Determine any data values that are missing from the table, assuming that the data represent a linear function.
| x | y |
| --- | --- |
| -9 | 3 |
| -6 | 2 |
| -3 | |
| | 0 |
A. Missing x: -2, Missing y: 0
B. Missing x: 0, Missing y: 1
C. Missing x: -2, Missing y: 2
D. Missing x: -1, Missing y: 1
Please select the best answer from the choices provided
Answer (2)
Calculate the slope using the given points: m = − 6 − ( − 9 ) 2 − 3 = − 3 1 .
Determine the equation of the line: y = − 3 1 x .
Find the missing y value when x = − 3 : y = − 3 1 ( − 3 ) = 1 .
Find the missing x value when y = 0 : x = 0 .
The missing values are x = 0 and y = 1 , so the answer is x : 0 , y : 1 .
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 x and y values in the table.
Calculating the Slope First, we need to find the slope of the linear function. We can use the points ( − 9 , 3 ) and ( − 6 , 2 ) to calculate the slope m using the formula: m = x 2 − x 1 y 2 − y 1 Substituting the given values, we get: m = − 6 − ( − 9 ) 2 − 3 = 3 − 1 = − 3 1 So, the slope of the linear function is − 3 1 .
Finding the Equation of the Line Next, we need to find the equation of the line. Since the y-intercept is 0, the equation of the line is simply: y = − 3 1 x We can verify this by plugging in one of the points, for example, ( − 9 , 3 ) :
3 = − 3 1 ( − 9 ) = 3 This confirms that our equation is correct.
Finding the Missing y-value Now, we can find the missing y value when x = − 3 . Substituting x = − 3 into the equation, we get: y = − 3 1 ( − 3 ) = 1 So, the missing y value is 1 .
Finding the Missing x-value Finally, we can find the missing x value when y = 0 . Substituting y = 0 into the equation, we get: 0 = − 3 1 x Multiplying both sides by − 3 , we get: x = 0 So, the missing x value is 0 .
Conclusion Therefore, the missing x value is 0 and the missing y value is 1 . Comparing this to the options provided, we see that option b matches our result.
Examples
Linear functions are used in many real-world applications. For example, if you are tracking the distance you travel over time at a constant speed, the relationship between distance and time can be modeled by a linear function. Similarly, if you are calculating the cost of a taxi ride based on a fixed initial fee and a per-mile charge, the total cost can be represented by a linear function. Understanding linear functions helps in making predictions and analyzing trends in various scenarios.
The missing values from the table are x = 0 and y = 1, corresponding to option B. We found these values by calculating the slope of the linear function and using the linear equation derived from the given points. This process involved substituting known x-values to find missing y-values and vice versa.
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