Problem 58 - College Algebra with Modeling and Visualization

Question

Graph the parabola. Label the vertex, focus, and directrix. $$ -2(y+1)=(x+3)^{2} $$

Step-by-Step Solution

Answer

Vertex: (-3, -1); Focus: (-3.5, -1); Directrix: x = -2.5.

Step 1: Converting to Standard Form

We start by writing the equation in standard form of a parabola that opens left or right: \((y - k)^2 = 4p(x - h)\). Here, our equation is \(-2(y+1) = (x+3)^2\). Divide both sides by -2, we get:\((y + 1) = -\frac{1}{2}(x + 3)^2\).This indicates that the parabola opens horizontally.

Step 2: Identifying the Vertex

In the standard form \((x - h)^2 = 4p(y - k)\) or \((y - k)^2 = 4p(x - h)\), \((h, k)\) is the vertex.For \((y + 1) = -\frac{1}{2}(x + 3)^2\), we have:\((h, k) = (-3, -1)\). Thus, the vertex of the parabola is at \((-3, -1)\).

Step 3: Finding the Focus

The focus formula for a parabola \((y - k)^2 = 4p(x - h)\) is \((h + p, k)\). Here, \(4p = -2\), so \(p = -\frac{1}{2}\). Beginning at the vertex \((-3, -1)\), the focus is:\((-3 + (-\frac{1}{2}), -1) = (-3.5, -1)\).

Step 4: Determining Directrix

The directrix of a parabola \((y - k)^2 = 4p(x - h)\) is \(x = h - p\). Using our \(p = -\frac{1}{2}\):The directrix is:\(x = -3 - (-\frac{1}{2}) = -2.5\).

Step 5: Graphing Elements

1. Plot the vertex at \((-3, -1)\).2. Plot the focus at \((-3.5, -1)\).3. Draw the line \(x = -2.5\) as the directrix.4. Sketch the parabola opening to the left, as determined by the negative \(p\) value.

Key Concepts

Vertex Identification

The vertex of a parabola is a crucial point and serves as the peak or the lowest point, depending on the parabola's orientation. In this exercise, our original equation is given, \(-2(y+1)=(x+3)^2\). To identify the vertex, we need to convert this equation to the standard form for a horizontal parabola, \((y-k)^2=4p(x-h)\).

After dividing both sides by -2, we have \((y + 1) = -\frac{1}{2}(x + 3)^2\). The form clearly presents the vertex at \((h, k)\), which here translates to \(-3, -1\). This means the vertex is located at \((-3, -1)\), a point on the coordinate plane where the parabola turns.

• The vertex acts as a reference point for the rest of the parabola's features.
• In a horizontally opening parabola, the vertex gives the horizontal shift, and the vertical position from which calculations for the focus and directrix are made.

Focus Calculation

Determining the focus of a parabola gives us insight into its reflective properties and helps in broader visualizations. The focus is a single, particular point inside the parabola that 'directs' the curve.

For our parabola, the focus can be calculated using the information from its standard form \((y - k)^2 = 4p(x - h)\). The value \(p\) represents the distance from the vertex to the focus along the axis of symmetry.

• From the equation \((y + 1) = -\frac{1}{2}(x + 3)^2\), we identify that \(4p = -2\), giving \(p = -\frac{1}{2}\).
• This means the focus lies \(-\frac{1}{2}\) units horizontally from the vertex in the direction opposite the parabola opening (to the left for positive values of \(4p\), and right for negatives).
• Thus, starting from the vertex \((-3, -1)\), the focus is located at \((-3.5, -1)\).

The focus serves as a pivotal point that defines the parabola's shape and orientation.

Directrix Determination

The directrix of a parabola provides a geometrical boundary that complements the focus. It is a line to which the parabola maintains an equal distance from any point on the parabola to the focus. Understanding how to locate the directrix is essential for sketching and analyzing parabolas.
In this particular problem, the parabola's directrix is calculated with the formula \(x = h - p\), as the parabola opens horizontally.

• Given \(p = -\frac{1}{2}\), from our equation \((y + 1) = -\frac{1}{2}(x + 3)^2\), calculate the directrix as \(x = -3 - ( -\frac{1}{2}) = -2.5\).
• The directrix is a vertical line at \(x = -2.5\) on the coordinate plane.

The significance of the directrix lies in how it allows us to visualize the symmetry and balance of the parabola: every point on the parabola is equally distant from the focus and this line.