Graphing Circle Functions - Algebra II

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Question

The graph of the equation

$x^{2} + y^{2} + 10x - 8y = 84$

is a circle with what as the length of its radius?

Answer

Rewrite the equation of the circle in standard form

$(x - h)^{2} + \left ( y - k\right )^{2} = r^{2}$

as follows:

$x^{2} + y^{2} + 10x - 8y = 84$

$x^{2} + 10x+ y^{2} - 8y = 84$

Since $\left (\frac{10}{2} \right ) ^{2} = 5 ^{2} = 25$ and $\left (\frac{-8}{2} \right ) ^{2} = (-4)^{2} = 16$ , we complete the squares by adding:

$x^{2} + 10x+ 25 + y^{2} - 8y + 16 = 84 + 25 + 16$

$(x+5) ^{2} + \left (y- 4 \right ) ^{2} = 125$

The standard form of the equation sets

$r^{2} = 125$ ,

so the radius of the circle is

$r = \sqrt{125} = \sqrt{25} \cdot \sqrt{5} = 5 \sqrt{5}$

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Question

Determine the graph of the equation

Answer

The equation of a circle in standard for is:

Where the center and the radius of the cirlce is .

Dividing by 4 on both sides of the equation yields

$(x-2)^2+(y-3)^2 = \frac{1}{4}$

or

$(x-2)^2+(y-3)^2 = \bigg(\frac{1}{2}\bigg)^2$

an equation whose graph is a circle, centered at (2,3) with radius = .5

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Question

Give the radius and the center of the circle for the equation below.

$(x-3)^{2}+(y + 4)^2 = 16$

Answer

Look at the formula for the equation of a circle below.

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

Here is the center and is the radius. Notice that the subtraction in the center is part of the formula. Thus, looking at our equation it is clear that the center is and the radius squared is . When we square root this value we get that the radius must be .

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Question

Determine the equation of a circle whose center lies at the point $\left ( 3,4\right )$ and has a radius of .

Answer

The equation for a circle with center and radius is :

Our circle is centered at with radius , so the equation for this circle is :

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Question

$\small f(x)=(x-2)^2+(y+5)^2=36$

What is the radius of the circle?

Answer

The parent equation of a circle is represented by $\small f(x)=(x-h)^2+(y-k)^2=r^2$ . The radius of the circle is equal to $\small r$ . The radius of the cirle is $\small \sqrt{36}$ .

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Question

What is the center of the circle expressed by the funciton $\small x^2+4x+y^2=5$ ?

Answer

The equation can be rewritten so that it looks like the parent equation for a circle $\small (x-h)^2+(y-k)^2=r^2$ . After completeing the square, the equation changes from $\small x^2+4x+y^2=5$ to $\small x^2+4x+4+y^2=9$ . From there it can be expressed as $\small (x+2)^2+y^2=9$ . Therefore the center of the circle is at $\small (-2,0)$ .

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Question

What are the coordinates of the center of a circle with the equation ?

Answer

The equation of a circle is , in which (h, k) is the center of the circle. To derive the center of a circle from its equation, identify the constants immediately following x and y, and flip their signs. In the given equation, x is followed by -1 and y is followed by -6, so the coordinates of the center must be (1, 6).

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Question

What is the radius of a circle with the equation ?

Answer

To convert the given equation into the format , complete the square by adding to the x-terms and to the y-terms.

The square root of 4 is 2, so the radius of the circle is 2.

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Question

What is the radius of a circle with the equation ?

Answer

To convert the given equation into the format , complete the square by adding to the x-terms and to the y-terms.

The square root of 25 is 5, so the radius of the circle is 5.

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Question

Which equation does this graph represent?

Answer

The equation of a circle is , in which (h, k) is the center of the circle and r is its radius. Because the graph of the circle is centered at (0, 0), h and k are both 0. Because the radius is 3, the right side of the equation is equal to 9.

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Question

Which equation does this graph represent?

Answer

The equation of a circle is , in which (h, k) is the center of the circle and r is its radius. Because the graph of the circle is centered at (2, -3), h and k are -2 and 3. Because the radius is 4, the right side of the equation is equal to 16.

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