Problem 6 - Calculus Early Transcendentals

Question

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$ x-1=(y+5)^{2} $$

Step-by-Step Solution

Answer

Vertex: \((1, -5)\), Focus: \(\left(\frac{5}{4}, -5\right)\), Directrix: \(x = \frac{3}{4}\).

Step 1: Identify the Given Equation

The given equation is \( x - 1 = (y + 5)^2 \). This equation is not in the standard form for parabolas. Notice that the squared term is \( (y+5)^2 \), indicating a parabola that opens horizontally.

Step 2: Rewrite the Parabola Equation

For parabolas that open horizontally, the standard form is \( (y-k)^2 = 4p(x-h) \). We rewrite the given equation using this standard: \( (y+5)^2 = 4p(x-1) \). From the given equation, it's clear that 4p is equal to 1 since the right side of the equation already matches the form \( 4p(x-1) \). So, \( 4p = 1 \), which gives us \( p = \frac{1}{4} \).

Step 3: Identify Parabola Components

- **Vertex**: Compare \((y+5)^2 = 4p(x-1)\) with the standard form \((y-k)^2 = 4p(x-h)\). The vertex \((h, k)\) is \((1, -5)\).- **Focus**: Since the parabola opens to the right and \( p = \frac{1}{4} \), the focus is \( (h+p, k) = (1 + \frac{1}{4}, -5) = (\frac{5}{4}, -5) \).- **Directrix**: The directrix is the vertical line \( x = h - p \). So, \( x = 1 - \frac{1}{4} = \frac{3}{4} \).

Step 4: Sketch the Graph

Draw the horizontal parabola with the vertex at \( (1, -5) \). Since \( p > 0 \), the parabola opens to the right. Plot the focus at \( (\frac{5}{4}, -5) \), and draw the directrix line at \( x = \frac{3}{4} \). Ensure that the parabola is equidistant from the focus and directrix at any point along its curve.

Key Concepts

Vertex

The vertex of a parabola is the point where it changes direction. In other words, it is the turnaround point or the "tip" of the parabola. For the equation given, which is \( x - 1 = (y + 5)^2 \) we have reformulated it into the standard form for a horizontally opening parabola: \( (y-k)^2 = 4p(x-h) \). From this, we deduce that the vertex \((h, k) \) is \((1, -5) \). The vertex is critical because it provides a starting point for graphing the parabola. Knowing the vertex tells us where the parabola lies on the coordinate plane, and with horizontally opening parabolas, the y-coordinate will guide how high or low the vertex is located, while the x-coordinate tells us the left-right placement.

• The vertex is always a half-way point between the focus and directrix.
• In our example, the vertex is located at \((1, -5)\), putting it firmly in the fourth quadrant of a coordinate plane.

Understanding the vertex is essential in graphing and identifying the nature of the parabola. It indicates the minimum or maximum point and helps you determine the parabola's opening direction.

Focus

The focus of a parabola is an interior point where light rays that are parallel to the axis of symmetry are reflected towards. For horizontally opening parabolas, this point lies along the horizontal line that passes through the vertex. The focus is highly important as it tells us the degree of "openness" or width of the parabola. For the equation \( x - 1 = (y + 5)^2 \), we have determined that the focus is located at \((h+p, k) = (\frac{5}{4}, -5)\). This underscores the horizontal shift \( p \) that the parabola undergoes from the vertex to the focus.

• The focus is always located inside the curve of the parabola.
• For a parabola that opens to the right, the focus moves to the right from the vertex.

Notably, the distance to the vertex \( p \) helps us decide how wide or narrow the parabola is. A small \( p \) means a sharper, narrow curve, while a larger \( p \) results in a flatter, wider curve. Understanding the focus lets us draw the parabola with precision and helps ensure that all points on the parabola are equidistant from both the focus and directrix.

Directrix

The directrix of a parabola acts as a guideline and is a line that is perpendicular to the axis of symmetry. It serves as a boundary that the parabola curves away from. For our horizontal parabola, the directrix is a vertical line. This line helps us understand the geometry of the parabola and ensures that each point on the parabola is equidistant to both the directrix and the focus.From the equation \( x - 1 = (y + 5)^2 \), the vertical directrix line is at \( x = \frac{3}{4} \).

• The directrix line is always positioned opposite to the focus with respect to the vertex.
• This line does not intersect the parabola but serves as an imaginary boundary to create symmetry.

By understanding the directrix, we can ensure that each plotted point of the parabola is correctly placed to reflect equal distances from both the focus and the directrix, centered over the vertex. This symmetry is crucial for sketching an accurate parabola graph and determining its properties and dimensions.