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Magazine Chapter 2: Problem 34
19鈥�44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ x^{2}+y^{2}=9 $$
Short Answer
Expert verified
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 by step solution
01
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 \(r = 3\).
02
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 \(x\).
03
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.
04
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 \((-3, 0)\).
05
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 \((0, -3)\).
06
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 \(-y\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
- \(x^2 + 0^2 = 9\)
- \(x^2 = 9\)
- \(x = \pm 3\)
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:
- \(0^2 + y^2 = 9\)
- \(y^2 = 9\)
- \(y = \pm 3\)
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.
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 \(y = \pm \sqrt{9 - x^2}\).
- \(x = 0\): \(y = \pm 3\)
- \(x = 1\): \(y = \pm \sqrt{8}\)
- \(x = 2\): \(y = \pm \sqrt{5}\)
- \(x = 3\): \(y = 0\)
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.
