Problem 54 - Algebra 2

Question

Error Analysis A student claims that the circle \((x+7)^{2}+(y-7)^{2}=8\) is a translation of the circle \(x^{2}+y^{2}=8,7\) right and 7 down. What is the student's mistake?

Step-by-Step Solution

Answer

The student's mistake is that they have the direction of the translation wrong. The circle \((x+7)^{2}+(y-7)^{2}=8\) is a translation of the circle \(x^{2}+y^{2}=8,7\) 7 units to the left and 7 units up, not right and down.

Step 1: Understand the standard form for a circle

In a standard form for a circle \( (x - h)^{2} + (y - k)^{2} = r^{2} \), the center of the circle is at the point (h, k) and the radius is r. The student is correct about the circle \( (x + 7)^{2} + (y - 7)^{2} = 8 \) being a translation of the circle, but the direction and the magnitude of the translation must be verified.

Step 2: Analyze the given circles

For the circle \( (x + 7)^{2} + (y - 7)^{2} = 8 \), the center is at (-7, 7) and the radius is \( \sqrt{8} \), while for the circle \( x^{2} + y^{2} = 8,7 \), the center is at (0, 0) and the radius is approximately \( \sqrt{8.7} \). This indicates that the circle has shifted from the origin to the point (-7, 7), meaning it has moved 7 units to the left and 7 up.

Step 3: Identify the student's mistake

The student's mistake lies in the direction of the translation of the circle. Instead of being translated right and down by 7 units as the student's claim, the circle is actually translated left and up by 7 units.

Key Concepts

Standard Form of a Circle

The standard form of a circle is expressed as \((x - h)^2 + (y - k)^2 = r^2\). In this formula, the center of the circle is located at the point \((h, k)\), and \(r\) represents the radius of the circle. This format is extremely important when analyzing circle equations as it easily shows the center and radius, allowing students to identify the circle's position and size on the coordinate plane. For example, in the equation \((x + 7)^2 + (y - 7)^2 = 8\), the center is at \((-7, 7)\) and the radius is \(\sqrt{8}\). Understanding this form helps students decipher transformations and shifts in graphs.

Coordinate Geometry

Coordinate geometry is the study of geometric figures using the coordinate plane. It allows us to describe shapes using algebraic equations. This method bridges algebra and geometry, making it easier to visualize and calculate geometric properties. By using coordinates, we can determine distances, angles, and other essential elements of geometric figures. Circles, in particular, are often examined using their center and radius, which are derived from their standard form. This makes calculations, such as finding intersection points or graphing circles, more straightforward.

Error Analysis

Error analysis involves identifying and understanding mistakes in solving mathematical problems. In the context of circle translations, errors often arise from misunderstanding the direction or magnitude of shifts. In this exercise, the mistake was in translating the circle in the wrong direction. Recognizing that the equation \((x + 7)^2 + (y - 7)^2 = 8\) indicates a left and up movement, rather than right and down, highlights the importance of careful analysis. Students can avoid such errors by correctly interpreting signs and understanding the geometric implications of the equations they work with. This piece of understanding is crucial for building confidence and accuracy in solving problems.

Translation of Circles

Translating a circle involves moving it horizontally or vertically on the coordinate plane without changing its size or shape. The change is reflected in the circle’s equation by altering the values of \(h\) and \(k\) in the standard form. Translating \((x + 7)^2 + (y - 7)^2 = 8\) involves understanding that the circle’s center moves to \((-7, 7)\), indicating a leftward and upward shift from the origin \((0, 0)\). Correctly identifying the directions of these translations requires careful attention to the signs: positive signs in the equation indicate movement in the opposite direction (e.g., \(+7\) in \((x + 7)^2\) means moving left by 7). This foundational concept helps in performing transformations and can be applied to many other figures in geometry.