Problem 7 - 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 D: Circle; center \((-3,4)\); radius 5.

Step 1: Identify the Equation Type

First, let's identify if the equation is that of a circle or a parabola. Given equation is \((x+3)^{2}+(y-4)^{2}=25\). This is in the general form of a circle equation \((x-h)^{2}+(y-k)^{2}=r^{2}\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Determine the Center of the Circle

From the equation \((x+3)^{2} + (y-4)^{2} = 25\), we can identify the center \((h, k)\) by comparing it to \((x-h)^{2} + (y-k)^{2}\). This gives us the center as \((-3, 4)\).

Step 3: Find the Radius of the Circle

The equation \((x+3)^{2}+(y-4)^{2}=25\) is in the form \((x-h)^{2} + (y-k)^{2} = r^{2}\), where \(r^{2} = 25\). Thus, the radius \(r\) is the square root of \(25\), which is \(5\).

Step 4: Match the Description

Compile the information from the previous steps. We have a circle with center \((-3, 4)\) and radius \(5\). According to Column II, this matches description D: Circle; center \((-3,4)\); radius 5.

Key Concepts

Center of a Circle

The center of a circle is a fundamental aspect when working with circle equations. In the standard form of a circle equation, \[ (x - h)^2 + (y - k)^2 = r^2 \] where

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

To find the center of a circle from its equation, recognize and rewrite the equation in the standard form, if necessary, and identify the values of \(h\) and \(k\). For instance, in the equation \[ (x+3)^{2}+(y-4)^{2}=25, \] the center is

• \((-3, 4)\)

because by comparing with \((x-h)^2 + (y-k)^2\), the transformations on \(x\) and \(y\) indicate a shift from \((0,0)\) to the center’s coordinates.
By knowing the center, you can easily sketch the circle's position on a coordinate plane, which is crucial in geometry and various applications.

Radius of a Circle

The radius of a circle is another key element, especially when trying to understand its properties and equation representation. By definition, the radius is the distance from the center of a circle to any point on its edge. In the circle equation \[(x - h)^2 + (y - k)^2 = r^2,\] \(r\) represents the radius.
To determine the radius from the circle equation, you need to take the square root of the right side of the equation. For example, given \( (x+3)^{2}+(y-4)^{2}=25,\) the equation is already in the standard form, so you know that

• \(r^2 = 25.\)

Therefore, the radius \(r\) is

• \(5\).

Knowing the radius is important when you are calculating the circle's circumference or area, and it helps visualize the circle effectively.

Parabola Descriptions

Parabolas are curves that are well-known in both algebra and geometry. They can open in four different directions depending on their equation: up, down, left, or right. A regular quadratic equation in the vertex form for parabolas is: \[ y = a(x - h)^2 + k\] which opens up or down, depending on the sign of \(a\).

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

For parabolas that open left or right, the equation is \[ x = a(y - k)^2 + h\].

• If \(a > 0\), it opens right.
• If \(a < 0\), it opens left.

Understanding these basic forms and conditions is essential when trying to match an equation to its graphical description, such as determining in which direction a parabola opens based on its equation. Recognizing these patterns in equations allows one to predict and sketch the shape and orientation of a parabola with ease.