Problem 10 - Precalculus Mathematics for Calculus

Question

Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$(y+5)^{2}=-6 x+12$$

Step-by-Step Solution

Answer

Vertex: (2, -5); Focus: \(\left(\frac{1}{2}, -5\right)\); Directrix: \(x = \frac{7}{2}\).

Step 1: Rewrite the Equation in Vertex Form

Start by rewriting the given equation \((y+5)^2 = -6x + 12\). Rearrange the equation to isolate the quadratic term on one side:\((y+5)^2 = -6(x - 2)\).This equation is now in the standard form of a parabola that opens horizontally, \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex.

Step 2: Identify the Vertex

From the equation \((y+5)^2 = -6(x-2)\), identify the vertex \((h, k)\). Comparing with the standard form, we see \(h = 2\) and \(k = -5\). Thus, the vertex of the parabola is \((2, -5)\).

Step 3: Determine the Parameter p

The standard form of the parabola is \((y - (-5))^2 = 4p(x - 2)\), which simplifies to \(4p = -6\).Solve for \(p\) by dividing both sides by 4:\(p = \frac{-6}{4} = -\frac{3}{2}.\)The negative value of \(p\) indicates that the parabola opens to the left.

Step 4: Find the Focus

The focus of a parabola is \((h + p, k)\).Substituting the values:\((h, k) = (2, -5)\) and \(p = -\frac{3}{2}\), the focus is \((2 - \frac{3}{2}, -5) = (\frac{1}{2}, -5)\).

Step 5: Find the Directrix

The directrix of a horizontally opening parabola is given by the line \(x = h - p\).Substituting the values:\(h = 2\) and \(p = -\frac{3}{2}\), close to directrix is \(x = 2 + \frac{3}{2} = \frac{7}{2}.\)So, the equation of the directrix is \(x = \frac{7}{2}\).

Step 6: Sketch the Graph

To sketch the graph, draw the vertex at \((2, -5)\).Next, plot the focus at \((\frac{1}{2}, -5)\).Then, draw the directrix line vertically at \(x = \frac{7}{2}\).Since the parabola opens to the left due to the negative \(p\), sketch a curve passing through the vertex and approaching the focus, constantly being equidistant to the directrix.

Key Concepts

Vertex of a Parabola

In a parabola, the vertex is a crucial point that represents the "turning" or "stopping" part of the curve. It's where the parabola changes direction in its plane. To determine the vertex from an equation, such as \((y+5)^2 = -6(x - 2)\), it's helpful to express the equation in vertex form: \((y - k)^2 = 4p(x - h)\). Here, the vertex data is denoted by \((h, k)\).

• For the given equation, solve for the vertex: \(h = 2\) and \(k = -5\).
• This means that the vertex of the parabola is \((2, -5)\).

Understanding the vertex is essential because it provides a starting point for sketching the parabola's shape and position on the graph. Since the vertex can be a minimum or maximum point, depending on whether the parabola opens upwards or downwards, it helps in visualizing the general shape of the curve.

Focus of a Parabola

The focus is a special point inside the parabola, where all points on the curve are equidistant from the directrix. It plays a significant role in understanding the parabola's reflective properties. The formula to find the focus is straightforward when you know the vertex and parameter \(p\). For a parabola in the form \((y - k)^2 = 4p(x - h)\), the focus is located at \((h + p, k)\).

• In this exercise, calculate the \(p\) value: \(p = -\frac{3}{2}\).
• Use the vertex \((2, -5)\) to find the focus: \((2 - \frac{3}{2}, -5) = (\frac{1}{2}, -5)\).

The negative sign of \(p\) indicates that the parabola opens to the left, impacting the location of the focus. With the focus found, you can understand not only the structure of the parabola but also the way it curves, as the vertex and focus provide guidance for shaping the graph accurately.

Directrix of a Parabola

The directrix is an essential part of a parabola's anatomy, playing the role of a "mirror" line that is considered equidistant from all points on the curve relative to the focus. The directrix helps define the set of points that constitute the parabolic shape. For parabolas that open horizontally, the directrix is a vertical line.

• The directrix equation is calculated as \(x = h - p\).
• Substituting the given values for this problem: \(h = 2\) and \(p = -\frac{3}{2}\), the directrix is \(x = 2 + \frac{3}{2} = \frac{7}{2}\).

This equation determines where the vertical line of the directrix sits relative to the vertex and focus. In graphically illustrating a parabola, the directrix ensures accuracy in reflecting symmetry and distance from the parabola's points to both focus and directrix.