A line with a negative slope goes downward as you move from left to right. Translate that into the table: what must happen to the $y$-values as $x$ gets larger?

For the tables where $y$ always decreases, compute how much $y$ changes each time $x$ increases by 1. Are those changes roughly the same, or do they vary a lot?

A linear model means the points would lie close to a straight line. Based on the changes in $y$, which table’s values would most closely line up on a straight, downward-sloping line?

For a given answer choice, type its four ordered pairs into Desmos as one expression, for example for a table you might use {(1,20),(2,25),(3,30),(4,35)}. Repeat with separate expressions for each of the other answer choices you want to compare.

Look at each set of plotted points and decide whether they lie close to a straight line and whether that line slopes downward (from left to right).

For a more precise check, you can store the $x$-values and $y$-values of one table as lists (for example, x1={1,2,3,4} and y1={40,35,25,24}), then type y1 ~ m x1 + b to perform a linear regression. The more closely the points follow this line and the more clearly $m$ is negative, the better that table is modeled by a linear function with negative slope.

A linear function means the data should follow (or closely follow) a straight-line pattern, which shows a constant rate of change. A negative slope means that as $x$ increases, $y$ decreases. So we want a table where:

• The $y$-values go down as $x$ goes up, and
• The amount they go down each time is about the same.

Look at how $y$ changes as $x$ increases from 1 to 4 in each answer choice:

• A) $20, 25, 30, 35$ → $y$ goes up (positive slope).
• B) $28, 29, 27, 28$ → $y$ goes up, then down, then up (mixed).
• C) $40, 35, 25, 24$ → $y$ goes down every time (negative direction).
• D) $35, 33, 30, 27$ → $y$ goes down every time (negative direction).

Only C and D have $y$ decreasing each time, so only those could have a negative slope.

Now we decide which of C or D is more linear by looking at the change in $y$ between each step in $x$.

For C (40, 35, 25, 24):

• From 40 to 35: change is $-5$.
• From 35 to 25: change is $-10$.
• From 25 to 24: change is $-1$. These changes are very different, so the rate of change is not close to constant.

For D (35, 33, 30, 27):

• From 35 to 33: change is $-2$.
• From 33 to 30: change is $-3$.
• From 30 to 27: change is $-3$. These changes are all close to each other, so the rate of change is nearly constant.

We need data that:

• Decrease as $x$ increases (negative direction), and
• Change by about the same amount each time (linear behavior).

Table D, with

has $y$ decreasing at an almost constant rate, so it is best modeled by a linear function with a negative slope. Therefore, the correct choice is D.