Problem 4 - 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 $$y^{2}=-3 x$$

Step-by-Step Solution

Answer

B. Parabola; opens left.

Step 1: Identify the Given Equation

We are given the equation \( y^2 = -3x \). This equation can be classified based on its structure.

Step 2: Recognize the Equation Type

The given equation is of the form \( y^2 = -kx \), where \( k \) is a constant and \(-3x\) indicates a negative coefficient on the \( x \) term. Equations of this form represent parabolas.

Step 3: Determine the Parabola's Orientation

For parabolas represented by \( y^2 = -kx \), they open horizontally based on the sign of \( k \). When \( k \) is negative, as in this case, the parabola opens to the left.

Step 4: Match the Description with Column II

From Column II, the description that matches a parabola opening to the left is 'B. Parabola; opens left.'

Key Concepts

Circle Equations

Circle equations might seem confusing at first, but they are quite simple once you understand the basic structure. A circle's equation is derived from the distance formula and is used to determine every point that lies on the circle. In mathematical terms, we use the standard form of \[(x - h)^2 + (y - k)^2 = r^2\]where:

• \( (h, k) \) is the center of the circle.
• \( r \) is the radius of the circle.

For instance, the equation \[(x - 3)^2 + (y + 4)^2 = 25\]could represent a circle with center at \((3, -4)\) and a radius of \(5\). Here the radius is determined by taking the square root of \(25\), which is \(5\).

Whenever you see an equation in this format, remember it's revealing the unique layout of a circle in the coordinate plane. Understanding how to identify the circle's center and radius from its equation is crucial for solving many problems.

Parabola Orientation

Parabolas are fascinating curves, and their orientation depends on their equation. For a parabola, its orientation, i.e., the direction it opens, can be determined by the equation's structure. When dealing with equations of the form \( y^2 = -kx \), two main factors decide orientation:

• If \( k \) is positive, the parabola opens to the right.
• If \( k \) is negative, as with our example \( y^2 = -3x \), it opens to the left.

In contrast, vertical parabolas (those involving \( x^2 \)) open upward if \( k \) is positive, and downward if it's negative.

Understanding the direction a parabola faces helps in graphing it on a coordinate plane. We can predict the shape and better analyze the behavior of its graph, determining how it stretches and where its focus and directrix are situated.

Equation Classification

Classifying equations is the first step in solving mathematical problems involving graphs. Each equation type relates to a distinct geometric shape, like circles or parabolas. Here's a quick guide:

• Circle equations are in the form \( (x-h)^2 + (y-k)^2 = r^2 \).
• Horizontal parabolas, like \( y^2 = -kx \), open right or left.
• Vertical parabolas, in the form \( x^2 = ky \), open up or down.

Recognizing these forms at a glance allows you to match equations with their corresponding graph shapes. In our exercise, understanding that \( y^2 = -3x \) represents a horizontally opening parabola was the key to identifying it as a left-opening parabola.

This skill of classification can step up your problem-solving efficiency, making complex equations far less intimidating.