Problem 16 - Algebra 2

Question

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ x^{2}+y^{2}+4 x-6 y=-4 $$

Step-by-Step Solution

Answer

The equation represents a circle centered at \((-2, 3)\) with radius 3.

Step 1: Rearrange and Complete the Square for x

Start with the given equation: \[x^2 + y^2 + 4x - 6y = -4\]Rearrange it to group x-terms together:\[x^2 + 4x + y^2 - 6y = -4\]To complete the square for \(x\), notice the \(x\) terms: \(x^2 + 4x\). Take half of 4, square it, and add it inside the group: (\(4/2=2\), \(2^2=4\)).Add and subtract 4 within the equation:\[(x^2 + 4x + 4) + y^2 - 6y = -4 + 4\]This can be rewritten as:\[(x + 2)^2 + y^2 - 6y = 0\]

Step 2: Complete the Square for y

Next, take the y-terms \(y^2 - 6y\). To complete the square, take half of -6, square it, and add it inside the group: (\(-6/2 = -3\), \((-3)^2 = 9\)).Add and subtract 9:\[(x + 2)^2 + (y^2 - 6y + 9) = 0 + 9\]This can be rewritten as:\[(x + 2)^2 + (y - 3)^2 = 9\]

Step 3: Identify the Conic Sections

The equation now is in the form \[(x + 2)^2 + (y - 3)^2 = 9\]This equation matches the standard form of a circle:\[(x - h)^2 + (y - k)^2 = r^2\]where \(h = -2\), \(k = 3\), and the radius \(r = 3\).

Step 4: Graph the Circle

On a coordinate plane, plot the center of the circle at \((-2, 3)\). Using the radius \(r = 3\), draw a circle around this center. The circle should pass through the points \((-5, 3)\), \((1, 3)\), \((-2, 0)\), and \((-2, 6)\).

Key Concepts

Complete the Square

Completing the square is a powerful algebraic method used to transform a quadratic expression into a perfect square trinomial. This technique is particularly useful in the context of conic sections, such as circles and ellipses, to rewrite equations in standard form. In our exercise, we started with the equation: \[x^2 + y^2 + 4x - 6y = -4\]The goal was to isolate the terms so that both the x and y parts can resemble the pattern \((x-h)^2 + (y-k)^2 = r^2\), the equation of a circle. Here’s how it works:

• First, rearrange the terms to group x and y terms: \(x^2 + 4x + y^2 - 6y = -4\).
• For the x terms, complete the square by taking half of the coefficient of x (which is 4), squaring it to get 4, and adding/subtracting this square: \((x+2)^2\).
• Do the same for the y terms with coefficient -6: Take half (-3), square it to get 9, and use it in the equation: \((y-3)^2\).

By completing the square, you get a neat and tidy expression \((x+2)^2 + (y-3)^2 = 9\) which is easier to interpret and graph.

Circle Equation

Once the equation is put in the form \((x+2)^2 + (y-3)^2 = 9\), it aligns perfectly with the standard form of a circle's equation. The general form of a circle is represented as: \[(x-h)^2 + (y-k)^2 = r^2\].This format provides clear information about the structure of the circle:

• \(h\) and \(k\) are the coordinates of the center, in this case, \((-2, 3)\).
• \(r^2\) is the radius squared. Here, \(r^2\) is 9, so \(r = 3\).

Having the equation in this form makes it straightforward to identify key features of the circle.The circle's location and size are easily determined:

• The center at \((-2, 3)\) tells us the exact point from where the circle is equidistant.
• The radius of 3 indicates how far from the center to plot the circle.

Understanding this form simplifies the graphing process immensely.

Graphing Equations

Graphing a circle from its equation is much easier when you know the center and radius. With the equation \((x+2)^2 + (y-3)^2 = 9\), we can readily translate these into graphing directions. Let’s break it down:Once identified as a circle, the next step is to plot its center point, which is \((-2, 3)\) on the coordinate plane. This point acts like the anchor for drawing the circle.From the center, use the radius to determine the circle's boundary. For instance, in this case, a radius of 3 implies you can mark points 3 units to the left, right, above, and below the center:

• Left: \((-5, 3)\)
• Right: \((1, 3)\)
• Up: \((-2, 6)\)
• Down: \((-2, 0)\)

Once these are plotted, gently draw a curved line connecting these, sketching the circumference. Graphing a circle becomes intuitive by visualizing it through its central symmetry and consistent radius from the focal point.