Graphing Circles: A Step-by-Step Guide

by Viktoria Ivanova 39 views

Hey guys! Let's dive into the world of circles and graphs! Today, we're going to break down how to sketch the graph of a circle given its equation. Specifically, we'll be working with the equation (x−4)2+(y−3)2=36(x-4)^2 + (y-3)^2 = 36. Don't worry, it might look a little intimidating at first, but we'll take it one step at a time. We will find the center and radius and then use that information to accurately sketch the circle. So, grab your pencils and let's get started!

(a) Finding the Center of the Circle

Okay, first things first, let's find the center of the circle. The equation we're working with, (x−4)2+(y−3)2=36(x-4)^2 + (y-3)^2 = 36, is in the standard form of a circle's equation. This standard form is super helpful because it tells us directly what the center and radius are. Remember the standard form equation: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) represents the center of the circle and rr represents the radius.

Now, let's compare our equation, (x−4)2+(y−3)2=36(x-4)^2 + (y-3)^2 = 36, with the standard form. Notice how the xx term is (x−4)(x - 4)? This means that hh, the x-coordinate of the center, is 4. Similarly, the yy term is (y−3)(y - 3), so kk, the y-coordinate of the center, is 3. Therefore, the center of our circle is at the point (4,3)(4, 3). Easy peasy, right? To make it crystal clear, think of it this way: the numbers inside the parentheses with xx and yy are the opposite of the coordinates of the center. So, (x−4)(x - 4) gives us a +4 for the x-coordinate, and (y−3)(y - 3) gives us a +3 for the y-coordinate. This is a crucial step because the center is the anchor point from which we'll draw the entire circle. Getting the center wrong means the whole graph will be off, so double-check your work here! Remember, understanding the connection between the equation and the center is far more important than just memorizing a formula. This understanding will help you tackle any circle equation that comes your way. We've successfully located the heart of our circle – the point (4, 3). Now, we're one step closer to sketching the complete graph. Let's move on to the next exciting part: finding the radius!

(b) Determining the Radius of the Circle

Alright, we've nailed down the center of the circle. Now, let's figure out the radius. Remember that the standard form equation of a circle is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where rr represents the radius. In our equation, (x−4)2+(y−3)2=36(x-4)^2 + (y-3)^2 = 36, the number on the right side, 36, is equal to r2r^2. So, to find the radius, we need to figure out what number, when squared, equals 36.

Think back to your perfect squares. What number times itself gives you 36? If you said 6, you're absolutely correct! The square root of 36 is 6. Therefore, the radius of our circle is 6 units. That means from the center of the circle, we can move 6 units in any direction (up, down, left, right, and all points in between) to reach a point on the circle. This radius is super important because it dictates the size of our circle. A larger radius means a bigger circle, and a smaller radius means a smaller circle. Knowing the radius allows us to accurately represent the circle's dimensions on our graph.

It's also worth noting that the radius is always a positive value. We're talking about a physical distance, and distances can't be negative. So, even though mathematically, -6 squared also equals 36, we only consider the positive root, which is 6, for the radius. Now, we have two key pieces of information: the center, which is (4, 3), and the radius, which is 6. With these two elements, we're fully equipped to sketch the circle's graph. We know where the circle is positioned on the coordinate plane (the center), and we know how far the circle extends from that center (the radius). Let's move on to the exciting part of putting it all together and drawing the circle!

(c) Sketching the Graph of the Circle

Okay, let's put everything together and graph the circle! We know the center is at the point (4, 3) and the radius is 6 units. Grab your graph paper (or a digital graphing tool) and let's get started.

  1. Plot the Center: First, locate the point (4, 3) on your coordinate plane. Remember, the x-coordinate is 4, so move 4 units to the right along the x-axis. Then, the y-coordinate is 3, so move 3 units up along the y-axis. Mark this point clearly – this is the heart of our circle!
  2. Use the Radius to Find Key Points: Now that we have the center, we can use the radius to find a few key points on the circle. Since the radius is 6, we can move 6 units in each of the four cardinal directions (up, down, left, and right) from the center.Let's do it step by step:
    • 6 Units to the Right: Starting from (4, 3), move 6 units to the right. This brings us to the point (10, 3).
    • 6 Units to the Left: Starting from (4, 3), move 6 units to the left. This brings us to the point (-2, 3).
    • 6 Units Up: Starting from (4, 3), move 6 units up. This brings us to the point (4, 9).
    • 6 Units Down: Starting from (4, 3), move 6 units down. This brings us to the point (4, -3). These four points give us a good sense of the circle's shape and size.
  3. Sketch the Circle: Now for the fun part! Using the center as your guide and the four points you just found, sketch a smooth, round circle. Try to make the circle as symmetrical as possible. It can be helpful to lightly sketch the circle first and then go over it again to make it darker and more defined. Imagine you're drawing a perfect loop around the center point, maintaining a consistent distance (the radius) from the center at all times. If you're using a digital tool, there's likely a circle tool that will make this step even easier. If you're drawing by hand, don't worry if it's not perfectly round – the key is to show that you understand the concept and have correctly plotted the center and radius. A slightly wobbly circle is perfectly fine!
  4. Double-Check: Once you've sketched your circle, take a moment to double-check your work. Does the center look like it's in the right place? Does the circle appear to have a radius of approximately 6 units? If everything looks good, congratulations! You've successfully graphed the circle represented by the equation (x−4)2+(y−3)2=36(x-4)^2 + (y-3)^2 = 36.

And there you have it! We've taken an equation, found its center and radius, and used that information to sketch the graph of the circle. This process might seem like a lot of steps at first, but with practice, it becomes second nature. The key is to understand the connection between the equation, the center, the radius, and the visual representation of the circle on the coordinate plane. So, keep practicing, and you'll be graphing circles like a pro in no time!