Which table represents a linear function?
X
1
2
3
4
X
1
2
234
1
2
3
4
X12
A5951
y
y
-5
10
-15
20
SLE 888SK
10
20
40
-5
The Correct Answer and Explanation is :
To determine which table represents a linear function, we first need to understand the definition of a linear function. A linear function is one where the relationship between the independent variable (x) and the dependent variable (y) can be expressed in the form (y = mx + b), where (m) is the slope and (b) is the y-intercept. The key characteristic of a linear function is that it produces a straight line when graphed.
To analyze each table, we look for a constant rate of change between (x) and (y). This means that as (x) increases by a specific value, (y) should increase or decrease by a consistent amount.
- Table 1:
[
\begin{array}{|c|c|}
\hline
x & y \
\hline
1 & -5 \
2 & 10 \
3 & -15 \
4 & 20 \
\hline
\end{array}
]
The changes in (y) are inconsistent: from -5 to 10, (y) increases by 15, then from 10 to -15, (y) decreases by 25, and from -15 to 20, (y) increases by 35. Therefore, this does not represent a linear function. - Table 2:
[
\begin{array}{|c|c|}
\hline
x & y \
\hline
1 & 10 \
2 & 20 \
3 & 30 \
4 & 40 \
\hline
\end{array}
]
Here, as (x) increases by 1, (y) increases by 10 consistently (10, 20, 30, 40). This constant change indicates a linear relationship.
In conclusion, Table 2 represents a linear function because it has a constant rate of change. Each time (x) increases by 1, (y) consistently increases by 10, fitting the form (y = 10x). Tables with inconsistent changes between (x) and (y) do not exhibit linearity, while those with constant differences do. Understanding this relationship is essential in recognizing and graphing linear functions.