Caitlin is designing a railing for a set of stairs. the railing will begin at a height of 36 inches and follow the slant of the stairs, which decreases 9 inches for every 12 horizontal inches. which function can represent the height, y, of the railing in inches according to the horizontal distance in inches, x, from the top of the stairs? y = –x 36 y = –3x 36 y = x 36 y = 3x 36

Step-by-step explanation:

Since we are given that the railing will begin at a height of 36 inches and follow the slant of the stairs, which decreases 9 vertical inches for every 12 horizontal inches.

Thus the rate of change :  

Rate of change represents the slope of line.

Since we are given that height, y, of the railing in inches according to the horizontal distance in inches, x, from the top of the stairs

And the initial height of the railing was 36 inches .

So,the function which would represent the situation can be shown using equation of line :

y = mx+c

So

c = 36 inches

So, the function :

Hence The function which can represent the height, y, of the railing in inches according to the horizontal distance in inches, x, from the top of the stairs :

*Keyword:

Slope, proportion, relationship, decrease, line equation

The key to answering this question is to understand what means that the height of the railing decreases 9 vertical inches per 12 inches horizontal. This ratio represents the slope of the railing and therefore the slope of the line equation. We can write the relationship as follows:

Then the slope .

In general, the equation of a line is:

.

Where:

m = slope

b = cut point with the y axis

Then, we know that the initial height of the railing is 36 inches. This means that the greatest value that the variable y can take is 36. This value must occur when the value of x (the horizontal length) is minimum (equal to 0). Therefore 36 is the cutoff point with the y axis. b = 36

Finally, the equation sought is:

 Answer

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.

answer

see attachment

explanation

the given function is

this function has y-intercept at

-4 and a slope of 2.

note that, the graph has a positive slope.

the graph that represents this function is the straight line that moves upward from left to right.

the graph touches the y-axis at -4.

the correct choice is the graph shown in the attachment.