Problem 89 - College Algebra Essentials

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph of \((x-2)^{2}+(y+1)^{2}=16\) is my graph of \(x^{2}+y^{2}=16\) translated two units right and one unit down.

Step-by-Step Solution

Answer

Yes, the statement makes sense. Given circle equation represents the translation of the initial circle two units right and one unit down.

Step 1: Identify the special form of circle equations

Recognize that the circle equation is given in special form as \((x-h)^{2}+(y-k)^{2}=r^{2}\). Here, \((h, k)\) are the coordinates of the circle's center and \(r\) is the radius.

Step 2: Identify the translation parameters

From the given equations, identify that the shift parameter \(h\) is equal to 2 (implying a rightward shift) and \(k\) is equal to -1 (implying a downward shift).

Step 3: Analyze the statement

Analyze the statement that the graph of \((x-2)^{2}+(y+1)^{2}=16\) is an exact translation of the graph \(x^{2}+y^{2}=16\), moved two units to the right and one unit down.

Step 4: Determine if the statement makes sense

Determine that the statement does make sense, since the values of \(h\) and \(k\) in the given circle equation do represent a movement of the circle's center which matches the parameters described in the statement.

Key Concepts

Translation of Graph

When we talk about the translation of a graph, we're essentially discussing how the graph moves around on the coordinate plane. Translation involves shifting the entire graph in various directions without altering its shape or orientation. For circles on a coordinate graph, this means moving the entire circle, allowing us to identify shifts based on changes in its equation.

In terms of equations, the standard form of a circle's equation is \[(x-h)^2 + (y-k)^2 = r^2\]where \((h, k)\) are the new coordinates of the center after translation. If compared to the base circle \(x^2 + y^2 = r^2\), any non-zero \(h\) or \(k\) indicates a shift.

• \(h > 0\) indicates a shift right; \(h < 0\), a shift left.
• \(k > 0\) indicates a shift up; \(k < 0\), a shift down.

Translation makes sense when the shifts specified in the equation align with the directions you expect based on the equation's modification.

Center of a Circle

The center of a circle on a coordinate plane is crucial because it determines the circle's position. In the general equation of a circle, \[(x-h)^2 + (y-k)^2 = r^2\], the center is located at coordinates \( (h, k) \.\)

Understanding the center's role helps you track where the circle is placed and how it moves with translations. The initial circle's center in the form \(x^2 + y^2 = r^2\) is at the origin \( (0,0) \.\) When transformed by translation to \( (h, k) \,\) you must adjust the graph visually by moving the entire figure to the new coordinates. For instance, if \(h=2\) and \(k=-1\), this indicates the circle is translated to the right by 2 units and down by 1 unit, landing at \((2, -1)\).

Emphasizing these changes helps in understanding how coordinate adjustments directly affect graph positions.

Radius of a Circle

The radius of a circle is a fundamental feature that measures the distance from its center to any point on its circumference. In the circle's equation \[(x-h)^2 + (y-k)^2 = r^2\], \(r^2\) represents the square of the radius. You can find the radius by taking the square root of \r^2\.

The radius affects the circle's size, but not its position. In both the original and translated circle equations \((x-2)^2+(y+1)^2=16\) and \(x^2+y^2=16\), the radius calculated is the same, resulting in a consistent size for the circle. This consistency demonstrates that while the circle's location can change (due to translation), its size, as determined by the radius, remains unchanged.

• Example: Given \(r^2=16\), the radius \(r\) will be \sqrt{16}\, which is 4.

Thus, understanding the radius helps in maintaining the circle's dimensions constant despite translations on the graph.