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Magazine Chapter 1: Problem 34
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. $$ x^{2}-6 x+y^{2}-y=-9 $$
Short Answer
Expert verified
The graph is a circle centered at (3, 0.5) with radius ~3.04.
Step by step solution
01
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: \( (x - 3)^2 \).
02
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 \( \frac{37}{4} \).
03
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 \(\sqrt{\frac{37}{4}} \approx 3.04\).
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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鈥攖his ensures the equation remains balanced. So, we write \( x^2 - 6x + 9 - 9 \).
- Rewriting the expression as a perfect square: \((x-3)^2 \).
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:
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.
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:
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.
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:
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.
