Question

a function is represented by the values in the table.

x	y
20	22
16	20
14	18
10	14

choose from the drop - down menu to complete the statement.
the function represented in the table choose... linear.

Explanation:

Step1: Calculate the slope between pairs of points.

For points \((22, 26)\) and \((20, 22)\):
Slope \(m_1 = \frac{22 - 26}{20 - 22} = \frac{-4}{-2} = 2\).

Step2: Check another pair of points.

For points \((20, 22)\) and \((16, 20)\):
Slope \(m_2 = \frac{20 - 22}{16 - 20} = \frac{-2}{-4} = 0.5\).
Since \(m_1 eq m_2\), the function is not linear. Wait, wait, let's recalculate. Wait, maybe I mixed up x and y. Let's do it correctly: slope is \(\frac{y_2 - y_1}{x_2 - x_1}\).
For \((22,26)\) and \((20,22)\): \(\frac{22 - 26}{20 - 22} = \frac{-4}{-2} = 2\).
For \((20,22)\) and \((16,20)\): \(\frac{20 - 22}{16 - 20} = \frac{-2}{-4} = 0.5\). Wait, that's inconsistent. Wait, maybe I made a mistake. Let's check another pair: \((16,20)\) and \((14,18)\): \(\frac{18 - 20}{14 - 16} = \frac{-2}{-2} = 1\). Wait, no, this is confusing. Wait, maybe I should check the differences in x and y. Let's list the x and y values:
x: 22, 20, 16, 14, 10
y: 26, 22, 20, 18, 14
Differences in x (Δx): 20 - 22 = -2; 16 - 20 = -4; 14 - 16 = -2; 10 - 14 = -4
Differences in y (Δy): 22 - 26 = -4; 20 - 22 = -2; 18 - 20 = -2; 14 - 18 = -4
Now, check the ratio \(\frac{\Delta y}{\Delta x}\):
For first pair: \(\frac{-4}{-2} = 2\)
Second pair: \(\frac{-2}{-4} = 0.5\)
Third pair: \(\frac{-2}{-2} = 1\)
Fourth pair: \(\frac{-4}{-4} = 1\)
Wait, this is inconsistent. Wait, maybe the table is misread. Wait, maybe the x and y are swapped? No, the table is x and y. Wait, maybe I made a mistake. Wait, let's check again. Wait, the problem is to determine if the function is linear. A linear function has a constant slope. Let's take (22,26), (20,22): slope 2. (20,22), (16,20): slope 0.5. (16,20), (14,18): slope 1. (14,18), (10,14): slope (14 - 18)/(10 - 14) = (-4)/(-4) = 1. Wait, this is not constant. Wait, but maybe I messed up the order. Wait, maybe the x values are not in order? No, they are decreasing. Wait, maybe the correct way is to check if the rate of change is constant. Let's list the x and y:
x: 10,14,16,20,22 (sorted)
y:14,18,20,22,26
Now, Δx: 14-10=4; 16-14=2; 20-16=4; 22-20=2
Δy:18-14=4; 20-18=2; 22-20=2; 26-22=4
Now, \(\frac{\Delta y}{\Delta x}\) for each:
4/4=1; 2/2=1; 2/4=0.5; 4/2=2. Wait, no, this is still inconsistent. Wait, maybe the problem is that the function is not linear? But that contradicts. Wait, maybe I made a mistake. Wait, let's check (10,14) and (14,18): Δx=4, Δy=4, slope=1. (14,18) and (16,20): Δx=2, Δy=2, slope=1. (16,20) and (20,22): Δx=4, Δy=2, slope=0.5. (20,22) and (22,26): Δx=2, Δy=4, slope=2. Oh! Wait, the x differences are 4,2,4,2 and y differences are 4,2,2,4. So when Δx is 4, Δy is 4 (slope 1) or 2 (slope 0.5). When Δx is 2, Δy is 2 (slope 1) or 4 (slope 2). So the slopes are not constant. Therefore, the function is not linear. Wait, but the initial calculation was wrong. Wait, no, the key is that for a linear function, the slope between any two points must be the same. Since the slopes are different, the function is not linear. Wait, but the problem says "the function represented in the table [Choose...] linear". So the answer is "is not". Wait, but let's re-express the table:

x: 22, 20, 16, 14, 10

y: 26, 22, 20, 18, 14

Let's check the equation of a line. Suppose we take two points, say (10,14) and (14,18). The slope is (18-14)/(14-10)=4/4=1. So the equation is y -14 = 1*(x -10) → y = x +4. Let's check if this holds for other points. For x=16: y=16+4=20. Which matches (16,20). For x=20: y=20+4=24. But the table has y=22 for x=20. So that's a problem. Wait, x=20, y=22. 20+4=24≠22. So that's not good. Wait, another point: x=22, y=26. 22+4=26. Oh! Wait, x=22, y=26: 22+4=26. x=10,14: 10+4=14, 14+4=18. x=16:16+4=20. x=20:20+4=24, but table has 22. Wait, that's a mistake. Wait, x=20, y=22. 20+4=24≠22. So why does x=22,16,14,10 work but x=20 doesn't? Wait, maybe I misread the table. Let me check again:

The table is:

x | y

22 | 26

20 | 22

16 | 20

14 | 18

10 | 14

Ah! Wait, x=20, y=22. Let's check the equation with (22,26) and (20,22). Slope is (22-26)/(20-22)= (-4)/(-2)=2. So equation: y -26=2(x -22) → y=2x -44 +26 → y=2x -18. Let's check x=20: y=40 -18=22. Correct. x=16: y=32 -18=14? No, table has y=20. Wait, no. Wait, x=16, y=20. 2*16 -18=32-18=14≠20. So that's wrong. Wait, (20,22) and (16,20): slope (20-22)/(16-20)= (-2)/(-4)=0.5. Equation: y-22=0.5(x-20) → y=0.5x -10 +22 → y=0.5x +12. Check x=16: 0.5*16 +12=8+12=20. Correct. x=14: 0.5*14 +12=7+12=19. But table has y=18. No. Wait, (16,20) and (14,18): slope (18-20)/(14-16)= (-2)/(-2)=1. Equation: y-20=1*(x-16) → y=x +4. Check x=14:14+4=18. Correct. x=10:10+4=14. Correct. x=22:22+4=26. Correct. x=20:20+4=24. But table has y=22. So x=20 is an outlier? Wait, no, maybe the table is correct and I made a mistake. Wait, the key is that for a linear function, the slope between any two points must be the same. Let's check all consecutive points:

Between (22,26) and (20,22): slope 2.

Between (20,22) and (16,20): slope 0.5.

Between (16,20) and (14,18): slope 1.

Between (14,18) and (10,14): slope 1.

Since the slopes are not constant, the function is not linear. Wait, but earlier when we checked (22,26), (16,20), (14,18), (10,14), they fit y=x+4, but (20,22) doesn't. So there's an inconsistency. Wait, maybe the table has a typo, but according to the given table, the slopes are not constant, so the function is not linear. Wait, but wait, let's recalculate the slopes correctly:

Slope between (22,26) and (20,22): (22-26)/(20-22) = (-4)/(-2) = 2.

Slope between (20,22) and (16,20): (20-22)/(16-20) = (-2)/(-4) = 0.5.

Slope between (16,20) and (14,18): (18-20)/(14-16) = (-2)/(-2) = 1.

Slope between (14,18) and (10,14): (14-18)/(10-14) = (-4)/(-4) = 1.

Since the slopes are not the same, the function is not linear. Therefore, the statement is "The function represented in the table is not linear."

Answer:

is not