Problem 16 - Algebra 2

Question

Graph each equation. $$ (y-8)^{2}=-4(x-4) $$

Step-by-Step Solution

Answer

The parabola is centered at (4, 8) and opens to the left.

Step 1: Identify the Equation Type

The given equation \((y-8)^2 = -4(x-4)\) is in the form \((y-k)^2 = 4p(x-h)\). This represents a conic section equation, specifically a parabola. Since the term involving \(x\) is not squared, and the coefficient of \((x-4)\) is negative, it indicates a parabola opening to the left.

Step 2: Locate the Vertex

The general form \((y-k)^2 = 4p(x-h)\) gives the vertex of the parabola as the point \((h, k)\). From the given equation, \(h = 4\) and \(k = 8\). Therefore, the vertex is at \((4, 8)\).

Step 3: Determine the Distance p and Direction

In the equation \((y-8)^2 = -4(x-4)\), the term \(-4\) represents \(4p\). Thus, \(4p = -4\), giving \(p = -1\). Since \(p\) is negative, the parabola opens left.

Step 4: Sketch the Graph

With the vertex at \((4, 8)\) and knowing that the parabola opens to the left, plot the vertex. Use the value of \(p = -1\) to find additional points. From the vertex, move 1 unit to the left along the x-axis because \(p = -1\), keeping the y-coordinate the same. Choose a point on the y-axis, such as \((3, 8)\) when \(x = 3\), for orientation. Sketch the parabola.

Key Concepts

Conic Sections

Conic sections are an important part of algebra and geometry that deal with the ways a plane can intersect with a cone. These intersections can create different shapes, such as circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties and equations that describe them. In the case of the given problem, the equation \((y-8)^2 = -4(x-4)\) represents a conic section specifically known as a parabola. This equation fits the general form \((y-k)^2 = 4p(x-h)\), which describes a parabola that opens either left or right along the x-axis. Parabolas are distinct among conic sections because they have a curved 'U' shape that can open in different directions based on the specific details of their equation. Understanding the general forms of conic section equations is crucial in graphing and interpreting these curves.

Vertex of Parabola

The vertex of a parabola is a critical focal point where the curve changes direction. This point is crucial because it gives you an anchor for sketching and understanding the parabola's path. Given an equation of the form \((y-k)^2 = 4p(x-h)\), the vertex is easily identified as \((h, k)\). For the equation \((y-8)^2 = -4(x-4)\), the vertex is at the point \((4, 8)\). Identifying the vertex helps in plotting the parabola accurately on a graph. Start at this point when drawing the parabola and consider the vertex as a central feature that defines the other aspects of the graph, like direction and width. Always remember that the vertex is the point where the parabola will either bend upwards, downwards, left, or right, depending on the orientation of the curve.

Direction of Parabola Opening

The direction in which a parabola opens is a defining characteristic that tells us how the parabola is oriented on the graph. For a parabola described by the equation \((y-k)^2 = 4p(x-h)\), the sign and value of \(p\) determine the direction:

• If \(p > 0\), the parabola opens towards the positive x-axis (right).
• If \(p < 0\), the parabola opens towards the negative x-axis (left).

In our example, the coefficient of \((x-4)\) is -4, which means \(4p = -4\) leading to \(p = -1\). This negative \(p\) value indicates the parabola will open to the left. Knowing the direction of opening helps in plotting the parabola and predicting its behavior on the graph. This is not just a technical detail but crucial for understanding the entire parabola's scope in its graphing context.