Question

Which of the following tables represents a linear function?

a.
x = −2 −1 0 2 4
y = −4 negative two thirds −1 two thirds 1

b.
x = −3 −1 0 1 5
y = −7 negative nine halves negative thirteen fourths −2 3

c.
x = −2 −1 0 2 4
y = −4 −2 −1 0 1

d.
x = −4 −1 0 1 2
y = −4 2 −4 0 2

Answer 1

Option c, which shows a consistent increase of y by 1 for each increase in x by 1, represents a linear function with a slope of 1.

Step-by-step explanation:

To determine which table represents a linear function, we look for a constant rate of change between x and y. This means that for each increase in x, y should change by a constant amount, which is the definition of a linear function. To find this, we can calculate the differences in y-values divided by the differences in x-values (the slope).

• For option c, the changes in x are 1, and the corresponding changes in y are +2, +2, +1, +1. This is not constant, so it cannot be a linear function.
• For option d, the changes in y do not correspond to consistent changes in x. Hence, it is not linear.
• Options a and b have inconsistent changes in y-values for changes in x-values and therefore are not linear functions either.

However, based on the practice test concepts, we are reminded that a linear equation is of the form y = mx + b, where m is the slope and b is the y-intercept. Looking at the tables, we can see that only option c represents a function where y increases by a constant amount for a constant increase in x, which is characteristic of a linear function with a positive slope.

Therefore, the correct answer is option c:

x = − 2 − 1 0 2 4,

y = − 4 − 2 − 1 0 1, which consistently shows an increase of the y-value by 1 for each increase of the x-value by 1, indicating a slope of 1 and therefore represents a linear function.