Problem 1 - A Graphical Approach to College Algebra

Question

Skills Match each equation in Column I with the appropriate description in Column II. Do not use a calculator. A. Circle; center \((3,-4) ;\) radius 5 B. Parabola; opens left C. Parabola; opens upward D. Circle; center \((-3,4)\); radius 5 E. Parabola; opens right F. Circle; center \((0,0) ;\) radius \(\sqrt{5}\) G. No points on its graph H. Parabola; opens downward $$x=2 y^{2}$$

Step-by-Step Solution

Answer

The equation \(x = 2y^2\) matches with (E) Parabola; opens right.

Step 1: Identify Equation Type

Examine the equation given, which is \(x = 2y^2\). Notice that the equation involves \(y^2\), indicating it is a parabola as circles involve both \(x^2\) and \(y^2\). This helps us focus on parabola options from Column II.

Step 2: Determine Parabola Orientation

Observe the structure of the equation \(x = 2y^2\). In standard form, a parabola \(y^2 = 4ax\) opening right or left has \(y^2\) on one side and \(x\) on the other. A positive coefficient implies the parabola opens towards increasing \(x\) values (right). Therefore, \(x = 2y^2\) opens right.

Step 3: Match with Description

Based on our analysis, the equation \(x = 2y^2\) is a parabola opening towards the right. We look for a description matching a parabola that opens right, which is (E) Parabola; opens right.

Key Concepts

Equation Identification

Identifying the type of equation you are dealing with is crucial in understanding how it will behave graphically. In this exercise, we are given the equation \(x = 2y^2\). To determine what kind of shape this equation represents, we need to examine its terms.

The presence of the term \(y^2\) and the absence of \(x^2\) is a tell-tale sign that we are dealing with a parabola. If an equation had both \(x^2\) and \(y^2\) with the same or symmetrical coefficients, it could represent a circle. Knowing these key features helps us quickly identify the type of graph without needing to delve into complex calculations.

Remember, the first step in equation identification is to look for square terms and their coefficients.

• If you see \(y^2\) or \(x^2\), suspect a parabola.
• If both \(x^2\) and \(y^2\) are present and they have equal coefficients, you might have a circle.

By understanding these distinct features, you can swiftly classify the equation type by sight.

Parabola Graphs

Parabolas are U-shaped graphs that can open in different directions depending on the form and coefficients of the equation. The equation \(x = 2y^2\) represents a specific type of parabola.

**Understanding Parabola Orientation**
The orientation of a parabola is determined by how it's written. For the equation \(x = 2y^2\), it takes the form \(y^2 = 4ax\), indicating that the parabola opens sideways.

- If the equation is \(x = by^2\), the parabola might open horizontally (right or left).
- If the equation is \(y = bx^2\), it generally opens vertically (up or down).

**Determining Direction**
The coefficient in front of the squared term indicates the parabola's direction:

• A positive coefficient like in \(x = 2y^2\) means it opens towards positive x values (right).
• If the coefficient was negative, the graph would open towards negative x values (left).

Understanding how different factors affect parabolas can help you sketch them easily and accurately.

Graph Matching Skills

Once we've identified and understood the features of the equation, the next step is to match it with the correct description or graph. This skill not only tests our understanding but also reinforces it.

When tasked with matching graphs, it's important to:

• Focus on key characteristics like orientation and axis of symmetry.
• Use what you know: if the equation \(x = 2y^2\) opens right, eliminate options that don't fit.
• Remember previous calculations or recognitions of form. For \(x = 2y^2\), we've determined that it corresponds to 'parabola; opens right'.

This step may seem straightforward but can become complex as equations and graphs become more intricate. Always return to basic principles: identify the type, determine the orientation, and use these insights to make the match. Practice regularly with different equations to hone this skill further.