Question

if two graphs each have a horizontal line and have the same y - intercept, are they the same linear function? why or why not?
yes: a horizontal line has a slope of zero, and if it intersects the y - axis in the same place as another horizontal line, then the linear function they represent is the same. if two lines have the same slope and y - intercept, they are the same function.
yes: a horizontal line has an equation in the form x = k, where k is any real number. this type of line crosses the y - axis in one place and if another line in the same form does as well, then the linear function they represent is the same.
no: a horizontal line has a slope of zero and wont intersect the y - axis at all, so it will not have a y - intercept. therefore, two graphs each containing a horizontal line wont have y - intercepts at all.
no: a horizontal line has an undefined slope and wont intersect the y - axis at all, so it will not have a y - intercept. therefore, two graphs each containing a horizontal line wont have y - intercepts at all.

Explanation:

A horizontal line has a slope of 0, and its equation is $y = k$, where $k$ is the y-intercept (the point where it crosses the y-axis). For two linear functions to be identical, they must have the same slope and same y-intercept. Since both horizontal lines have a slope of 0 and share the same y-intercept, they are the same function. The first option correctly states this logic, while the second option has a typo (writing $z=k$ instead of $y=k$), and the last two options incorrectly claim horizontal lines have no y-intercept or undefined slope.

Answer:

Yes; a horizontal line has a slope of zero, and if it intersects the y-axis in the same place as another horizontal line, then the linear function they represent is the same. If two lines have the same slope and y-intercept, they are the same function.