Problem 21 - College Algebra with Corequisite Support

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: Identify the given equation of the parabola

The given equation is \((x+4)^2 = 24(y+1)\). This equation is in the form of \((x-h)^2 = 4p(y-k)\) which describes a vertical parabola.

Step 2: Write the equation in standard form

The standard form of a vertical parabola is \((x-h)^2 = 4p(y-k)\). Comparing it with the given equation \((x+4)^2 = 24(y+1)\), we can see it is already in the standard form.

Step 3: Determine the vertex of the parabola

The vertex \(V(h,k)\) is obtained directly from the standard form equation \((x-h)^2 = 4p(y-k)\). From \((x+4)^2 = 24(y+1)\), we get \(h = -4\) and \(k = -1\). Therefore, the vertex is \V(-4, -1)\.

Step 4: Calculate the parameter \(p\)

The given equation \((x+4)^2 = 24(y+1)\) can be compared with \(4p(y-k)\). Since \(4p = 24\), solving for \(p\) gives \(p = \frac{24}{4} = 6\).

Step 5: Determine the focus of the parabola

The focus \(F\) of a vertical parabola is located at \(\left(h, k + p\right)\). Substituting the known values \(h = -4\), \(k = -1\), and \(p = 6\), the focus is \F(-4, -1+6) = F(-4, 5)\.

Step 6: Find the directrix of the parabola

The directrix \(d\) of a vertical parabola is given by \(y = k - p\). Substitute \(k = -1\) and \(p = 6\) to get \(y = -1 - 6 = -7\).

Key Concepts

Standard Form

The standard form of a **parabola** is essential for understanding its properties. For a **vertical parabola**, the standard form equation is \[ (x-h)^2 = 4p(y-k) \]where

• \( (h, k) \) represents the **vertex** of the parabola.
• \( p \) is a parameter that determines the shape and direction of the parabola.
Comparing the equation \((x+4)^{2}=24(y+1)\) with the standard form, we notice it already fits perfectly, making it straightforward to identify key characteristics of the parabola.
This form helps in easily determining other crucial features like the **vertex**, **focus**, and **directrix**.

Vertex

The **vertex** of a parabola is its turning point, the location where the parabola changes direction. For our vertical parabola equation in standard form:\[ (x-h)^2 = 4p(y-k) \] we directly identify the vertex as \((h, k)\).
In the example provided, \((x+4)^2 = 24(y+1)\), by setting

• \(h = -4\)
• \(k = -1\)

Thus, the vertex is at \((-4, -1)\).
Understanding the vertex is useful for graphing and interpreting the parabola's symmetry.

Focus

The **focus** of a parabola is a point that helps define its specific shape. For a vertical parabola, the focus is located at:\[ (h, k+p) \]where

• \(p\) is determined by equating \(4p\) to the coefficient next to \((y-k)\) in the equation.

Returning to our equation \((x+4)^2 = 24(y+1)\), solve \(4p = 24\) to find \(p = 6\).
Substitute:

• \(h = -4\)
• \(k = -1\)
• \(p = 6\)

This gives us a focus at \((-4, 5)\).
The focus is crucial as it represents the point to which all points on the curve are equidistant from the corresponding directrix.

Directrix

The **directrix** of a parabola is a line associated with its geometric properties. For a vertical parabola, the equation for the directrix is \[ y = k - p \]This line, alongside the focus, defines the set of points that form the parabola.
In our example:

• \(k = -1\)
• \(p = 6\)

We find the directrix to be \(y = -1 - 6 = -7\).
Conceptually, every point on the parabola is equidistant from the focus and its directrix. Thus, understanding the directrix helps clarify the symmetrical property of the parabola.