Problem 5 - A Graphical Approach to Precalculus with Limits

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x-1=-3(y-4)^{2}$$

Step-by-Step Solution

Answer

The equation represents a parabola that opens to the left.

Step 1: Identify the General Form

The given equation is \( x - 1 = -3(y - 4)^2 \). This equation resembles the standard form of a conic section. To identify the type of conic, we need to compare it with the standard forms.

Step 2: Standard Form for Parabolas

In conic sections, a parabola can be expressed in the form \((x - h)^2 = 4p(y - k)\) or \((y - k)^2 = 4p(x - h)\). The given equation resembles the latter, i.e., \( (y - k)^2 = 4p(x - h) \) with transformations.

Step 3: Match the Equation to Parabola Form

Rewriting the given equation, \( x - 1 = -3(y - 4)^2 \), we see it matches the form \((y - k)^2 = 4p(x - h)\) if we solve for \(x\): \( y - 4 = \pm \sqrt{\frac{x - 1}{-3}} \). The negative sign before the \(3\) indicates a parabola that opens horizontally and to the left.

Step 4: Determine Type of Conic Section

Since the equation fits the pattern \((y - k)^2 = 4p(x - h)\), this indicates it is a parabola. The negative coefficient \(-3\) implies that the parabola opens leftward.

Key Concepts

Parabola

A parabola is a specific type of conic section with a distinct U-shaped curve. It can open upwards, downwards, left, or right, based on the orientation of the equation. When we talk about parabolas, we often refer to their standard forms:

• Vertical parabolas: \((x - h)^2 = 4p(y - k)\)
• Horizontal parabolas: \((y - k)^2 = 4p(x - h)\)

Here, \((h, k)\) represents the vertex of the parabola, and \(p\) is a constant that determines the parabola's width and the direction in which it opens.
For instance, in the given equation \(x - 1 = -3(y - 4)^2\), we identify that it fits the structure of a horizontal parabola. The signs and coefficients inform us more about its orientation and movement.

Equation Transformation

Equation transformation is a key concept in recognizing and rewriting expressions to better understand their geometrical properties. The given equation \(x - 1 = -3(y - 4)^2\) needs some tweaking to reveal its underlying conic form.
By transforming the equation, we rewrite it as \((y - 4) = \pm \sqrt{\frac{(x - 1)}{-3}}\), showcasing its relationship to the standard parabola form \((y - k)^2 = 4p(x - h)\).

• The presence of the square root indicates the dependency on \(x\), aligning with a horizontal orientation.
• Moreover, the negative sign in front of 3 points to the direction the parabola opens, which is towards the left.

Equation transformation aids in revealing these directional and positional attributes of parabolas. It offers a clearer understanding of the type and orientation of the conic section observed.

Graph Analysis

Although the task doesn't require graphing, understanding how the equation relates to a graph is crucial in identifying its attributes. In graph analysis, the vertex \((h, k)\) is the focal point, guiding us in visualizing direction and shape.
The transformation derived: \(y - 4 = \pm \sqrt{\frac{x - 1}{-3}}\) corresponds to a leftward-opening parabola with a vertex at \((1, 4)\). This information tells us:

• The vertex, situated at \(x = 1\) and \(y = 4\), acts as the starting point for the parabola.
• The negative coefficient indicates horizontal westward movement, which opposes the usual right or up directions.

Graph analysis simplifies the geometrical visualization and provides deeper insight into the structural elements, assisting learners to conceptualize the parabola's behavior and placement in the coordinate system.