To graph the equation $x^2 + y^2 = 72$, follow these steps:

1. Recognize the Equation Type

The given equation $x^2 + y^2 = 72$ represents a circle centered at the origin (0, 0) with a radius $r$.

2. Determine the Radius

The general form of a circle's equation is $x^2 + y^2 = r^2$. Here, $r^2 = 72$. To find the radius $r$:

$r = \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} \approx 8.49$

3. Plot the Circle

• Center: The center of the circle is at (0, 0).
• Radius: The radius is approximately 8.49 units.

To graph the circle:

• Plot the center at (0, 0).
• Mark points 8.49 units away from the center in all directions (up, down, left, and right).
• Draw a smooth curve that passes through these points to form a circle.

4. Graph Representation

The circle will touch the points approximately at (±8.49, 0) and (0, ±8.49) on the Cartesian plane.

Would you like to see a detailed step-by-step graphical representation of this process, or do you have any specific questions?

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Relative Questions:

1. What are the coordinates of points where the circle intersects the x-axis and y-axis?
2. How would the graph change if the equation were $x^2 + y^2 = 100$?
3. Can you graph the equation $(x - 2)^2 + (y + 3)^2 = 25$?
4. What is the significance of the radius in the graph of a circle?
5. How would you determine if a point lies inside, on, or outside the circle?

Tip: The radius of a circle from its equation $x^2 + y^2 = r^2$ is always $\sqrt{r^2}$, which helps determine the size and scale of the graph.