The table represents a linear function. X -2 -1 0 1 2 y 8 2 -4 -10 -16
What is the slope of the function?
-6
-4
4
6
The Correct Answer and Explanation is :
The correct answer is -6.
To find the slope of a linear function, we use the formula:
[
\text{slope} = \frac{{\text{change in } y}}{{\text{change in } x}} = \frac{{y_2 – y_1}}{{x_2 – x_1}}
]
A linear function has a constant slope, meaning the rate of change between the ( x ) and ( y ) values is consistent across the table. We can calculate the slope by picking any two points from the table and applying the formula. Let’s use the points ( (-2, 8) ) and ( (-1, 2) ).
- Change in ( y ):
[
y_2 – y_1 = 2 – 8 = -6
] - Change in ( x ):
[
x_2 – x_1 = -1 – (-2) = -1 + 2 = 1
]
Now, calculate the slope:
[
\text{slope} = \frac{{-6}}{{1}} = -6
]
The slope of the function is -6, which indicates that for every 1 unit increase in ( x ), the value of ( y ) decreases by 6 units. This makes sense because as you look at the table, each time ( x ) increases by 1, ( y ) decreases by 6. For example:
- When ( x ) changes from -2 to -1, ( y ) decreases from 8 to 2 (a change of -6).
- When ( x ) changes from -1 to 0, ( y ) decreases from 2 to -4 (a change of -6).
Thus, the function has a constant rate of change, confirming that the slope is indeed -6. The slope being negative shows that the function is decreasing as ( x ) increases.