Problem 34 - 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. $$ x^{2}-6 x+y^{2}-y=-9 $$

Step-by-Step Solution

Answer

The graph is a circle centered at (3, 0.5) with radius ~3.04.

Step 1: Complete the square for the x terms

To complete the square for the x terms in the equation \( x^2 - 6x + y^2 - y = -9 \), focus on the expression \( x^2 - 6x \). Calculate \( (\frac{-6}{2})^2 = 9 \). Add and subtract 9 inside the equation to form a complete square: \( (x - 3)^2 \).

Step 2: Complete the square for the y terms

Next, complete the square for the y terms \( y^2 - y \). Calculate \( (\frac{-1}{2})^2 = \frac{1}{4} \). Add and subtract \( \frac{1}{4} \) inside the equation: \( (y - \frac{1}{2})^2 \). The equation becomes: \( (x - 3)^2 + (y - \frac{1}{2})^2 = -9 + 9 + \frac{1}{4} \), simplifying to \( \frac{37}{4} \).

Step 3: Simplify and set up standard circle equation

Rewriting the equation from Step 2, we have: \( (x - 3)^2 + (y - \frac{1}{2})^2 = \frac{37}{4} \). This equation represents a circle with center at \((3, \frac{1}{2})\) and radius \(\sqrt{\frac{37}{4}} \approx 3.04\).

Step 4: Sketch the graph

Draw the coordinate axes. Plot the center at \((3, \frac{1}{2})\) and draw a circle with radius approximately 3.04. Label the circle's center on your graph. Note these as the "X" and "Y" axes shifted from the original center (0,0). Original x and y axes remain as drawn.

Key Concepts

Completing the Square

Completing the square is a method used to take a quadratic expression and turn it into a perfect square trinomial. This is particularly useful when dealing with quadratic equations because it allows us to rewrite the equation in a form that is easier to interpret or solve.

Consider when you have a quadratic in the form of \( ax^2 + bx + c \). By completing the square, you rewrite it as \( a(x-h)^2 + k \), where \( h \) and \( k \) are constants. This involves:

• Finding the value to complete the square. For the expression \( x^2 - 6x \), calculate \( (\frac{-6}{2})^2 = 9 \).
• Adding and subtracting that value inside the expression—this ensures the equation remains balanced. So, we write \( x^2 - 6x + 9 - 9 \).
• Rewriting the expression as a perfect square: \((x-3)^2 \).

This method simplifies solving equations and also helps graph them more effectively by translating them to readable shapes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. By using algebraic techniques, we can solve geometric problems and find the relationship between points, lines, and shapes in a plane.

In a 2D space, each point can be described as a pair of numbers (\(x, y\)), representing its horizontal and vertical location. For example, the point \((3, \frac{1}{2})\) lies at 3 units horizontally to the right and half a unit up from the origin.

Coordinate geometry allows us to:

• Find the distance between points using the distance formula \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
• Determine midpoints and lines equations.
• Graph shapes and interpret their defining equations.

This branch of mathematics helps in visualizing algebraic equations by showing them as geometric shapes on the coordinate plane.

Circle Equations

A circle's equation in a coordinate plane is often given in the form \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle, and \(r\) is its radius.

From the exercise, we derived the circle equation: \((x - 3)^2 + (y - \frac{1}{2})^2 = \frac{37}{4}\).

This tells us:

• The circle's center is \((3, \frac{1}{2})\).
• The radius is the square root of \(\frac{37}{4}\), approximately 3.04 units.

Understanding this form allows us to easily identify the circle's geometric properties from algebraic equations. Graphing these equations gives a visual representation of the circle and helps in comprehending the spatial relationship between points.

Axis Translation

Axis translation involves shifting the position of a graph within the coordinate plane. By translating the axis, we can more easily analyze or sketch an equation within a new context.

For the equation derived as \((x - 3)^2 + (y - \frac{1}{2})^2 = \frac{37}{4}\), we're effectively translating the standard axes (at origin \((0,0)\)) to a new set of axes centered at \((3, \frac{1}{2})\).

This means:

• Shifting the graph's center horizontally 3 units to the right.
• Shifting it vertically \(\frac{1}{2}\) units upwards.
• The original equation \((x^2 - 6x + y^2 - y = -9)\) for this graph moves with respect to its new axis.

Understanding axis translation helps in re-centering graphs to new positions, making it easier to interpret their behaviors or significant features.