Problem 50 - Intermediate Algebra

Question

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$x=y^{2}+2$$

Step-by-Step Solution

Answer

The graph is a right-opening parabola with vertex (2, 0).

Step 1: Identify the Type of Graph

The given equation is \( x = y^2 + 2 \). Recognize that it resembles the form \( x = a(y - k)^2 + h \), which is the standard form of a parabola. This indicates the graph is a parabola.

Step 2: Determine the Orientation

In the standard form \( x = a(y - k)^2 + h \), the parabola opens left or right. Because \( a > 0 \), since it is positive \( 1 \) (from \( y^2 \)), the parabola opens to the right.

Step 3: Find the Vertex of the Parabola

The vertex form \( x = a(y - k)^2 + h \) indicates the vertex is at \( (h, k) \). Compare \( x = y^2 + 2 \) to \( x = (y - 0)^2 + 2 \) to find that \( h = 2 \) and \( k = 0 \). Thus, the vertex is at the point \((2, 0)\).

Step 4: Sketch the Graph

Plot the vertex \((2, 0)\) on a coordinate plane. Since the parabola opens to the right, it will extend symmetrically with respect to the horizontal line going through the vertex. Additional points such as \((3, 1)\) and \((3, -1)\) can be plotted to ensure the shape.

Key Concepts

Parabola

A parabola is a curve formed by plotting quadratic equations. In the context of graphing equations, it represents expressions of the type \( y = ax^2 + bx + c \) or involving \( x \) and \( y^2 \) like \( x = ay^2 + bx + c \).
Parabolas have distinct features:

• They are symmetrical.
• They have a single vertex, which serves as a turning point.
• They can open either up, down, left, or right depending on the equation's structure.

The direction a parabola opens depends on the terms involved:

• If \( y \) is expressed in terms of \( x^2 \), it opens vertically.
• If \( x \) is expressed in terms of \( y^2 \), it opens horizontally.

In our example, the equation \( x = y^2 + 2 \) forms a parabola that opens horizontally. Understanding this basic shape helps when sketching graphs.

Vertex

The vertex of a parabola is an important point. It's the point where the curve turns.
Depending on the direction the parabola opens, the vertex is the highest or lowest point for vertically opening parabolas, or the most left or right point for horizontally opening parabolas.
For graphs like \( x = a(y-k)^2 + h \), the vertex can be easily identified as \( (h, k) \). This is because the terms \( h \) and \( k \) shift the parabola away from the origin where it would normally be centered.

• In our example \( x = y^2 + 2 \), comparing it to \( x = (y - 0)^2 + 2 \) shows that the vertex is located at \( (2, 0) \).
• The vertex here represents the point at which the graph changes its direction and moves symmetrically outward as it opens to the right.

Finding the vertex is crucial for sketching because it allows you to know the starting point from which the parabola expands.

Equation Types

In mathematics, understanding different equation types is crucial for graphing. The main types include:

• Linear equations \( y = mx + b \), which graph as straight lines.
• Quadratic equations \( y = ax^2 + bx + c \), forming parabolas.
• Circular equations \( (x-h)^2 + (y-k)^2 = r^2 \), which describe circles.

Our focus is on parabolas, deriving from quadratic relationships.
These can often be transformed into a standard form, illuminating key characteristics like the vertex and opening direction. In this exercise, the equation \( x = y^2 + 2 \) represents a parabola, showing off its dependency on \( y^2 \).
Recognizing these patterns helps drastically simplify the graphing process and ensures the geometry is understood properly.