Problem 78 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(x^{2}+y^{2}=16\)

Step-by-Step Solution

Answer

The graph of the equation is a circle centered at the origin with a radius of 4. The x-intercepts and y-intercepts are \(-4\) and \(4\). The graph is symmetric about the x-axis, y-axis, and the origin.

Step 1: Sketching the graph

Sketch a circle centered at the origin (0,0) with a radius of 4. This is obtained by taking the square root of the constant term on the right of the equation, which gives the radius of the circle.

Step 2: Identifying the intercepts

The x-intercepts are the points where the graph intersects the x-axis (where \(y=0\)). Substituting \(y=0\) in the equation, we get \(x^{2}=16\), therefore \(x=\pm4\). The y-intercepts are the points where the graph intersects the y-axis (where \(x=0\)). Substituting \(x=0\) in the equation, we get \(y^{2}=16\), therefore \(y=\pm4\). So, the x-intercepts are \(-4\) and \(4\), and the y-intercepts are \(-4\) and \(4\).

Step 3: Testing for symmetry

To test for symmetry about the x-axis, replace \(y\) with \(-y\) and see if the equation remains the same. Doing so gives, \(x^{2}+(-y)^{2}=16\). The equation remains the same, so it is symmetric about the x-axis. To test for symmetry about the y-axis, replace \(x\) with \(-x\). This also yields the original equation, so the graph is also symmetric about the y-axis. If both x and y are replaced with their negatives, the original equation is still valid which denotes that it is symmetric about the origin.

Key Concepts

Graphing Equations

Mastering how to graph equations is foundational for students delving into algebra and higher mathematics. The equation in our exercise, {x^2+y^2=16}, represents a circle. To graph it, we find its center and radius. Since the terms x^2 and y^2 have no coefficients or constants attached, the circle is centered at the origin, (0,0). The number 16 is the square of the radius; thus, the radius is 4.

When graphing, begin by plotting the center. Then, use the radius to mark points at equal distances in all directions from the center—up, down, left, and right. These points will align with the intercepts if you're graphing correctly. Connect these points using a smooth, round curve to complete the circle. A common mistake is to draw an oval or misshapen circle; ensure your graph reflects the equal radius at all points.

Finding Intercepts

A critical step in sketching graphs is finding intercepts. Intercepts are points where the graph crosses the x-axis and y-axis. To find the x-intercepts, set y to zero and solve for x. Conversely, to find the y-intercepts, set x to zero and solve for y.

For our equation {x^2+y^2=16}, let's calculate them. For x-intercepts: when y=0, we have x^2=16. Thus, x equals ±4. For y-intercepts: when x=0, we have y^2=16, yielding y as ±4. Hence, the intercepts are at the points (-4,0), (4,0), (0,-4), and (0,4). Marking intercepts first on your graph can offer a helpful structure when drawing curves or lines.

Symmetry in Algebra

Symmetry in algebra helps us understand the balance and proportions inherent in equations and their graphs. A graph is symmetric with respect to an axis if reflecting it across that axis yields the same graph. For the circle equation {x^2+y^2=16}, we use algebraic manipulation to test symmetry.

Substitute y with -y to test the x-axis symmetry, and x with -x for y-axis symmetry. If the equation remains unchanged, the graph is symmetric about that axis. In this equation, replacing x or y with their negatives still satisfies the original equation, confirming symmetry about both axes. Additionally, this circle is symmetric about the origin because swapping both x and y with their negatives doesn't change the equation—this type of symmetry is known as 'central symmetry'.

Circle Equations

Circle equations take the form {x^2+y^2=r^2}, where r stands for the radius. The elements involved tell us much about the circle. A circle's equation without x or y terms outside the squares indicates a center at the origin; coefficients other than one multiply the x or y terms may suggest a stretch or compression.

To sketch the graph of a circle, use the radius found in the equation for precision. In our case, {r^2=16} suggests a radius of 4. Always cross-check your plotted circle with the given equation to confirm its accuracy. A complete understanding of circle equations can greatly enhance geometric intuition and provide a stepping stone to more complex algebraic topics.