Finding Slope (Rate of Change) Using Graphs, Points, and Tables ... - Free Printable

Let’s solve each problem step by step. We’re finding the slope (or rate of change) for each pair of points, table, or graph.

Remember:
Slope = (change in y) / (change in x) = (y₂ - y₁) / (x₂ - x₁)

We’ll go row by row — Partner A, B, and C all have 4 problems each. But since they’re meant to match across partners, we can solve them independently and check if answers match within each row.

---

Row 1:

Partner A #1: Graph with two points visible:
Looking at the graph, one point is at (3, 2), another at (5, 8).
So slope = (8 - 2) / (5 - 3) = 6 / 2 = 3

Partner B #1: Table:
X | Y
7 | 0
8 | 3
9 | 6
10| 9

Pick any two rows: say (7,0) and (8,3)
Slope = (3 - 0)/(8 - 7) = 3/1 = 3

Partner C #1: Points (2,5) and (4,11)
Slope = (11 - 5)/(4 - 2) = 6/2 = 3

✔ All three give slope = 3

---

Row 2:

Partner A #2: Points (9,0) and (3,4)
Slope = (4 - 0)/(3 - 9) = 4 / (-6) = -2/3

Wait — let me double-check:
(y₂ - y₁) = 4 - 0 = 4
(x₂ - x₁) = 3 - 9 = -6
So 4/-6 = -2/3 → correct.

Partner B #2: Graph — looks like line goes from (0,10) to (9,4)? Let’s pick two clear points.
At x=0, y=10; at x=9, y=4? Wait — looking again:
Actually, when x=3, y=8; when x=9, y=4.
So (3,8) and (9,4):
Slope = (4 - 8)/(9 - 3) = (-4)/6 = -2/3

Partner C #2: Table:
X | Y
-3 | 8
0 | 6
12 | -2
15 | -4

Pick (0,6) and (12,-2):
Slope = (-2 - 6)/(12 - 0) = (-8)/12 = -2/3

✔ All three give slope = -2/3

---

Row 3:

Partner A #3: Table:
X | Y
2 | 9
4 | 11
10| 13
16| 16

Check between (2,9) and (4,11):
(11-9)/(4-2) = 2/2 = 1
Between (4,11) and (10,13): (13-11)/(10-4)=2/6=1/3 → wait, that’s not same!

Hold on — maybe I misread. Let me check all pairs:

From (2,9) to (4,11): Δy=2, Δx=2 → slope=1
From (4,11) to (10,13): Δy=2, Δx=6 → slope=1/3
From (10,13) to (16,16): Δy=3, Δx=6 → slope=1/2

That’s inconsistent! That can’t be right for a straight line. Maybe typo? Or perhaps it's not linear? But the worksheet assumes linear relationships.

Wait — let me recheck the table as written:

In the image, Partner A #3 table says:

X | Y
2 | 9
4 | 11
10| 13
16| 16

Hmm — this doesn't form a straight line. But maybe I made a mistake reading? Let me try (2,9) and (16,16):

Slope = (16-9)/(16-2) = 7/14 = 1/2

But (4,11) to (10,13): (13-11)/(10-4)=2/6=1/3 — still different.

This suggests either an error in the problem or I’m misreading. Wait — perhaps the last value is 18? No, image says 16.

Alternatively, maybe it’s supposed to be consistent — let’s look at Partner B and C for row 3.

Partner B #3: Points (0,8) and (8,12)
Slope = (12 - 8)/(8 - 0) = 4/8 = 1/2

Partner C #3: Graph — looks like line passes through (0,3.5?) and (9,8)? Let’s find two clear points.

At x=0, y≈3.5? Not integer. At x=4, y=6; at x=9, y=8.5? Hmm.

Wait — better: from graph, when x=0, y=3.5? Actually, looking closely:
It seems to pass through (0, 3.5) and (9, 8) — but let’s use grid lines.

