Determine any data values that are missing from the table, assuming that the data represent a linear function.
| x | y |
| --- | --- |
| -1 | 5 |
| 0 | 7 |
| 2 | |
A. 5
B. 9
C. 13
D. 11
Answer (2)
Calculate the slope using the points ( − 1 , 5 ) and ( 0 , 7 ) : m = 0 − ( − 1 ) 7 − 5 = 2 .
Determine the equation of the line using the slope and y-intercept: y = 2 x + 7 .
Substitute x = 2 into the equation to find the missing y-value: y = 2 ( 2 ) + 7 = 11 .
The missing y-value is 11 .
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 = 2 . We have two points on the line: ( − 1 , 5 ) and ( 0 , 7 ) .
Calculating the Slope First, we need to find the slope of the linear function. The slope, m , is calculated as the change in y divided by the change in x . Using the points ( − 1 , 5 ) and ( 0 , 7 ) , we have: m = x 2 − x 1 y 2 − y 1 = 0 − ( − 1 ) 7 − 5 = 1 2 = 2
Finding the Equation of the Line Now that we have the slope, we can find the equation of the line. Since we have the point ( 0 , 7 ) , we know that the y -intercept, b , is 7. So, the equation of the line in slope-intercept form ( y = m x + b ) is: y = 2 x + 7
Finding the Missing y-value Finally, we substitute x = 2 into the equation of the line to find the corresponding y -value: y = 2 ( 2 ) + 7 = 4 + 7 = 11
Conclusion Therefore, the missing y -value is 11.
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 or plan how long it will take to reach a financial goal. Another example is calculating the distance traveled at a constant speed. The distance is a linear function of time, where the speed is the slope.
The missing y-value for x = 2, given the linear function represented in the table, is 11, which corresponds to option D. To find this, we calculated the slope between the known points and derived the linear equation. We then substituted x = 2 into the equation to find the corresponding y-value.
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