Problem 5 - Algebra 2

Question

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ y^{2}-x-10 y+34=0 $$

Step-by-Step Solution

Answer

The graph is a parabola.

Step 1: Recall the general forms of conic sections

Conic sections are defined by their general equations. A quadratic equation involving two variables can be characterized as:1. Parabola: Only one squared term, either \(x^2\) or \(y^2\).2. Circle: Both \(x^2\) and \(y^2\) with equal coefficients.3. Ellipse: Both \(x^2\) and \(y^2\) with unequal coefficients showing the same sign.4. Hyperbola: Both \(x^2\) and \(y^2\) with opposite signs.

Step 2: Identify the given equation

The given equation is \(y^2 - x - 10y + 34 = 0\). This equation includes the terms \(y^2\), \(x\), \(-10y\), and a constant.

Step 3: Analyze the squared terms

In the equation \(y^2 - x - 10y + 34 = 0\), the only squared term is \(y^2\). There is no \(x^2\) term in the equation.

Step 4: Determine the type of conic

Since there is only one squared term \(y^2\) and no \(x^2\), the graph of the equation \(y^2 - x - 10y + 34 = 0\) represents a parabola.

Key Concepts

Parabola

A parabola is one of the most commonly encountered conic sections, and its graph forms a symmetrical curve. You can easily recognize a parabola from its equation, which features only one squared variable term. This means, in equations like \[ y^2 = 4ax \]or\[ x^2 = 4ay \],you see just one of the variables—either \(x\) or \(y\)—is squared.
Distinct features of parabolas include:

• Vertex: The highest or lowest point on the parabola, depending on its direction.
• Axis of Symmetry: A line that runs through the vertex, and the parabola is a mirror image on either side of this line.
• Focus and Directrix: Points that help define and construct the parabola, such that every point on the parabola is equidistant from the directrix and the focus.

Converting a general equation into vertex form can make it easier to identify these features, but recognizing that an equation is a parabola allows you to anticipate the shape and orientation of its graph.

Quadratic Equation

Quadratic equations form the mathematical backbone of parabolas, represented typically as\[ ax^2 + bx + c = 0 \]for a single variable. When dealing with two variables, as with conic sections, you might see equations such as \[ ax^2 + bxy + cy^2 + dx + ey + f = 0 \].
What uniquely identifies a parabola among conic sections is having just one squared term—in other words, either \(x^2\) or \(y^2\) but not both. Recognizing which variable is squared can help determine the parabola's direction:

• Vertical Parabola: If \(y^2\) is squared, the parabola opens horizontally.
• Horizontal Parabola: If \(x^2\) is squared, the parabola opens vertically.

Indeed, in the equation \(y^2 - x - 10y + 34 = 0\), the lone squared term \(y^2\) signifies it describes a parabola opening in a direction parallel to the \(x\)-axis. Gathering these insights can facilitate graph sketching even before detailed plotting.

Graph Analysis

Graphing conic sections like parabolas requires understanding the relationships between algebraic expressions and their geometric representations. When analyzing the graph of a quadratic equation, you should:

• Identify the squared term(s) to determine the type of conic section. Only one squared term (like \(y^2\) in our example) signifies you are dealing with a parabola.
• Locate the vertex, where the curve leans or pivots—the critical middle point providing symmetry.
• Determine the axis of symmetry, which for a standard form equation, often aligns with one of the coordinate axes.

Beyond these steps, addressing constants and linear terms provides refinements:

• The linear terms (like \(-10y\) in our equation) alter the position and orientation of the curve.
• The constant term shifts the parabola along its axis of symmetry but doesn't affect its overall shape.

Each aspect plays a part in crafting the graph, and careful analysis ensures an accurate representation of the equation's visual form.