Problem 34 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ x^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

A circle centered at (0,0) with radius 3; x-intercepts at (3,0) and (-3,0); y-intercepts at (0,3) and (0,-3); symmetric about both axes and the origin.

Step 1: Understand the Equation

The equation given is \(x^2 + y^2 = 9\). This is the equation of a circle centered at the origin \((0,0)\) with a radius of 3, because the equation follows the type \((x-h)^2 + (y-k)^2 = r^2\) where \(h=k=0\) and \(r^2 = 9\), so \(r = 3\).

Step 2: Make a Table of Values

To create a table of values, select various values of \(x\) and solve for \(y\): For \(x=0\), \(y=\pm3\); For \(x=1\), \(y=\pm\sqrt{8}\); For \(x=2\), \(y=\pm\sqrt{5}\); For \(x=3\), \(y=0\); likewise for negative \(x\).

Step 3: Sketch the Graph

Plot the points obtained from the table of values on the Cartesian plane. Connect these points smoothly to sketch a circle centered at the origin with radius 3.

Step 4: Find the x-intercepts

To find the \(x\)-intercepts, set \(y=0\) in the equation: \(x^2 + 0^2 = 9\), which simplifies to \(x^2 = 9\). Thus, \(x = \pm3\). The intercepts are \((3, 0)\) and \((-3, 0)\).

Step 5: Find the y-intercepts

To find the \(y\)-intercepts, set \(x=0\) in the equation: \(0^2 + y^2 = 9\), which simplifies to \(y^2 = 9\). Thus, \(y = \pm3\). The intercepts are \((0, 3)\) and \((0, -3)\).

Step 6: Test for Symmetry

The equation is symmetric with respect to the x-axis, y-axis, and the origin due to its form. Symmetry about the x-axis and y-axis is evident as \(y^2 = 9 - x^2\) and \(x^2 = 9 - y^2\) remain unaltered when \(x\) and \(y\) are respectively replaced by \(-x\) and \(-y\).

Key Concepts

Understanding X-Intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis. At these points, the value of \(y\) is zero. To find the x-intercepts of the circle equation \(x^2 + y^2 = 9\), set \(y = 0\) and solve for \(x\). For this problem:

• \(x^2 + 0^2 = 9\)
• \(x^2 = 9\)
• \(x = \pm 3\)

Therefore, the x-intercepts are \((3, 0)\) and \((-3, 0)\). It's essential to understand that finding x-intercepts helps in graphing as they indicate where the circle crosses the horizontal axis.

Exploring Y-Intercepts

Just like x-intercepts, y-intercepts occur where the graph crosses the y-axis. At these points, the value of \(x\) is zero. To find the y-intercepts of our equation \(x^2 + y^2 = 9\), set \(x = 0\) and solve for \(y\). Here's how it's computed:

• \(0^2 + y^2 = 9\)
• \(y^2 = 9\)
• \(y = \pm 3\)

Thus, the y-intercepts are \((0, 3)\) and \((0, -3)\). These points are crucial for sketching the circle because they give us insights into where it touches the vertical axis.

Understanding Symmetry in Circles

Symmetry is an important property in graphing. A graph is symmetric about the x-axis, y-axis, or the origin if flipping over these lines does not change the graph. The equation \(x^2 + y^2 = 9\) describes a circle that is symmetric with respect to:

• **The x-axis:** Flipping over the x-axis results in a graph that looks the same. This is because replacing \(y\) with \(-y\) does not alter the equation.
• **The y-axis:** Similarly, replacing \(x\) with \(-x\) keeps the equation unchanged.
• **The origin:** This means the graph looks the same if rotated 180 degrees around the origin.

Recognizing symmetry helps to simplify graphing by reducing how much of the circle you need to manually plot.

Creating a Table of Values

A table of values is a helpful tool for graphing, especially when dealing with circles. By selecting different x-values and computing the corresponding y-values, we gather concrete points that lie on the graph.Steps for using \(x^2 + y^2 = 9\) in a table of values:

• Choose x-values within the radius. E.g., \(x = 0, 1, 2, 3\).
• Calculate y-values using \(y = \pm \sqrt{9 - x^2}\).

For this circle:

• \(x = 0\): \(y = \pm 3\)
• \(x = 1\): \(y = \pm \sqrt{8}\)
• \(x = 2\): \(y = \pm \sqrt{5}\)
• \(x = 3\): \(y = 0\)

Repeating this for negative x-values provides a complete set of points for sketching.

Sketching the Graph of a Circle

Once you have considered the x- and y-intercepts, symmetry, and created a table of values, sketching the graph of a circle becomes manageable. With the equation \(x^2 + y^2 = 9\), you know:

• Center is at \((0, 0)\).
• Radius is 3, as \(r^2 = 9\) gives \(r = 3\).
• Intercepts guide the initial points for the circle.
• Symmetry tells you the circle is evenly round.

Plot all calculated points from the table of values, ensuring they form a smooth curve. As you connect these points, a perfect circle centered at the origin should emerge, respecting all components of the circle's symmetry.