Determine if the following represent linear functions or not and explain.

1)

| x | y |
| --- | --- |
| -3 | 1 |
| -2 | 2 |
| -1 | 3 |
| -1 | 4 |
| 1 | 5 |

2) [tex]y=\frac{3}{x}-2[/tex]

Which table represents the equation [tex]y=-2 x+17[/tex]?

A.

| x | y |
| -- | -- |
| 8 | 13 |
| 10 | 12 |
| 12 | 11 |
| 14 | 10 |
| 16 | 9 |

B.

| x | y |
| -- | -- |
| 2 | 18 |
| 4 | 19 |
| 6 | 20 |
| 8 | 21 |
| 10 | 22 |

C.

| x | y |
| -- | -- |
| 2 | 21 |
| 4 | 25 |
| 6 | 29 |
| 8 | 33 |
| 10 | 37 |

D.

| x | y |
| -- | -- |
| 8 | 1 |
| 9 | -1 |
| 10 | -3 |
| 11 | -5 |
| 12 | -7 |

The table 1.017 does not represent a linear function because the slope is not constant and there are repeated x-values with different y-values.
The equation 1.018 does not represent a linear function because it cannot be written in the form y = m x + b .
Table D represents the equation y = − 2 x + 17 .
Therefore, the correct answer is that 1.017 and 1.018 are not linear functions, and Table D represents the equation y = − 2 x + 17 .

Explanation

Problem Analysis Let's analyze the given problems to determine if they represent linear functions and to identify the correct table for the given equation.

Analyzing Table 1.017 1.017) To determine if the table represents a linear function, we need to check if the rate of change (slope) between consecutive points is constant. The slope is calculated as Δ x Δ y ​ . From the table, we have the points (-3, 1), (-2, 2), (-1, 3), (-1, 4), and (1, 5). Calculating the slopes between consecutive points:



Between (-3, 1) and (-2, 2): − 2 − ( − 3 ) 2 − 1 ​ = 1 1 ​ = 1
Between (-2, 2) and (-1, 3): − 1 − ( − 2 ) 3 − 2 ​ = 1 1 ​ = 1
Between (-1, 3) and (-1, 4): − 1 − ( − 1 ) 4 − 3 ​ = 0 1 ​ which is undefined.
Between (-1, 4) and (1, 5): 1 − ( − 1 ) 5 − 4 ​ = 2 1 ​

Since the slope is not constant and we have a repeated x-value (-1) with different y-values (3 and 4), the table does not represent a linear function.

Analyzing Equation 1.018 1.018) A linear function can be written in the form y = m x + b , where m and b are constants. The given equation is y = x 3 ​ − 2 . This equation cannot be written in the form y = m x + b because of the term x 3 ​ . Therefore, the equation does not represent a linear function.

Finding the Correct Table 1.019) We need to find the table that represents the equation y = − 2 x + 17 . We can substitute the x-values from each table into the equation and check if the resulting y-values match the table.



Table A:

x = 8: y = − 2 ( 8 ) + 17 = − 16 + 17 = 1 . The table shows y = 13, which does not match.
x = 10: y = − 2 ( 10 ) + 17 = − 20 + 17 = − 3 . The table shows y = 12, which does not match.


Table B:

x = 2: y = − 2 ( 2 ) + 17 = − 4 + 17 = 13 . The table shows y = 18, which does not match.
x = 4: y = − 2 ( 4 ) + 17 = − 8 + 17 = 9 . The table shows y = 19, which does not match.


Table C:

x = 2: y = − 2 ( 2 ) + 17 = − 4 + 17 = 13 . The table shows y = 21, which does not match.
x = 4: y = − 2 ( 4 ) + 17 = − 8 + 17 = 9 . The table shows y = 25, which does not match.


Table D:

x = 8: y = − 2 ( 8 ) + 17 = − 16 + 17 = 1 . The table shows y = 1, which matches.
x = 9: y = − 2 ( 9 ) + 17 = − 18 + 17 = − 1 . The table shows y = -1, which matches.
x = 10: y = − 2 ( 10 ) + 17 = − 20 + 17 = − 3 . The table shows y = -3, which matches.
x = 11: y = − 2 ( 11 ) + 17 = − 22 + 17 = − 5 . The table shows y = -5, which matches.
x = 12: y = − 2 ( 12 ) + 17 = − 24 + 17 = − 7 . The table shows y = -7, which matches.



Table D represents the equation y = − 2 x + 17 .
Examples
Linear functions are used in many real-world applications, such as calculating the cost of a taxi ride based on the distance traveled, determining the amount of interest earned on a savings account over time, or modeling the relationship between the number of hours studied and the grade received on a test. Understanding linear functions helps us make predictions and solve problems in these and other situations. For example, if a taxi charges a fixed fee of $5 plus 2 p er mi l e , t h e t o t a l cos t c anb ere p rese n t e d b y t h e l in e a r f u n c t i o n y = 2x + 5 , w h ere x i s t h e n u mb ero f mi l es t r a v e l e d an d y i s t h e t o t a l cos t . S imi l a r l y , t h ee q u a t i o n y = -2x + 17$ can model the temperature decreasing by 2 degrees every hour from an initial temperature of 17 degrees.

Answered by GinnyAnswer | 2025-07-03

Neither the table nor the equation represents linear functions. However, Table D correctly matches the equation y = − 2 x + 17 .
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Answered by Anonymous | 2025-07-04