Question

x | y
-18 | -13
-15 | -11
-9 | -7
one group offers this explanation: \for the first two sets of ordered pairs, as the x - values increase by 2, the y - values increase by 2. this is a slope of \\(\frac{2}{2}\\). for the last two sets of ordered pairs, as the x - values increase by 6, the y - values increase by 4. this is a slope of \\(\frac{4}{6}\\), which is the same as \\(\frac{2}{3}\\). since the slope stays the same, these three points could be graphed and there would be one line through all of them. also, each x - value pairs with exactly one y - value. therefore, this table of values represents a linear function.\
is the group correct? why or why not?
no; although the given x - and y - values meet the definition of a function, the group incorrectly identified the slope as \\(\frac{2}{2}\\) when it should have been 3/2.
no; the group didnt recognize that without an x - intercept or y - intercept in the table, there cannot be a linear function.
no; the group didnt show a graph of the three points, so they cannot be sure the values represent a linear function.
yes; the group explained that the three points would be connected by a single line of slope \\(\frac{2}{3}\\), and the given x - and y - values meet the definition of a function.

Explanation:

Step1: Verify slope between first two points

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_1=\frac{-11-(-13)}{-15-(-18)}=\frac{2}{3}$

Step2: Verify slope between last two points

$m_2=\frac{-7-(-11)}{-9-(-15)}=\frac{4}{6}=\frac{2}{3}$

Step3: Check function definition

Each $x$-value maps to exactly one $y$-value, so it is a function.

Step4: Evaluate group's reasoning

The group correctly identified constant slope $\frac{2}{3}$ and confirmed the function definition.

Answer:

Yes; the group explained that the three points would be connected by a single line of slope $\frac{2}{3}$, and the given x-and y-values meet the definition of a function.