Problem 40 - Algebra 2

Question

For Exercises \(40-43,\) use the following information. since a circle is not the graph of a function, you cannot enter its equation directly into a graphing calculator. Instead, you must solve the equation for \(y .\) The result will contain a \pm symbol, so you will have two functions. Solve \((x+3)^{2}+y^{2}=16\) for \(y\)

Step-by-Step Solution

Answer

\(y = \pm \sqrt{16 - (x + 3)^2}\)

Step 1: Identify the Circle Equation

The given equation is \((x + 3)^2 + y^2 = 16\), which represents a circle centered at \((-3, 0)\) with a radius of \(4\).

Step 2: Rearrange the Equation

To solve for \(y\), start by isolating \(y^2\). Subtract \((x + 3)^2\) from both sides: \(y^2 = 16 - (x + 3)^2\).

Step 3: Solve for \(y\)

Take the square root on both sides to solve for \(y\): \[y = \pm \sqrt{16 - (x + 3)^2}\].

Step 4: Identify the Two Functions

The solution \(y = \pm \sqrt{16 - (x + 3)^2}\) indicates two functions: \(y = \sqrt{16 - (x + 3)^2}\) and \(y = -\sqrt{16 - (x + 3)^2}\). These represent the upper and lower halves of the circle, respectively.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for visualizing mathematical equations, especially those involving complex shapes like circles. In the context of solving circle equations, graphing calculators help us approximate visual outputs. However, the nature of a circle not being a function of a single variable presents a challenge.

When given an equation like \((x + 3)^2 + y^2 = 16\), entering it directly into a graphing calculator isn't feasible. This is because circles don’t fit the standard y = f(x) format. Instead, we must solve for \(y\), resulting in two separate functions, \(y = \sqrt{16 - (x + 3)^2}\) and \(y = -\sqrt{16 - (x + 3)^2}\). These equations each represent half of the circle on the graph.

• Understand your calculator’s limitations with non-function graphs.
• Use separate inputs for the upper and lower halves of the circle.
• Visualize circles through piecewise function representations.

Circle Equation

A circle equation is typically represented in the standard form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius. In the given exercise, \((x + 3)^2 + y^2 = 16\) follows this format with the center being \((-3, 0)\) and the radius \(r = 4\). This particular representation shows the circle lying symmetrically across the origin, shifted left by 3 units.

Understanding the components helps in visualizing the circle:

• Center point: Identifies where the circle is pinned on the coordinate plane.
• Radius: Defines the size of the circle, influencing how wide or narrow it appears.

The symmetry and algebraic manipulation of circle equations are crucial in transforming these into graphable forms, particularly with tools like graphing calculators.

Coordinate Geometry

Coordinate geometry provides a systematic method for discovering properties of geometric figures using algebraic equations plotted on a coordinate plane. This field of mathematics is pivotal when dealing with circle equations like \((x + 3)^2 + y^2 = 16\).

Using coordinate geometry, we establish relationships between the equation and its geometric representation. The equation reveals the circle's graphical traits, indicating how it is positioned and how it intersects with the axes. For instance:

• The equation’s alteration, \((x + 3)\), shifts the circle left along the x-axis.
• The constant value, \(16\), gives insights into the radius, equating to \(r^2\).

This often facilitates better comprehension and visualization, making abstract mathematical concepts more tangible.

Function Transformation

Function transformation refers to various operations that alter the appearance of a function’s graph. In the context of circle equations, these transformations are vital for graphing purposes when using a graphing calculator.

For the problem \((x + 3)^2 + y^2 = 16\), rewriting as two functions \(y = \pm \sqrt{16 - (x + 3)^2}\) is a form of decomposition. This breaks the circle into two functions:

• \(y = \sqrt{16 - (x + 3)^2}\) - representing the top half.
• \(y = -\sqrt{16 - (x + 3)^2}\) - representing the bottom half.

By viewing a circle this way, each half can be graphed separately. This transformation allows each "halved" function to be entered into a graphing tool as input functions, helping students visualize and comprehend the complete circular graph. The transformed functions assist graphically in manifesting the entire geometric shape in parts, which are then pieced back together visually.