The coordinates of the vertices of triangle ABC are A(1,6), B(2,9), and C(7,10). Identify A'''B'''C''' after the following transformations; reflection y=x, reflection x-axis, and translation (x -7, y + 4) respectively.

A'''(-1,3), B'''(2,2) and C'''(3,-3)

Step-by-step explanation:

Hello There!

So the first thing we need to do is reflect the coordinates given over the y = x line which would be a straight line across the origin with a rate of change of 1

The rule for a reflection over the y = x line is  (x,y) > (y,x)

so pretty much we're just flipping the y and x values

(1,6) ---> (6,1)

(2,9) ---> (9,2)

(7,10) ---> (10,7)

so the new coordinates of A'B'C' are A'(6,1) , B'(9,2), and C'(10,7)

Now we want to reflect the points over the x axis

The rule for reflection over the x axis is (x,y) > (x,-y)

so the x value stays the same and the y values sign just get flipped so if the y value was originally negative then it will change to positive vise versa

(6,1) ---> (6,-1)

(9,2) ---> (9,-2)

(10,7) ---> (10,-7)

So the coordinates of A''B''C'' are  A''(6,-1) , B''(9,-2), and C''(10,-7)

Our final step is to translate 7 units to the left 4 units up

so the rule would be (x,y) > (x - 7, y + 4)

6-7=-1 | -1+4=3

(6,-1) ---> (-1,3)

9-7=2 | -2+4=2

(9,-2) ---> (2,2)

10-7=3 | -7+4=-3

(10,-7) ---> (3,-3)

so the after the three transformations the coordinates would be

A'''(-1,3), B'''(2,2) and C'''(3,-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.