Problem 18 - 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. $$ (y-2)^{2}=\frac{4}{5}(x+4) $$

Step-by-Step Solution

Answer

Vertex: \((-4, 2)\), Focus: \((-\frac{19}{5}, 2)\), Directrix: \(x = -\frac{21}{5}\).

Step 1: Recognize the Equation Format

The given equation \((y-2)^2 = \frac{4}{5}(x+4)\) is in the vertex form of a parabola that opens horizontally, which is \((y - k)^2 = 4p(x - h)\). Here, \(h = -4\) and \(k = 2\).

Step 2: Convert to Standard Form

To convert to standard form, first identify the values of \(h\), \(k\), and \(4p\). The equation is \((y - 2)^2 = (x + 4)\frac{4}{5}\), where \(4p = \frac{4}{5}\). Thus, \(p = \frac{1}{5}\). The standard form with variables substituted can be written as \((y - 2)^2 = \frac{4}{5}(x - (-4))\).

Step 3: Identify the Vertex

The vertex \((V)\) of the parabola is given by \((h, k)\) based on the vertex form of the equation. Therefore, the vertex here is at \((-4, 2)\).

Step 4: Determine the Focus

The focus \((F)\) of a horizontally opening parabola is located at the distance \(p\) from the vertex along the axis of symmetry (horizontal in this case). Therefore, the focus is \((-4 + \frac{1}{5}, 2) = \left(-\frac{19}{5}, 2\right)\).

Step 5: Determine the Directrix

The directrix \((d)\) of a parabola is a line perpendicular to the axis of symmetry, located \(p\) units away from the vertex in the opposite direction of the focus. For a horizontally opening parabola, the directrix is \(x = h - p\). Hence, the directrix is \(x = -4 - \frac{1}{5} = -\frac{21}{5}\).

Key Concepts

Vertex Form

The vertex form of a parabola is a specialized way of expressing its equation to focus on its crucial point, the vertex. When dealing with a parabola that opens horizontally, the vertex form equation is \[ (y - k)^2 = 4p(x - h) \] where:

• \( h \) and \( k \) represent the coordinates of the vertex \((h, k)\).
• \( 4p \) is a coefficient relating to the distance from the vertex to the focus and directrix.

The vertex is one of the most significant features of a parabola because it represents the point where the parabola changes direction. Identifying \( h \) and \( k \) in the equation allows us to locate this turning point easily.
For example, in the equation \((y - 2)^2 = \frac{4}{5}(x + 4)\), the vertex is at \((-4, 2)\). This is derived directly from the formula by setting \( h = -4 \) and \( k = 2 \).
Understanding the vertex form is fundamental as it connects to other properties of the parabola, such as the focus and directrix.

Focus of Parabola

The focus of a parabola is a fixed point that, along with the directrix, defines the parabola's unique shape. For a parabola opening horizontally, like in our equation example, the focus is the point situated a distance \( p \) away from the vertex along the axis of symmetry. To find the focus, you use the formula:\( F(h + p, k) \)

• \( h \) and \( k \) are the vertex coordinates.
• \( p \) is derived from the equation's coefficient, \( 4p \).

In our case, we recognize \( 4p = \frac{4}{5} \), so \( p = \frac{1}{5} \). The vertex is \((-4, 2)\), making the focus \( \left(-4 + \frac{1}{5}, 2\right) = \left(-\frac{19}{5}, 2\right) \).
By comprehending the location of the focus, you understand how the parabola is influenced, specifically how the curve opens and at what angle.
This concept is crucial for graphing parabolas and comprehending their geometric properties.

Directrix of Parabola

The directrix of a parabola is a straight line that, together with the focus, helps guide the formation of the parabola by maintaining an equal distance between any point on the parabola and both the directrix and the focus.For a horizontally opening parabola like the one we're discussing, the directrix can be defined as:\( x = h - p \)

• Here, \( h \) is the x-coordinate of the vertex.
• \( p \) is the distance we determined from the equation \( 4p = \frac{4}{5} \). Thus, \( p = \frac{1}{5} \).

In this particular situation, with \( h = -4 \), the directrix becomes \( x = -4 - \frac{1}{5} = -\frac{21}{5} \).
The line of the directrix is essential as it is perpendicular to the axis of symmetry of the parabola and influences its geometric properties.
In problems involving optimization, reflections, or trajectory, understanding the directrix's role and placement is key to solving complex mathematical and physical challenges.