Problem 53 - Intermediate Algebra

Question

Use a graphing calculator to graph each equation. $$ (x+1)^{2}+y^{2}=16 $$

Step-by-Step Solution

Answer

The graph is a circle centered at \((-1, 0)\) with radius 4.

Step 1: Identify the Equation Type

The given equation \((x+1)^2 + y^2 = 16\) is the equation of a circle centered at \((-1, 0)\) with a radius of 4, as it is in the form \((x-h)^2 + (y-k)^2 = r^2\).

Step 2: Setup for Graphing

Prepare your graphing calculator. Ensure that you can input equations directly in the form \((x+1)^2 + y^2 = 16\). Most graphing calculators will allow you to graph implicitly or transform the equation based on this form.

Step 3: Input the Equation

Enter the equation \((x+1)^2 + y^2 = 16\) into the graphing calculator. If it requires explicit functions, solve for \(y\) as \(y = \pm \sqrt{16 - (x+1)^2}\) to graph the upper and lower halves separately.

Step 4: Adjust Viewing Window

Set an appropriate window on your calculator to visualize the circle. Since the radius is 4, a window from \(-6\) to \(4\) for \(x\) and \(-4\) to \(4\) for \(y\) should work well.

Step 5: Graph the Equation

Execute the graphing function to display the circle on the screen. Check that it appears as a circle with center \((-1, 0)\) and radius \(4\).

Step 6: Verify and Interpret

Verify that the graph matches the expected circle by checking the center and radius. The circle should pass through points like \((3, 0)\) and \((-5, 0)\) along the \(x\)-axis, and reach \(y = 4\) and \(y = -4\) vertically.

Key Concepts

Circle Equations

The equation \((x+1)^2 + y^2 = 16\) represents a circle in the coordinate plane. When identifying circle equations, the standard form is \((x-h)^2 + (y-k)^2 = r^2\). Here,

• \((h, k)\) is the center of the circle, and
• \(r\) is the radius.

In our given equation, by comparing it to the standard form, we recognize

• The center is \((-1, 0)\), because the equation can be rewritten to match the standard form with \(h = -1\) and \(k = 0\).
• The radius is \(4\), since the equation equals \(16\), which is \(r^2\).

This implies that any point \((x, y)\) that satisfies this equation will lie on the perimeter of a circle centered at \((-1, 0)\) with a radius of \(4\). Understanding how to identify the center and radius is crucial for graphing circles correctly.

Coordinate Geometry

In coordinate geometry, every equation like \((x+1)^2 + y^2 = 16\) has its story in the spatial setup of a graph. Points are arranged in the coordinate plane using coordinates \((x, y)\), and geometrical shapes like circles are defined by their specific equations.
For example, you can use coordinate geometry to trace the path of a circle defined by our equation. The algebraic structure tells you everything:

• The "plus 1" in \((x+1)^2\) indicates a horizontal shift of the circle to the left of the origin by 1 unit.
• The \(y^2\) part means there is no vertical shift; the circle remains centered along the horizontal line \(y=0\).

Knowing how to interpret these components allows you to visualize and sketch the circle accurately on a coordinate grid.
Coordinate geometry transforms algebraic expressions into visual concepts, providing deeper insights into relationships and distances in the plane.

Algebraic Graphing Methods

Algebraic graphing methods allow us to convert equations into visual graphs using tools like graphing calculators. To graph an equation such as \((x+1)^2 + y^2 = 16\), follow a series of methodical steps:
1. **Identify the Type of Equation**: Recognize that this is a circle equation, as discussed.2. **Prepare for Graphing**: With a graphing calculator, ensure it can handle implicit equations. If needed, convert to explicit form: solve for \(y\) in terms of \(x\), giving \(y = \pm \sqrt{16 - (x+1)^2}\).
3. **Input the Equation**: Enter the equation into the calculator. If using the explicit form, graph top and bottom parts separately. 4. **Adjusting Viewing Settings**: Set the viewing window to capture the whole circle. For example, use the window settings of \(-6\) to \(4\) for \(x\) and \(-4\) to \(4\) for \(y\). This ensures the circle fits well within the screen.
5. **Graph and Verify**: After inputting and setting the window, execute the graphing function. Verify the graph resembles a circle with the correct center and radius by observing key points like where the circle intercepts the \(x\) and \(y\) axes.
Utilizing algebraic graphing methods like this enhances our ability to visualize complex equations effectively.