Question

determine whether the function is linear. if it is, identify the rate of change.

x	y
-7	11
-5	14
-3	17
-1	20
0	23

a) not linear
b) linear, 0
c) linear, 2
d) linear, 4

Explanation:

Step1: Calculate the differences in x

The x-values are -7, -5, -3, -1, 0. The differences between consecutive x-values: $-5 - (-7) = 2$, $-3 - (-5) = 2$, $-1 - (-3) = 2$, $0 - (-1) = 1$. Wait, no, the rate of change for a linear function is $\frac{\Delta y}{\Delta x}$. Let's calculate $\Delta y$ and $\Delta x$ for consecutive points.

First pair: x1 = -7, y1 = 11; x2 = -5, y2 = 14. $\Delta x = -5 - (-7) = 2$, $\Delta y = 14 - 11 = 3$. Rate of change: $\frac{3}{2} = 1.5$? Wait, no, maybe I made a mistake. Wait, second pair: x2 = -5, y2 = 14; x3 = -3, y3 = 17. $\Delta x = -3 - (-5) = 2$, $\Delta y = 17 - 14 = 3$. Rate: $\frac{3}{2} = 1.5$? Wait, third pair: x3 = -3, y3 = 17; x4 = -1, y4 = 20. $\Delta x = -1 - (-3) = 2$, $\Delta y = 20 - 17 = 3$. Rate: $\frac{3}{2} = 1.5$? Wait, fourth pair: x4 = -1, y4 = 20; x5 = 0, y5 = 23. $\Delta x = 0 - (-1) = 1$, $\Delta y = 23 - 20 = 3$. Rate: $\frac{3}{1} = 3$. Wait, that's different. Wait, no, maybe I misread the table. Wait, the x-values: -7, -5, -3, -1, 0. The differences between x: -5 - (-7) = 2, -3 - (-5) = 2, -1 - (-3) = 2, 0 - (-1) = 1. Oh, the last $\Delta x$ is 1, not 2. Wait, but maybe the table has a typo, or I misread. Wait, no, let's check again. Wait, the y-values: 11, 14, 17, 20, 23. The differences between y: 14 - 11 = 3, 17 - 14 = 3, 20 - 17 = 3, 23 - 20 = 3. So $\Delta y$ is always 3. Now, $\Delta x$: first three intervals are 2 (from -7 to -5: 2, -5 to -3: 2, -3 to -1: 2), then from -1 to 0: 1. Wait, that would mean the rate of change is not constant, but wait, no—wait, maybe the last x is a mistake, or maybe I miscalculated. Wait, no, the problem is to determine if it's linear. Wait, for a function to be linear, the rate of change $\frac{\Delta y}{\Delta x}$ must be constant for all consecutive points. Let's check:

