For a linear relationship, the change in $y$ for each increase of 1 in $x$ should be about the same. Compute the differences between consecutive $y$ values and compare them.

When the differences are not similar, try comparing ratios like $y_{\text{next}} / y_{\text{current}}$. Are these ratios roughly the same each time?

After you decide whether the relationship is increasing or decreasing and whether it’s closer to a straight line or a curve with a constant ratio, choose the option whose description fits both observations.

In Desmos, create a table with $x$ and $y$ columns. Enter the $x$ values 0, 1, 2, 3, 4 in the first column and the corresponding $y$ values 80, 56, 39, 27, 19 in the second column. Look at the plotted points to see that they do not lie on a straight line.

Add a new expression line and type something like y1 ~ m x1 + b to do a linear regression. Observe the graph: notice that while a line can be drawn, the points bend away from a perfect straight line, especially at the ends, suggesting a non-linear pattern.

Add another expression line and type y1 ~ a * b^x1 to fit an exponential model. Compare how closely this curve follows the points versus the line from the linear regression. Also check the value of b; it should be less than 1, consistent with a repeated percentage decrease.

Based on whether the best-fit curve is roughly straight or clearly curved, and whether $y$ goes up or down as $x$ increases, match what you see in Desmos to the answer choice that names that type and direction of relationship.

Look at how $y$ changes as $x$ increases from 0 to 4:

• When $x$ goes from 0 to 4, $y$ goes from 80 down to 19.

So as $x$ increases, $y$ decreases. This rules out any "increasing" option.

For a linear relationship, when $x$ increases by 1, the change in $y$ should be (about) the same each time.

Compute the differences between consecutive $y$ values:

• From 80 to 56: $56 - 80 = -24$
• From 56 to 39: $39 - 56 = -17$
• From 39 to 27: $27 - 39 = -12$
• From 27 to 19: $19 - 27 = -8$

These changes are not constant; the drop in $y$ is getting smaller each step. That means the relationship is not linear, so the correct choice is not any linear option.

A common non-linear pattern is exponential, where each step multiplies $y$ by (about) the same factor instead of subtracting the same amount.

Compute ratios of consecutive $y$ values:

• $\dfrac{56}{80} = 0.70$
• $\dfrac{39}{56} \approx 0.70$
• $\dfrac{27}{39} \approx 0.69$
• $\dfrac{19}{27} \approx 0.70$

These ratios are all close to $0.7$, so each step multiplies $y$ by roughly $0.7$ (a $30\%$ decrease each time). This is the hallmark of an exponential relationship rather than a linear one.

We found that:

• $y$ decreases as $x$ increases (so the relationship is decreasing), and
• the ratio between consecutive $y$ values is roughly constant (so the relationship is exponential).

Therefore, the relationship between $x$ and $y$ is best described as decreasing exponential.