Problem 101 - A Graphical Approach to College Algebra

Question

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x^{2}+(y-2)^{2}=16$$

Step-by-Step Solution

Answer

Graph a circle with center at (0, 2) and radius 4.

Step 1: Identify the Equation Type

The given equation is in the form \(x^2 + (y-k)^2 = r^2\), which is the standard equation for a circle. Here, \(y-2\) indicates the center's y-coordinate is shifted, while \(x^2\) shows there's no shift on the x-coordinate.

Step 2: Determine the Center and Radius

From the equation \(x^2 + (y-2)^2 = 16\), we identify the center of the circle as \((0, 2)\) and the radius as \(r = 4\) since \(16 = 4^2\).

Step 3: Prepare to Graph the Circle

Locate the center of the circle at the point \((0,2)\). The radius of the circle is \(4\), meaning that from the center, the circle extends 4 units in all directions.

Step 4: Draw the Circle

From the center point \((0, 2)\), mark points at \((0+4, 2)\), \((0-4, 2)\), \((0, 2+4)\), and \((0, 2-4)\). These points represent the intersections of the circle with the horizontal and vertical lines of the grid.

Step 5: Complete the Circle

Using the points marked, sketch a circle that passes through all four points. This circle is centered at \((0,2)\) and has a consistent radius of 4 around the center.

Key Concepts

Standard Equation of a Circle

When graphing circles, a key element is the standard equation, which is expressed as \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle, while \(r\) stands for the radius. This format is particularly useful because it allows you to determine both the center and the radius directly from the equation. For instance, in the exercise problem \(x^2 + (y-2)^2 = 16\), we see that since \(x\) isn't shifted, \(h = 0\) and \(k = 2\), matching the structure of \((y-k)\). The right side of the equation \(16\) equals \(r^2\), from which we can deduce \(r = 4\). This straightforward representation makes it simple to identify critical components of the circle for graphing.

Center of a Circle

In the equation \((x-h)^2 + (y-k)^2 = r^2\), the center of the circle is denoted by the coordinates \((h, k)\). Identifying the center is crucial as it is the anchor point for the circle, marking its exact location on the coordinate plane. In our example equation \(x^2 + (y-2)^2 = 16\), the center is determined by identifying the constants \(h\) and \(k\). Here, \(h\) is 0, meaning there is no horizontal shift, and \(k\) is 2, indicating a vertical shift upwards by 2 units. So, the center is at the point \((0, 2)\). This means the circle is centered prominently on the y-axis, directly affecting how it is graphically represented.

Radius of a Circle

The radius is a fundamental feature in a circle's equation, symbolized by \(r\). It indicates the constant distance from the circle's center to any point on its edge. In the formula \((x-h)^2 + (y-k)^2 = r^2\), \(r\) is found by taking the square root of the number on the equation's right side. For the problem \(x^2 + (y-2)^2 = 16\), the number 16 is equal to \(r^2\). Thus, the radius \(r\) is \(\sqrt{16} = 4\). This tells us the circle extends equally 4 units in all directions from the center, ensuring symmetry and providing clear guidelines for sketching the circle accurately on a graph.

Graphing Circles

Graphing a circle starts with pinpointing its center and radius as derived from its standard equation. Using the center \((0, 2)\) and radius \(4\), you can mark specific points around the center that lie at a distance equivalent to the radius. Place these points directly up, down, left, and right of the center, such as \((0+4, 2)\), \((0-4, 2)\), \((0, 2+4)\), and \((0, 2-4)\). These guide dots help ensure the circle is symmetric and evenly sized.

• The initial point marks the center: \((0, 2)\).
• Next, determine boundary points along the x and y axes using the radius.
• Connect the dots smoothly to outline the circle.

For a more accurate sketch, continually check that the distances from the center to any point on the circle's edge remain uniform at 4 units. This process reaffirms the circle’s perfect symmetry.