What is the standard form of the equation of the circle x^2 - 2x + y^2 - 8y + 8 = 0?

Step-by-step explanation:

Begin by grouping the x terms and the y terms together and separating the constants out.

Now we'll complete the square on those x and y terms. Take half the linear term of each, square it, and add it to both sides. Our linear x term is 2, half of 2 is 1 and 1 squared is 1, so we add that in. Likewise, half the linear y term (which is 8) is 4, and 4 squared is 16, so we add that in, too. Like this:

Doing this gives us the perfect square binomials for each of the x and y terms, and then gives us the radius on the right:

This is a circle with a center of (1, 4) and a radius of 3.

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.