to extract more information about the behavior of the parabola.

• It gives insight into the parabola's direction and vertex position.
• It simplifies further processes such as finding focus and directrix.

Focus of Parabola

The focus of a parabola is a crucial point. It's used together with the directrix to define the parabola in geometric terms. In our equation \((y+7)^2=4(x+1)\), we find the focus point using \((h+p, k)\). Plugging in our values, the focus is at \((0, -7)\). This shows that every point on the parabola is equidistant from the focus and the directrix. The focus lies inside the parabola and helps to define its "direction".

• For a horizontally opening parabola, the focus shifts horizontally from the vertex.
• The distance between the vertex and focus is \(p\). In this scenario, \(p = 1\), indicating a shallow curve.

Directrix of Parabola

The directrix of a parabola is a line that, along with the focus, serves to define the parabola geometrically. For the equation \((y+7)^2 = 4(x+1)\), we determined the directrix using \(x = h - p\). In this case, the directrix is \(x = -2\). This vertical line lies opposite the focus relative to the vertex.
The directrix is essential for understanding the parabola's geometry—every point on the parabola is equally distant from the focus and the directrix.

• The directrix of a horizontal parabola is a vertical line.
• In contrast, a vertical parabola would have a horizontal directrix.

Graphing Parabolas

Graphing a parabola involves plotting specific key features like the vertex, focus, and directrix. For this parabola, begin by marking the vertex at \((-1, -7)\). Next, plot the focus \((0, -7)\), which lies to the right of the vertex. Then draw the directrix as a vertical line at \(x = -2\). The parabola will open to the right due to the positive \(p\). When graphing:

• Ensure the parabola is symmetric with respect to its axis of symmetry, in this case, \(y = -7\).
• The vertex is the turning point, and the parabola will "bowl" outwards towards the focus.

Graphing is a visual method that helps you understand the spatial arrangement of these features.