Problem 29 - Calculus with Concepts in Calculus

Question

Sketch the graph of the given equation with the help of a suitable translation. Show both the \(x\) and \(y\) axes and the \(X\) and \(Y\) axes. $$ x^{2}-2 x+y^{2}=3 $$

Step-by-Step Solution

Answer

The graph is a circle centered at (1, 0) with radius 2.

Step 1: Rewrite the equation

Start with the original equation: \( x^2 - 2x + y^2 = 3 \).This can be separated into components: \( (x^2 - 2x) + y^2 = 3 \). Our goal is to complete the square for the \(x\) part.

Step 2: Complete the square for x-terms

Take the \(x\) terms \(x^2 - 2x\) and complete the square. \(x^2 - 2x\) can be written as \((x^2 - 2x + 1) - 1\) to complete the square. So, express it as \((x - 1)^2 - 1\).

Step 3: Substitute back into the equation

Now replace \(x^2 - 2x\) in the original equation with \((x - 1)^2 - 1\), which turns the equation into: \((x - 1)^2 - 1 + y^2 = 3\). Simplify by adding 1 to both sides to get: \((x - 1)^2 + y^2 = 4\).

Step 4: Recognize the standard circle equation

The equation \((x - 1)^2 + y^2 = 4\) is the equation of a circle. The standard form of a circle's equation is \((X - h)^2 + (Y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Here, the center is \((1, 0)\) and the radius is \(2\).

Step 5: Translate the coordinate system

Translate the circle to align with the new coordinate system \(X = x-1\) and \(Y = y\). This means shifting the entire graph right by 1 unit from the origin, making the center of the circle \((h, k) = (1, 0)\) instead of \((0, 0)\).

Step 6: Plot the circle with new axes

Draw the circle with radius 2 centered at \((1, 0)\) on the \((X, Y)\) system. Ensure both the old \((x, y)\) axes and the new \((X, Y)\) axes are shown. Label the center, axis intersections, and draw the outer circle boundary.

Key Concepts

Completing the Square

Completing the square is a technique used to transform quadratic equations into a form that is easier to work with, often making it possible to recognize and manipulate the structure of the equation more clearly. In the given equation, we aimed to complete the square for the expression involving the variable \(x\), which initially appears as \(x^2 - 2x\).

• Find the coefficient of \(x\), which is \(-2\), and halve it to get \(-1\).
• Square this result to obtain \(1\). This is the value added and subtracted to complete the square.
• The expression \(x^2 - 2x\) can then be rewritten as \((x - 1)^2 - 1\).

By completing the square, the equation becomes easier to handle, allowing for a transformation into a recognizable form. This method highlights the transformation needed, setting up for further manipulations, such as coordinate translation.

Coordinate Translation

Coordinate translation involves shifting the entire graph of an equation by adding or subtracting constants from the variables. This concept is key to aligning an equation with a new coordinate system, as seen in the exercise.
Substitute \(X = x - 1\) and \(Y = y\). Here’s how this translation works:

• The \(x\)-coordinate in \((x−1)^2\) shows a horizontal rightward shift by \(1\).
• The \(y\)-coordinate remains unchanged in this case, indicating no vertical shift.

By translating the coordinate system, the graph can be effectively "moved" to a new position, making it simpler to interpret its relationship to the new axes and allowing better visualization and understanding of its properties.

Circle Equation

The equation of a circle can be easily identified when it’s in the standard form. For circles, the standard form is: \((X - h)^2 + (Y - k)^2 = r^2\)
Here, we've transformed our original equation into such a format, rendering it \((x - 1)^2 + y^2 = 4\). Here’s why recognizing this form matters:

• The center of the circle \((h, k)\) is \((1, 0)\).
• The radius \(r\) is deduced from the square root of \(4\), giving \(r = 2\).

Identifying the equation of a circle enables us to easily determine key attributes such as the center position and radius. Knowing these allows for accurate graph plotting and better understanding of the spatial layout of the circle on a coordinate plane.

Coordinate Axes

The coordinate axes refer to the horizontal and vertical lines that create a coordinate plane: the x-axis and y-axis in the standard system. An additional set of axes (X and Y) is introduced here after the translation of the circle. Understanding the role of both original and translated coordinate axes is crucial for visualizing movements within the plane.

• The x-axis acts as the baseline for horizontal measurement, while the y-axis takes vertical measurement.
• After translation, the new X-axis shifts right relative to the x-axis due to \(X = x - 1\).
• The Y-axis aligns vertically in the same position as the y-axis since \(Y = y\).

Visual representation of both sets of axes fosters deeper comprehension of the relative shifts and transformations performed on the function graph, bridging the gap between algebraic manipulation and geometric understanding.