From (-7,11) to (-5,14): $\Delta y = 3$, $\Delta x = 2$, rate = 3/2 = 1.5

From (-5,14) to (-3,17): $\Delta y = 3$, $\Delta x = 2$, rate = 3/2 = 1.5

From (-3,17) to (-1,20): $\Delta y = 3$, $\Delta x = 2$, rate = 3/2 = 1.5

From (-1,20) to (0,23): $\Delta y = 3$, $\Delta x = 1$, rate = 3/1 = 3

Wait, that's a problem. But that can't be. Wait, maybe the x-value for the last point is a mistake, or maybe I misread. Wait, the x-values: -7, -5, -3, -1, 0. The differences between x: -5 - (-7) = 2, -3 - (-5) = 2, -1 - (-3) = 2, 0 - (-1) = 1. So the last $\Delta x$ is 1. But the $\Delta y$ is 3 each time. So that would mean the rate of change is not constant, but that contradicts. Wait, no, maybe the last x is supposed to be 1, not 0? Or maybe I made a mistake. Wait, no, let's check the answer options. The options are: A) not linear, B) linear, 0, C) linear, 2, D) linear, 4. Wait, no, maybe I miscalculated $\Delta x$. Wait, -7 to -5: 2 units (since -5 - (-7) = 2). -5 to -3: 2 units. -3 to -1: 2 units. -1 to 0: 1 unit. But the y-values increase by 3 each time. So if we consider the first four points (x from -7 to -1), the $\Delta x$ is 2, $\Delta y$ is 3, so rate is 3/2. But the last point: x=0, y=23. From x=-1 (y=20) to x=0, $\Delta x=1$, $\Delta y=3$, rate=3. So that's inconsistent. But the answer options include "linear" with rate 1.5? No, the options are 0, 2, 4, or not linear. Wait, maybe I made a mistake in $\Delta x$. Let's check the x-values again: -7, -5, -3, -1, 0. The difference between -7 and -5 is 2 (since -5 - (-7) = 2). -5 to -3: 2. -3 to -1: 2. -1 to 0: 1. So the first three intervals have $\Delta x=2$, last has $\Delta x=1$. But the y-values: 11,14,17,20,23. $\Delta y$ between each is 3. So for the first three intervals, rate is 3/2=1.5, last is 3/1=3. So that would mean it's not linear. But the answer options have "linear" with rates 0,2,4. Wait, maybe I misread the table. Wait, maybe the x-values are -7, -5, -3, -1, 1? Then $\Delta x$ would be 2 each time. Let's check: if x=1, then from -1 to 1, $\Delta x=2$, $\Delta y=3$, rate=1.5. But the options don't have 1.5. Wait, maybe the table is: x: -7, -5, -3, -1, 0; y: 11, 14, 17, 20, 23. Wait, let's calculate the slope between the first and last point: (23 - 11)/(0 - (-7)) = 12/7 ≈ 1.714, which is not constant. But that can't be. Wait, maybe the problem is that the last x is a typo, and it's supposed to be 1, not 0. Then $\Delta x$ would be 2, and $\Delta y$=3, rate=1.5. But the options don't have that. Wait, maybe I made a mistake in the problem. Wait, the options are A) not linear, B) linear, 0, C) linear, 2, D) linear, 4. Wait, let's check the rate of change again. Wait, maybe the x-values are -7, -5, -3, -1, 0, and the y-values are 11, 14, 17, 20, 23. Let's check the slope between (-7,11) and (-5,14): (14-11)/(-5 - (-7)) = 3/2 = 1.5. Between (-5,14) and (-3,17): 3/2=1.5. Between (-3,17) and (-1,20): 3/2=1.5. Between (-1,20) and (0,23): (23-20)/(0 - (-1))=3/1=3. So the rate of change is not constant, so it's not linear? But that contradicts. Wait, no, maybe the last y is 21, not 23? Then $\Delta y$ would be 2, and rate=1. But no. Wait, maybe the problem is that I misread the y-values. Let me check again: the table says x: -7, -5, -3, -1, 0; y: 11, 14, 17, 20, 23. So the differences in y are 3,3,3,3. Differences in x are 2,2,2,1. So the rate of change is not constant, so it's not linear. But the options have A) not linear. So maybe that's the answer.

Wait, but maybe I made a mistake. Wait, let's check the definition of a linear function: a function is linear if it can be written as y = mx + b, where m is the slope (rate of change) and b is the y-intercept. For a linear function, the rate of change (slope) must be constant for all pairs of points. Let's check two points: (-7,11) and (-5,14). Slope m1 = (14-11)/(-5 - (-7)) = 3/2 = 1.5. (-5,14) and (-3,17): m2 = (17-14)/(-3 - (-5)) = 3/2 = 1.5. (-3,17) and (-1,20): m3 = (20-17)/(-1 - (-3)) = 3/2 = 1.5. (-1,20) and (0,23): m4 = (23-20)/(0 - (-1)) = 3/1 = 3. Since m1, m2, m3 are 1.5, but m4 is 3, which is different, the rate of change is not constant, so the function is not linear. Therefore, the answer is A) not linear.

Answer:

A) not linear