Problem 4 - A Graphical Approach to College Algebra

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. $$x=3 y^{2}+5 y-6$$

Step-by-Step Solution

Answer

The graph is a parabola.

Step 1: Identify the Standard Form

The given equation is \(x = 3y^2 + 5y - 6\). This equation resembles a conic section form where the variable \(x\) is in terms of \(y\). The typical forms for conic sections are: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).

Step 2: Determine the Type of Conic Section

Since the given equation is in the form \(x = ay^2 + by + c\), it implies no \(x^2\) or \(xy\) terms exist, indicating it's aligned along the y-axis. This configuration is indicative of a parabola.

Step 3: Analyze the Equation

The presence of only the \(y^2\) term and all coefficients being constant or attached to \(y\) shows there's no rotation or mixture of squared terms, further confirming it's a parabola, as when only one variable is squared, the graph is a parabola.

Key Concepts

Equation Analysis

When faced with the equation \(x = 3y^2 + 5y - 6\), the first step is identifying its structure to determine what kind of curve it represents. A good starting point is recognizing the standard form for conic sections: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). This specific equation doesn't fit perfectly into this format because it expresses \(x\) in terms of \(y\), rather than set to zero. However, this alone hints at a certain feature of the graph.

• Notice there are no \(x^2\) or \(xy\) terms.
• The squared term \(y^2\) takes precedence over \(y\), affecting the graph's curvature.

This tells us that the graph does not represent a circle or an ellipse, since these require both variables to be squared. The lack of an \(xy\) term also eliminates the possibility of it being a hyperbola. Pause to evaluate what this arrangement implies about the orientation and nature of the graph.

Parabola Identification

Given the form \(x = ay^2 + by + c\), the equation undoubtedly represents a parabola. Parabolas are one of the simpler conic sections characterized by having a single squared variable. Let's examine why our equation fits this category:

• No \(x^2\) term - so the parabola opens either horizontally or vertically.
• \(y^2\) is present, meaning it opens horizontally, specifically aligning along the x-axis.

In essence, if only one variable is squared (here \(y^2\)), the conic described is assuredly a parabola. By having the equation expressed as \(x = f(y)\), it implies the parabola's axis of symmetry is parallel to the x-axis, a key feature distinguishing parabolas from other conic sections.An important point here is the coefficient \(a = 3\). Since \(a > 0\), the parabola opens to the right.

Graphing Techniques

Once identified as a parabola, let's consider how to graph it effectively. Graphing a parabola with the equation \(x = 3y^2 + 5y - 6\) involves a few strategic steps:

• Determine the vertex by completing the square or using calculus-based methods.
• Note the direction in which the parabola opens. Here, it's horizontal.
• Plot key points such as the vertex and additional points on the parabola for accuracy.

While graphing, remember:- The coefficient \(3\) in front of \(y^2\) makes the parabola open toward the right side, as seen earlier.- Add a few points by plugging in values for \(y\) to calculate corresponding \(x\) values for an accurate sketch.- These steps create a scaffold allowing you to draw a precise, mirrored curve along its axis of symmetry.Understanding these techniques assures accurate graphing of conic sections, emphasizing the functional behavior and specific features of the parabola depicted by the given equation.