Problem 6 - 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 $$(x-3)^{2}+(y+4)^{2}=25$$

Step-by-Step Solution

Answer

The equation matches description A: Circle; center \((3,-4) ;\) radius 5.

Step 1: Recognize the Equation Type

The given equation is \((x-3)^{2}+(y+4)^{2}=25\), which is in the standard form of a circle equation: \((x-h)^{2}+(y-k)^{2}=r^{2}\).

Step 2: Identify the Center

Compare the given equation to the standard form, \((x-h)^{2}+(y-k)^{2}=r^{2}\). Here, \(h = 3\) and \(k = -4\), so the center is \((3, -4)\).

Step 3: Determine the Radius

The radius squared, \(r^{2}\), equals 25. Therefore, the radius \(r\) is the square root of 25, which is 5.

Step 4: Match with Description in Column II

The equation describes a circle with center \((3, -4)\) and radius 5. This corresponds to option A.

Key Concepts

Circle Equation

The circle equation in standard form provides a way to describe a circle on a coordinate plane with ease. An equation for a circle typically takes the form:

• \((x-h)^2 + (y-k)^2 = r^2\)

Understanding the Components

The components of this equation include:

• \((h, k)\), which represents the circle's center on the plane.
• \(r\), the circle's radius.

The terms \((x-h)^2\) and \((y-k)^2\) describe how far any point \((x, y)\) is from the center of the circle. The equation essentially ensures that this distance is always equal to \(r\). Recognizing this format will help identify a circle when matching it with a graphical description.

Parabola Equation

Parabolas, often seen in quadratic equations, take a distinct form. They can open in different directions, which influences their equations. A parabola equation in standard form looks like this:

• \(y = ax^2 + bx + c\) for those opening upwards or downwards.

Identifying Direction

To determine which way a parabola opens:

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.

Similarly, parabolas opening left or right depend on:

• \(x = ay^2 + by + c\)

These changes influence the orientation of the graph, influencing how you match them with descriptions.

Equation Identification

Identifying equations is crucial when trying to match them with graphs. Each type has distinct features:

• Circle Equation: Always involves squares of both \(x\) and \(y\), like \((x-h)^2 + (y-k)^2 = r^2\).
• Parabola Equation: Focuses on one variable being squared, like \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\).

Method of Comparison

Comparing equations to their standard forms helps quickly identify which type you are dealing with. This is crucial when matching with descriptions based on the components and structure noted in their equations.

Graph Descriptions

Describing graphs correctly can greatly aid in understanding the equation they represent. Graph descriptions often include:

• The direction a parabola opens (upwards, downwards, left, or right).
• The center and radius specifics for a circle.

Importance of Accurate Descriptions

Accurate graph descriptions are like roadmaps. They provide clear directions on how a graph should appear visually. Understanding these descriptions means you can interpret, and match, them to their respective equations efficiently. Matching requires noting all features specified, ensuring they align with the characteristics given by the equation.