Problem 38 - A Graphical Approach to College Algebra

Question

Graph each circle by hand if possible. Give the domain and range. $$(x-1)^{2}+(y+2)^{2}=16$$

Step-by-Step Solution

Answer

Center: (1, -2); Radius: 4; Domain: [-3, 5]; Range: [-6, 2].

Step 1: Identify the Circle's Center and Radius

The equation of a circle in standard form is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius. Comparing this with our equation \((x-1)^2 + (y+2)^2 = 16\), we can identify that the center of the circle is \((1, -2)\) and the radius is \(\sqrt{16} = 4\).

Step 2: Determine the Domain of the Circle

The domain of a circle refers to all possible x-values the circle can have. Since the radius is 4 and the center is at \((1, -2)\), the x-values range from \(1 - 4\) to \(1 + 4\), which gives the domain: \(-3 \leq x \leq 5\).

Step 3: Determine the Range of the Circle

Similarly, the range of a circle refers to all possible y-values the circle can have. Since the radius is 4 and the center is at \((1, -2)\), the y-values range from \(-2 - 4\) to \(-2 + 4\), which gives the range: \(-6 \leq y \leq 2\).

Step 4: Sketch the Circle

To graph the circle, start by plotting the center at \((1, -2)\). From this center, plot points at a distance of 4 units in all cardinal directions (up, down, left, right). This will help in sketching a circle. Ensure the circle passes through points \((-3, -2)\), \((5, -2)\), \((1, 2)\), and \((1, -6)\).

Key Concepts

Domain and Range

When graphing circles, it’s crucial to identify the circle's domain and range. These terms may sound mathematical, but they're simpler than you think!

• Domain refers to all the possible x-values a circle can have on a graph. When considering the circle equation \((x-1)^2 + (y+2)^2 = 16\), the circle's center is located at \((1, -2)\) with a radius of 4.
• This means the circle stretches 4 units from the center in each horizontal direction, so the domain is from \(1 - 4 = -3\) to \(1 + 4 = 5\). Therefore, the domain is \(-3 \leq x \leq 5\).

• Range is about the y-values. With the same logic, the circle’s vertical reach stretches 4 units from its center. Hence, \(-2 - 4 = -6\) to \(-2 + 4 = 2\).
• The range, therefore, becomes \(-6 \leq y \leq 2\).

Understanding these confinements helps you visualize how a circle behaves within its space on a graph, painting a clearer picture of its placement and size.

Circle Equation

The circle equation is your best friend in understanding the core structure of any circular graph. Let's break it down.

• Standard Form Equation: This is \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) is the circle's center, and \(r\) is the radius.
• Looking at our circle's equation, \((x-1)^2 + (y+2)^2 = 16\), identify that the circle's center is \((h, k) = (1, -2)\).
• The radius \(r\) is calculated by taking the square root of 16, which is 4.

This equation is powerful because it not only tells where the circle lies on the graph but also how big or small it is. A change in \(h\) or \(k\) shifts the circle's position, while \(r\) directly affects its size, encapsulating its graphical potential into simple numbers.

Graphing Techniques

Let's dive into some handy techniques to effortlessly sketch circles. Proper graphing leads to clarity in visual representations.

• Mark the Center: Start by plotting the circle's center derived from the equation, \((1, -2)\).
• Use the Radius: From the center, move 4 units in all four directions: left, right, up, and down. Plot these points as they mark the circle's boundaries, \((-3, -2), (5, -2), (1, 2), (1, -6)\).
• Draw the Curve: Connect these dots as smoothly as possible, forming a round shape. Remember, a circle should look continuous and looped rather than polygonal.

Once you get the hang of these steps, drawing circles will become intuitive. Practicing these techniques reinforces spatial awareness, making graphing less daunting, and helping you visually understand mathematical relationships.