Math worksheet for determining functions and calculating rate of change from tables.

Let’s go through each problem one by one. For each table, we need to:

1. Check if it’s a function — that means every X value must have only ONE Y value. If any X repeats with a different Y, it’s NOT a function.
2. If it IS a function, calculate the rate of change (also called slope). That’s:
> Rate of Change = (Change in y) / (Change in x)
We can pick any two points from the table and plug them into this formula.

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Problem 1)

Table:
X | Y
-6 | -4
-2 | 0
2 | 4
5 | 7
6 | 8

→ All X values are unique → Function? Yes

Pick two points: (-6, -4) and (-2, 0)

Change in y = 0 - (-4) = 4
Change in x = -2 - (-6) = 4
Rate of Change = 4/4 = 1

Check another pair to verify: (2,4) and (5,7)
Δy = 7-4=3, Δx=5-2=3 → 3/3=1 ✔

✔ Function? Yes
Rate of Change = 1

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Problem 2)

Table:
X | Y
5 | 5
7 | 3
9 | 1
11 | -1

All X unique → Function? Yes

Pick (5,5) and (7,3):
Δy = 3-5 = -2
Δx = 7-5 = 2
Rate = -2/2 = -1

Check (9,1) and (11,-1):
Δy = -1-1 = -2, Δx=11-9=2 → -2/2=-1 ✔

✔ Function? Yes
Rate of Change = -1

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Problem 3)

Table:
X | Y
-8 | -8
-5 | -5
-1 | 5
3 | -6
3 | 3 ← Uh oh! X=3 appears twice with different Y values (-6 and 3)

→ Not a function!

✘ Function? No
(No rate of change needed)

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Problem 4)

Table:
X | Y
60 | -6
-30 | 3
-60 | 6
-70 | 7
-90 | 9

All X unique → Function? Yes

Pick first two: (60, -6) and (-30, 3)
Δy = 3 - (-6) = 9
Δx = -30 - 60 = -90
Rate = 9 / (-90) = -1/10 or -0.1

Check next pair: (-30,3) and (-60,6)
Δy = 6-3=3, Δx=-60-(-30)= -30 → 3/-30 = -1/10 ✔

✔ Function? Yes
Rate of Change = -1/10

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Problem 5)

Table:
X | Y
-6 | -4
-5 | -9
-4 | -14
-3 | -19
-2 | -24

All X unique → Function? Yes

Pick (-6,-4) and (-5,-9):
Δy = -9 - (-4) = -5
Δx = -5 - (-6) = 1
Rate = -5/1 = -5

Check (-4,-14) and (-3,-19):
Δy = -19 - (-14) = -5, Δx = -3 - (-4) = 1 → -5/1 = -5 ✔

✔ Function? Yes
Rate of Change = -5

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Problem 6)

Table:
X | Y
-6 | 27
2 | 3
4 | -3
-8 | 33
9 | -18

All X unique → Function? Yes

Pick (-6,27) and (2,3):
Δy = 3 - 27 = -24
Δx = 2 - (-6) = 8
Rate = -24/8 = -3

Check (4,-3) and (9,-18):
Δy = -18 - (-3) = -15, Δx = 9-4=5 → -15/5 = -3 ✔

✔ Function? Yes
Rate of Change = -3

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Problem 7)

Table:
X | Y
72 | -27
79 | -25
104 | -15
154 | 5

All X unique → Function? Yes

Pick (72,-27) and (79,-25):
Δy = -25 - (-27) = 2
Δx = 79 - 72 = 7
Rate = 2/7

Check (104,-15) and (154,5):
Δy = 5 - (-15) = 20
Δx = 154 - 104 = 50
Rate = 20/50 = 2/5 ✘ Wait — not same as 2/7?

That means the rate is NOT constant → but wait, the question says “if the table IS a function, calculate the rate of change”. It doesn’t say it has to be linear. But typically in these problems, they expect you to check if it’s linear (constant rate), or just compute between two points? Let me re-read directions.

Directions: “IF the table IS a function, calculate the rate of change.”

It doesn’t specify which points. In most school contexts, if the rate isn’t constant, they might still want you to compute using first and last point? Or maybe assume it's linear? But here, let’s test all pairs.

From (72,-27) to (79,-25): rate = 2/7 ≈ 0.2857
From (79,-25) to (104,-15): Δy=10, Δx=25 → 10/25 = 2/5 = 0.4
Not same → so not linear.

But the problem doesn’t say “linear function” — just “function”. And since all X are unique, it IS a function.

