Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, determine which conic section is represented based on the given equation. \(4 y^{2}-5 x+9 y+1=0\)

Step-by-Step Solution

Answer

The equation represents a parabola.

Step 1: Identify the Equation Form

We start by inspecting the given equation: \(4y^2 - 5x + 9y + 1 = 0\). Our goal is to determine which conic section it represents by comparing it to the general form of a conic section equation, which is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Notably, this equation does not have an \(x^2\) term nor an \(xy\) term.

Step 2: Recognize the Form

Second, because we have a \(y^2\) term and no \(x^2\) term, and since the \(xy\) term is also absent, it implies that the conic is either a parabola, ellipse, or hyperbola. The absence of an \(x^2\) term in this context suggests that the conic section involves only quadratic behavior in terms of \(y\).

Step 3: Parallel the Standard Forms

Next, we compare it to the standard forms of conic sections. For a parabola, the general form for orientation along the x-axis would involve an \(x\) term not squared. Since our highest degree for \(y\) is 2 and there is no \(x^2\), we suspect a parabola with a vertical orientation.

Step 4: Confirm the Parabola Form

We confirm our suspicion by rewriting the equation in the possible form of a vertical parabola: rearrange as \(4y^2 + 9y = 5x - 1\) and solve for \(x\) in terms of \(y\). This makes the structure \(x = ay^2 + by + c\), a key characteristic of a parabola.

Key Concepts

Parabolas

Parabolas are a specific type of conic section that can be identified by their characteristic U-shape. Unlike other conic sections like circles, ellipses, or hyperbolas, a parabola consists of a single curve that extends infinitely in either direction.

In terms of equations, a parabola can be represented in various forms depending on its orientation:\[y = ax^2 + bx + c \]for a vertical parabola, or\[x = ay^2 + by + c \]for a horizontal one.
A key feature of all parabolas is that they contain only one squared term. This makes them distinct from ellipses and hyperbolas, which contain both an \(x^2\) and a \(y^2\) term.

To visualize a parabola:

• Look for the vertex, which is the peak or the lowest point, depending on its orientation.
• Check the axis of symmetry, which runs through the vertex and splits the parabola into two symmetrical parts.
• Identify the direction of the curve, which depends on the sign of the coefficient of the squared term.

Understanding these features aids in recognizing the parabola amidst other conic sections.

Quadratic Equations

Quadratic equations play a crucial role when dealing with parabolas. They form the foundation of the relationship between variables in a parabolic curve. A typical quadratic equation is written as \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\) to ensure that the equation represents a quadratic, rather than a linear equation.

Quadratic equations have distinct properties, such as:

• They always open upwards or downwards when graphed as a parabola on the coordinate plane.
• The solutions to these equations, also known as the roots, can be real or complex numbers depending on the discriminant, \(b^2 - 4ac\).
• The vertex form, another way to express a quadratic, highlights the maximum or minimum point more clearly.

These properties make quadratic equations invaluable for analyzing and predicting the behavior of parabolas.

Standard Form of Conic Sections

The standard form of conic sections is a universal framework used to categorize different types of curves based on the degree and presence of terms in their defining equations. A general conic section can be expressed as:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]
Different values of \(A\), \(B\), and \(C\) define the type of conic:

• Parabolas: Occur when either \(A = 0\) or \(C = 0\), but not both.
• Circles: Result from \(A = C\) and \(B = 0\).
• Ellipses: When \(A eq C\) but \(B = 0\), maintaining similar signs for \(A\) and \(C\).
• Hyperbolas: Distinct from ellipses by the opposite signs of \(A\) and \(C\).

The absence or presence of a specific term significantly influences the curve. For instance, if the \(x^2\) term is missing from an equation, it hints at the potential presence of a parabola, particularly if no \(xy\) term is involved. This systematic approach simplifies the process of identifying and analyzing the conic sections.