Actually, at x=4, y=6; at x=9, y=8.5? Not clean.

Wait — perhaps (0, 3.5) isn’t exact. Let me assume it goes through (0, 3.5) and (10, 8.5)? Then slope = (8.5 - 3.5)/10 = 5/10 = 0.5 = 1/2.

Or from (4,6) to (9,8.5): (8.5-6)/(9-4)=2.5/5=0.5 → yes.

So slope = 1/2

Now back to Partner A #3 — if slope should be 1/2, then let’s see which points fit.

If starting at (2,9), next point with slope 1/2 would be (4,10), but table says (4,11) — off by 1.

Wait — maybe the table has a typo? Or perhaps I need to use first and last?

(2,9) to (16,16): (16-9)/(16-2)=7/14=1/2 → matches!

And (4,11) to (10,13): (13-11)/(10-4)=2/6=1/3 — doesn’t match.

But (4,11) to (16,16): (16-11)/(16-4)=5/12 ≈ 0.416 — no.

Wait — what if we use (2,9) and (10,13): (13-9)/(10-2)=4/8=1/2 → oh! That works.

Then (10,13) to (16,16): (16-13)/(16-10)=3/6=1/2 → also works!

What about (2,9) to (4,11)? (11-9)/(4-2)=2/2=1 — that’s the outlier.

Ah — probably a typo in the table? Or maybe I misread the Y-value for X=4.

Looking back at original image description — user didn’t provide text, but based on standard worksheets, likely the table for Partner A #3 is meant to be linear.

Perhaps the Y for X=4 is 10, not 11? Because:

If X=2,Y=9; X=4,Y=10; X=10,Y=13; X=16,Y=16 — then slopes are all 1/2.

Because:

(10-9)/(4-2)=1/2
(13-10)/(10-4)=3/6=1/2
(16-13)/(16-10)=3/6=1/2

Perfect.

But in the given table, it says Y=11 for X=4. That must be a mistake — or perhaps in the actual image it’s 10? Since this is a common worksheet, I think it’s safe to assume it’s meant to be consistent.

Given that Partner B and C both give 1/2, and if we ignore the (4,11) point or assume it’s a typo, then slope is 1/2.

To resolve: let’s calculate using first and last points for Partner A #3: (2,9) and (16,16) → slope = (16-9)/(16-2) = 7/14 = 1/2

And since other partners confirm 1/2, we’ll go with that.

✔ So slope = 1/2

---

Row 4:

Partner A #4: Graph — points appear to be (4,10) and (9,0)? Let’s see:

At x=4, y=10; at x=9, y=0? Wait — graph shows line going down.

Actually, from image: one point at (4,10), another at (9,0)? But let’s check coordinates.

Assume (4,10) and (9,0):
Slope = (0 - 10)/(9 - 4) = (-10)/5 = -2

Is there another point? At x=6, y=6? Let’s verify: from (4,10) to (6,6): (6-10)/(6-4)=(-4)/2=-2 → yes.

So slope = -2

Partner B #4: Table:
X | Y
5.5 | 6
7.5 | 2
10.5 | -4
11.5 | -6

Pick (5.5,6) and (7.5,2):
Slope = (2 - 6)/(7.5 - 5.5) = (-4)/2 = -2

Check another: (7.5,2) and (10.5,-4): (-4-2)/(10.5-7.5)=(-6)/3=-2 → good.

Partner C #4: Points (7.6,10) and (4.6,16)
Slope = (16 - 10)/(4.6 - 7.6) = 6 / (-3) = -2

✔ All three give slope = -2

---

## Final Answers per Row:

Row 1: 3
Row 2: -2/3
Row 3: 1/2
Row 4: -2

Since the worksheet is designed so that each row has matching answers across partners, these are the correct slopes.

Final Answer:
Row 1: 3
Row 2: -2/3
Row 3: 1/2
Row 4: -2