Answer:
Find the slope for each of the points in the table. If all the slopes equal each other, the table represents a linear function. If not, then it's the opposite.
Step-by-step explanation:
(You didn't post the full picture, but I'll try to make sense of your problem and answer anyways.)
A linear function, or a line, has a constant slope. This means that no matter where you are on the line, it was always be "rising" some y and "running" some x. It will never be fluctuating in slope or anything like that because it's a line, not a curve, not a zigzag, etc. Just a line.
How do we observe this in tables? Well, remember that the slope m can be found with the following formula:
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
where (x1, y1) and (x2, y2) are two points on a line.
So, we can show that all these points have equal slopes to see if they form a line. For example, let's look at the table shown in the picture. We have five points, so let's find the slope for two of them at a time.
First, let's find the slope between (1, 5) and (2, 20).
[tex]m=\frac{20-5}{2-1} \\m=\frac{15}{1} \\m=15[/tex]
The slope between these two points is 15.
Now, let's find the slope between (2, 20) and (3, 45).
[tex]m=\frac{45-20}{3-2} \\m=\frac{25}{1} \\m=25[/tex]
And the slope is ... 25? This can't be right!
Well, it is. This means that these three points can't all form a line, and therefore, all of these points can't form a line. So, the table shown in the picture does not represent a linear function. Do this for the other tables in the question, and you got yourself your answer.