However, “rate of change” for non-linear functions isn’t single-valued. But in middle/high school worksheets like this, they usually imply linear relationships. Since the rates aren’t matching, perhaps I made a mistake?

Wait — let me recalculate:

Point A: (72, -27)
Point B: (79, -25) → Δy = 2, Δx = 7 → 2/7

Point C: (104, -15) → from B to C: Δy = -15 - (-25) = 10, Δx = 104-79=25 → 10/25 = 2/5

Point D: (154,5) → from C to D: Δy=5-(-15)=20, Δx=154-104=50 → 20/50=2/5

So from B to C to D, rate is 2/5, but from A to B it’s 2/7. So overall, not constant.

But maybe the worksheet expects us to use first and last point?

Try (72,-27) and (154,5):
Δy = 5 - (-27) = 32
Δx = 154 - 72 = 82
Rate = 32/82 = 16/41 ≈ 0.39

Still not nice number.

Alternatively, perhaps there’s a typo? Or maybe I should report that it’s a function but rate varies? But the blank asks for “Rate of Change ___”, implying one answer.

Looking back at other problems — all others had constant rate. Maybe this one does too? Let me check differences again.

List the points:

X: 72, 79, 104, 154
Y: -27, -25, -15, 5

Compute slopes between consecutive points:

Between 1st and 2nd: ( -25 +27 ) / (79-72) = 2/7
Between 2nd and 3rd: ( -15 +25 ) / (104-79) = 10/25 = 2/5
Between 3rd and 4th: (5 +15)/(154-104) = 20/50 = 2/5

So only first interval differs. Perhaps it’s meant to be linear? Or maybe accept that it’s a function and give average? But that’s not standard.

Wait — perhaps I miscalculated Y values? Let me see if there’s a pattern.

Assume linear: y = mx + b

Use first and last: m = (5 - (-27)) / (154 - 72) = 32/82 = 16/41

Then check middle points:

At x=79: y = (16/41)*79 + b → but we don’t know b.

This is getting messy. Given that in all other problems the rate was constant, and this one isn't, but the table still represents a function (since no repeated X), I think we should mark it as a function, and perhaps the expected answer is to use first and last point? Or maybe the problem has an error.

But looking at the numbers: from x=79 to 154, the rate is consistently 2/5. From 72 to 79, it's different. Maybe it's intentional to show non-linear? But then "rate of change" isn't defined uniquely.

In many curricula, when they ask for "rate of change" from a table without specifying, and if it's not constant, they might expect you to say it's not constant — but here the blank is for a number.

Perhaps I should proceed with the calculation between first and last point, as sometimes done.

So: Δy = 5 - (-27) = 32
Δx = 154 - 72 = 82
Rate = 32/82 = 16/41

But that’s ugly. Alternatively, maybe the Y values are wrong? Let me see if there’s a common difference.

Another idea: perhaps the rate is supposed to be calculated as total change over total change, even if not linear. In some contexts, that’s accepted.

I’ll go with that for now.

✔ Function? Yes
Rate of Change = 32/82 = 16/41 (or simplify later)

But let’s keep it as fraction: 16/41

Wait — 32 and 82 divided by 2: 16/41, yes.

But let me double-check the table: is it possible that Y at x=72 is -27, and at x=154 is 5, etc. Yes.

Perhaps the problem intends for us to notice it's not linear, but still call it a function and leave rate blank? But the instruction says "calculate the rate of change" if it is a function.

I think for consistency, since other problems have constant rate, and this one doesn't, but it is a function, I'll compute using first and last point.

So Rate of Change = (5 - (-27)) / (154 - 72) = 32/82 = 16/41

But let's see if 16/41 reduces? 16 and 41 coprime, so yes.

Alternatively, decimal? But fractions are better.

I'll put 16/41.

But let me confirm with another approach. Maybe the rate is constant if I misread the table.

Table is:

72 | -27
79 | -25
104 | -15
154 | 5

Let me plot mentally: from 72 to 79 (delta x=7), y from -27 to -25 (delta y=2)
79 to 104 (delta x=25), y from -25 to -15 (delta y=10) → 10/25=0.4
104 to 154 (delta x=50), y from -15 to 5 (delta y=20) → 20/50=0.4

So after the first interval, it's constant at 0.4 = 2/5. First interval is 2/7≈0.2857.

So overall, not constant. But perhaps in the context of the worksheet, they expect us to use any two points, or first and last.

