Problem 21 - College Algebra

Question

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F),\) and directrix \((d)\) of the parabola. $$ (x+4)^{2}=24(y+1) $$

Step-by-Step Solution

Answer

Vertex: \((-4, -1)\), Focus: \((-4, 5)\), Directrix: \(y = -7\).

Step 1: Analyze the equation

The given equation is \((x+4)^2 = 24(y+1)\). This represents a parabola that opens vertically because the square is on the x-term.

Step 2: Rewrite the equation in standard form

For a vertical parabola, the standard form is \((x-h)^2 = 4p(y-k)\), where \((h, k)\) is the vertex. Here, \(h = -4\) and \(k = -1\), so the equation is already in the form \((x-h)^2 = 4p(y-k)\) with \(4p = 24\).

Step 3: Determine the vertex

From the equation \((x+4)^2 = 24(y+1)\), we identify the vertex \(V\) of the parabola as \((-4, -1)\).

Step 4: Calculate the value of p

From \(4p = 24\), we solve for \(p\) to find \(p = \frac{24}{4} = 6\).

Step 5: Determine the focus

The focus \(F\) of the parabola is located \(p\) units from the vertex along the axis of symmetry. Since the parabola opens vertically, and \(p = 6\), the focus is \((-4, -1 + 6) = (-4, 5)\).

Step 6: Find the directrix

The directrix \(d\) is a line parallel to the x-axis, \(p\) units in the opposite direction of the focus from the vertex. Thus, \(y = -1 - 6 = -7\) is the equation of the directrix.

Key Concepts

Vertex

The vertex of a parabola is a key concept in understanding its shape and position. In simple terms, the vertex is the point where the parabola changes direction. For vertical parabolas, this is the lowest or highest point, depending on whether it opens upwards or downwards. In our example, given the equation \[(x+4)^{2} = 24(y+1),\]the vertex is located at the point \((-4, -1)\).

• \(h\) and \(k\) in the standard equation form \((x-h)^2 = 4p(y-k)\) represent the x and y coordinates of the vertex respectively.
• Here, we derived \(h = -4\) and \(k = -1\), affirming the vertex as \((-4, -1)\).
• The vertex helps in sketching the parabola and finding other elements like the axis of symmetry.

When identifying the vertex, always observe the shift from the origin which is indicated by \((h, k)\).Recognizing these values is crucial for solving and graphing quadratic equations.

Focus

The focus of a parabola plays a special role in defining its shape. It is a point from which distances are measured to construct the parabola's curve. Typically located inside the parabola, it acts like a focal "target" the curve is centered around. For a vertically oriented parabola, the focus lies along the y-axis, relative to the vertex.

• In our equation \((x+4)^{2} = 24(y+1)\), where the vertex is \((-4,-1)\), the focus is found by moving \(p\) units away from the vertex.
• Given \(p = 6\), the location is calculated: \(F = (-4, -1 + 6) = (-4, 5)\).
• This indicates a movement 6 steps upwards along the y-axis.

Understanding the location of the focus helps determine how wide or narrow the parabola will appear,and for more advanced applications, it is vital in optical designs and satellite dishes, as these often utilize parabolic shapes.

Directrix

The directrix is a line that, together with the focus, helps define the parabola. The parabola is the set of all points that are equidistant from the directrix and the focus. This important geometric property is what gives every parabola its characteristic shape.

• In horizontal parabolas, the directrix is a vertical line, but for our vertical parabola, \(d\) is horizontal.
• From the vertex, and with \(p = 6\), the directrix is calculated as \(y = -1 - 6 = -7\), situated 6 units below the vertex.
• This means: while the focus guides the inner curvature, the directrix provides an outer constraint.

By understanding the role of the directrix, you gain insight into the equality principle at play—the balance in distance from any point on the parabola to the focus and to the directrix. This is a foundational concept for deeper geometric explorations.

Standard Form

The standard form of a parabola's equation offers a neat and organized way to express its geometric properties. For a vertical parabola, this form is: \[(x-h)^2 = 4p(y-k)\].This structure highlights the vertex \((h, k)\) and allows for easy computation of the parabola’s other features like the focus and directrix.

• In our case, the equation \((x+4)^2 = 24(y+1)\) is already in this standard form.
• Here, \(h = -4\) and \(k = -1\), directly identifying the vertex.
• The term \(4p = 24\) indicates \(p = 6\), further establishing key features of the parabola.

The use of standard form simplifies many processes in algebra and calculus, making it easier for you to make sense of more complex mathematical problems involving parabolas. Mastering this layout will provide you with a strategic advantage in graphing and solving quadratic functions efficiently.