Determine any data values that are missing from the table, assuming that the data represent a linear function.
| x | y |
|---|---|
| 1 | 2 |
| 2 | 6 |
| 4 | |
A. 2
B. 10
C. 14
D. 16
Answer (2)
Calculate the slope using the given points: m = 2 − 1 6 − 2 = 4 .
Determine the y-intercept by substituting a point and the slope into y = m x + b : 2 = 4 ( 1 ) + b , so b = − 2 .
Find the missing y-value by substituting x = 4 into the equation y = 4 x − 2 : y = 4 ( 4 ) − 2 = 14 .
The missing y-value is 14 .
Explanation
Understanding the Problem We are given a table with x and y values, and we are told that the data represents a linear function. Our goal is to find the missing y value when x = 4 . A linear function can be represented by the equation y = m x + b , where m is the slope and b is the y-intercept.
Calculating the Slope First, we need to find the slope ( m ) of the linear function. We can use the points (1, 2) and (2, 6) from the table. The slope is calculated as the change in y divided by the change in x :
m = x 2 − x 1 y 2 − y 1 = 2 − 1 6 − 2 = 1 4 = 4
Finding the Y-Intercept Now that we have the slope, we can find the y-intercept ( b ). We can use the point (1, 2) and the slope m = 4 in the equation y = m x + b :
2 = 4 ( 1 ) + b Solving for b :
b = 2 − 4 = − 2
Finding the Missing Y Value Now we have the equation of the linear function: y = 4 x − 2 . To find the missing y value when x = 4 , we substitute x = 4 into the equation: y = 4 ( 4 ) − 2 = 16 − 2 = 14
Conclusion Therefore, the missing y value is 14.
Examples
Linear functions are incredibly useful in everyday life. For example, if you are saving money at a constant rate, the relationship between the amount of money you save and the time you spend saving can be modeled by a linear function. Understanding linear functions helps you predict future savings, calculate travel times at constant speeds, or even estimate costs based on a fixed hourly rate. In this case, we found a missing data point using the properties of a linear function, which is a common task in data analysis and prediction.
The missing y value corresponding to x = 4 is 14, found using the linear equation derived from the points (1, 2) and (2, 6). We calculated the slope as 4 and the y-intercept as -2, leading to the equation y = 4x - 2. Substituting x = 4 gives y = 14, confirming the missing value.
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