To avoid confusion, and since the problem likely assumes linearity, but it's not, I think the safe answer is to say it's a function, and for rate of change, use the consistent part or average. But I recall that in some systems, they define average rate of change as (f(b)-f(a))/(b-a).

So I'll go with that.

✔ Function? Yes
Rate of Change = (5 - (-27)) / (154 - 72) = 32/82 = 16/41

Simplify: 16/41 is already simplified.

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Problem 8)

Table:
X | Y
9 | 3
16 | -4
36 | -6
36 | 6 ← X=36 appears twice with Y=-6 and Y=6 → not a function!

✘ Function? No

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Problem 9)

Table is given horizontally:

X: 5, 8, 11, 14
Y: 10, 8, 6, 4

So pairs: (5,10), (8,8), (11,6), (14,4)

All X unique → Function? Yes

Pick (5,10) and (8,8):
Δy = 8-10 = -2
Δx = 8-5 = 3
Rate = -2/3

Check (11,6) and (14,4):
Δy=4-6=-2, Δx=14-11=3 → -2/3 ✔

✔ Function? Yes
Rate of Change = -2/3

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Problem 10)

Table:
X: 6, 14, 6, 16
Y: 0, 1, -2, -3

Look at X=6: appears twice, with Y=0 and Y=-2 → not a function!

✘ Function? No

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Now, compiling all answers:

1) Function? Yes, Rate of Change = 1
2) Function? Yes, Rate of Change = -1
3) Function? No
4) Function? Yes, Rate of Change = -1/10
5) Function? Yes, Rate of Change = -5
6) Function? Yes, Rate of Change = -3
7) Function? Yes, Rate of Change = 16/41 (but let me confirm if this is correct — perhaps I should use a different pair? Or maybe the problem has a typo. Another thought: perhaps the Y values are meant to be linear. Let me calculate what Y should be if rate is constant from start.

Suppose from (72,-27) to (154,5), slope m=32/82=16/41

Then at x=79, y = -27 + (16/41)*(79-72) = -27 + (16/41)*7 = -27 + 112/41 = (-27*41 + 112)/41 = (-1107 + 112)/41 = -995/41 ≈ -24.268, but actual Y is -25, close but not exact.

At x=104, y = -27 + (16/41)*(104-72) = -27 + (16/41)*32 = -27 + 512/41 = (-1107 + 512)/41 = -595/41 ≈ -14.512, actual is -15.

Not matching. So definitely not linear.

Perhaps the intended rate is 2/5, ignoring the first point? But that's arbitrary.

Maybe the first Y is wrong. Suppose at x=72, Y should be such that slope to next is 2/5.

From x=72 to 79, delta x=7, if slope=2/5, delta y= (2/5)*7 = 14/5 = 2.8, so Y at 79 would be Y72 + 2.8. But actual Y79 is -25, so Y72 = -25 - 2.8 = -27.8, but it's given as -27.

Close but not exact.

Given the context, and since other problems are clean, perhaps for problem 7, they expect us to use the majority rate or something. But I think for accuracy, I'll stick with the average rate of change from first to last.

So Rate of Change = 32/82 = 16/41

But let's reduce it: 32÷2=16, 82÷2=41, so 16/41.

Yes.

8) Function? No
9) Function? Yes, Rate of Change = -2/3
10) Function? No

Now, for the final answer, since the user didn't specify which problem, but the image has 10 problems, I need to provide all answers.

The response format says "Final Answer: [Provide only the final correct answer]", but since there are multiple, I should list them.

Looking back at the user's request: "Solve the problem accurately." and the image has 10 problems, so I need to solve all.

In the Final Answer section, I'll list each problem's answer.

Also, for problem 7, I'm a bit unsure, but I'll go with 16/41.

Perhaps I can write it as a mixed number or decimal, but fraction is fine.

Another idea: maybe calculate the rate between each consecutive pair and see if there's a pattern, but the blank is for one number.

I think 16/41 is correct for average rate of change.

So here we go.

Final Answer:

1) Function? Yes, Rate of Change = 1
2) Function? Yes, Rate of Change = -1
3) Function? No
4) Function? Yes, Rate of Change = -1/10
5) Function? Yes, Rate of Change = -5
6) Function? Yes, Rate of Change = -3
7) Function? Yes, Rate of Change = 16/41
8) Function? No
9) Function? Yes, Rate of Change = -2/3
10) Function? No

For problem 7, to make it neat, perhaps leave as 32/82 simplified to 16/41.

Yes.

I think that's it.