Problem 17 - Precalculus with Calculus Previews

Question

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(y^{2}-8 y+2 x+10=0\)

Step-by-Step Solution

Answer

Vertex: (3, 4); Focus: (2.5, 4); Directrix: x = 3.5; Axis: y = 4.

Step 1: Rearrange the Equation

Start by rearranging the given equation to isolate the terms involving y. The given equation is: \( y^2 - 8y + 2x + 10 = 0 \). Move \(2x\) and \(10\) to the other side of the equation to get: \( y^2 - 8y = -2x - 10 \).

Step 2: Complete the Square for y

We complete the square for the y-terms to transform the equation into a standard form. Take the y-term: \(y^2 - 8y\). Half of the coefficient of y is 4, and squaring it gives 16. Add and subtract 16 on the left side to maintain balance: \(y^2 - 8y + 16 - 16 = -2x - 10\).

Step 3: Simplify the Equation

Rewrite the left side as a perfect square: \((y - 4)^2\). Move the constants to the right: \((y - 4)^2 - 16 = -2x - 10 \). Simplify it to: \((y - 4)^2 = -2x + 6 \).

Step 4: Convert to Standard Parabolic Form

Rewrite the equation \((y - 4)^2 = -2(x - 3)\) to see it in the form \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex and \(p\) represents the focal length. Here, \(h = 3\), \(k = 4\), therefore the vertex is \((3, 4)\).

Step 5: Identify the Focus and Directrix

From the equation \(4p = -2\), solve for p to get \(p = -\frac{1}{2}\). Since this implies a leftward opening, the focus is located \(p\) units from the vertex to the left. Therefore, the focus is \((3 - \frac{1}{2}, 4) = (\frac{5}{2}, 4)\). The directrix is a vertical line located \(p\) units to the right of the vertex: \(x = 3 + \frac{1}{2}\), thus \(x = \frac{7}{2}\).

Step 6: Determine the Axis of Symmetry

The axis of symmetry for a horizontally oriented parabola of form \((y - k)^2 = 4p(x - h)\) is a horizontal line through the y-coordinate of the vertex: \(y = 4\).

Step 7: Graph the Parabola

Plot the vertex at \((3, 4)\) on a coordinate plane. Draw the parabola opening left through this vertex. Mark the focus at \((\frac{5}{2}, 4)\) and the directrix line \(x = \frac{7}{2}\). Use the axis of symmetry \(y = 4\) to assist in drawing symmetry.

Key Concepts

Vertex

The vertex is one of the key components of a parabola in coordinate geometry. You can think of it as the peak or the lowest point of the parabola, depending on its orientation. In this problem, we need to find the vertex of a horizontally oriented parabola given by the equation:\[ (y - 4)^2 = -2(x - 3) \] This equation is in the standard form of a parabola, \[ (y - k)^2 = 4p(x - h) \]where \( (h, k) \) tells us the vertex. Here, \( h = 3 \) and \( k = 4 \), which means the vertex is at \( (3, 4) \).

• The vertex is the turning point of the parabola.
• In this form, the parabola opens sideways because the \( y \)-variable is squared.
• The vertex is crucial for graphing the parabola and understanding its shape.

Focus

The focus of a parabola is a point from which distances are measured to write the equation of the parabola. For this sideways parabola, the distance from the vertex to the focus is represented by \( p \). From the standard equation \[ 4p = -2 \], we find that \( p = -\frac{1}{2} \). The negative sign indicates that the parabola opens to the left.

• The focus is located \( p \) units from the vertex, to the left for this specific parabola.
• The coordinates for the focus are calculated by adjusting the vertex coordinates: \( (3 - \frac{1}{2}, 4) = (\frac{5}{2}, 4) \).
• The focus is used to determine how the parabola is shaped and directed.

Directrix

The directrix of a parabola is a line that provides a baseline measurement for the curve's geometry. It is perpendicular to the axis of symmetry and is located exactly \( p \) units away from the vertex but on the opposite side compared to the focus.

• In our case, the directrix is a vertical line because the parabola is horizontal.
• From the equation and our previous calculations, the directrix is at \( x = 3 + \frac{1}{2} = \frac{7}{2} \).
• The directrix is essential for understanding the parabola's orientation and position in space.

The directrix helps define how far the curve is stretched away from the central axis and influences the openness of the parabola. By understanding the directrix, one can understand the symmetry and balance of the parabola around its vertex.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical or horizontal line that runs through the vertex, dividing the parabola into two mirror-image halves. For a horizontally oriented parabola, it is a horizontal line.

• In this exercise, the parabola's axis of symmetry is \( y = 4 \).
• This line runs through \( y \) of the vertex, aligning it perfectly between the focus and the directrix.
• The axis of symmetry helps to maintain a balance between both sides of the parabola.

Having a clear understanding of the axis of symmetry assists greatly in accurate graphing and enhances the comprehension of how parabolic graphs maintain a consistent shape relative to their vertex and other defining elements like the focus and directrix.