Sketch a graph of the circle based on the following equation: x 2 + ( y − 1 ) 2 = 9 x^2+\left(y-1\right)^2=9 x 2 + ( y − 1 ) 2 = 9

let's give this problem a try. In this problem, we are asked to sketch a graph of the circle based on this following equation. So to solve this problem, what I'm going to do is recognize the general form of any circle, which is x minus h squared, plus y minus k squared is equal to r squared. In this equation, r is the radius of the circle, and h and k are the horizontal and vertical position of the circle respectively. Now if I look at our equation up here, I notice that x squared is just by itself. In order for x squared to be by itself, that would mean h would have to equal zero. So we would end up with this whole thing being zero, meaning we would just have x squared. Now as for the y minus k portion, I noticed we have a y minus one. So that means k must be equal to one to give us y minus one squared. Now the last thing I notice here is this r squared, which is nine. And if r squared is equal to nine, that means r would simply be the square root of nine. So r would be the square root of nine which is three. But we could say that for this portion of the equation that r squared is nine. So this is what the equation would look like for the circle. That's the equation we have up here, and we have actually all the information we need about the circle. We have the horizontal and vertical position, and we also have the radius. So to go ahead and graph this, the horizontal position is zero. So on the x axis, we'd be at zero. And on the vertical position is one, so on the y axis, we'd be at one. So our position for the circle is zero one regarding the circle center. And then to find the size of this circle, we see that we have a radius of three. So I can go one, two, three units up, one, two, three units to the right, I can go three units to the left, and then one, two, three units down. And the circle is going to look something like this. So this is how you can graph a circle given the equation. Hope you found this helpful.

16. Parametric Equations & Polar Coordinates

Conic Sections

Calculus

16. Parametric Equations & Polar Coordinates

Conic Sections

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Multiple Choice

Sketch a graph of the circle based on the following equation: $x^2+\left(y-1\right)^2=9$

Graph of a circle centered at (0,1) with radius 3 in a coordinate plane.

Graph of a circle centered at (1,1) with radius 3 in a coordinate plane.

Verified step by step guidance

Step 1: Recognize the equation of the circle, which is given as

x^{2} + {(y - 1)}^{2} = 9

. This is in the standard form of a circle equation:

{(x - h)}^{2} + {(y - k)}^{2} = r^{2}

, where (h, k) is the center and r is the radius.

Step 2: Identify the center of the circle from the equation. The term

(y - 1)

indicates that the y-coordinate of the center is 1, and since there is no subtraction or addition in the x-term, the x-coordinate of the center is 0. Thus, the center is (0, 1).

Step 3: Determine the radius of the circle. The equation is equal to 9, which represents

r^{2}

. Taking the square root of 9 gives the radius r = 3.

Step 4: Sketch the graph of the circle. Start by plotting the center at (0, 1) on the coordinate plane. Then, draw a circle with a radius of 3 units, ensuring that the distance from the center to any point on the circle is 3 units.

Step 5: Verify the graph visually. The circle should be centered at (0, 1) and extend 3 units in all directions (up, down, left, and right). Compare the graph to the provided images to confirm accuracy.