Finding the Function Rule for a Table - Free Printable

Let’s solve each table one by one to find the equation that relates X and Y.

We’re looking for a pattern — usually, it’s a linear equation like:
Y = mX + b, where “m” is the slope (how much Y changes when X increases by 1), and “b” is the y-intercept (the value of Y when X = 0).

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Table 1 (top left):

X | Y
-2 | 6
-1 | 3
0 | 0 ← highlighted
1 | -3
2 | -6

When X goes from 0 to 1, Y goes from 0 to -3 → change of -3
From 1 to 2: -3 to -6 → also -3
So slope (m) = -3

At X=0, Y=0 → so b = 0

Equation: Y = -3X

Check:
X=-2 → -3*(-2)=6 ✔
X=-1 → -3*(-1)=3 ✔
X=1 → -3*1=-3 ✔
Perfect.

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Table 2 (top middle):

X | Y
-2 | 4
-1 | 5
0 | 6
1 | 7
2 | 8

From X=0 to 1: Y from 6 to 7 → +1
Slope = 1
At X=0, Y=6 → b=6

Equation: Y = X + 6

Check:
X=-2 → -2+6=4 ✔
X=-1 → -1+6=5 ✔
X=2 → 2+6=8 ✔
Good.

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Table 3 (top right):

X | Y
-2 | -4
-1 | -2
0 | 0
1 | 2
8 | 16 ← wait, this doesn’t fit the pattern? Let’s check.

From X=0 to 1: Y from 0 to 2 → +2
From X=1 to 2? Wait, next is X=8 → Y=16? That seems off. But let’s see the first few:

X=-2 → Y=-4 → ratio: (-4)/(-2)=2
X=-1 → Y=-2 → (-2)/(-1)=2
X=0 → Y=0 → 0/0 undefined, but fits if we assume Y=2X
X=1 → Y=2 → 2*1=2 ✔
X=8 → Y=16 → 2*8=16 ✔

Oh! So even though X jumps to 8, it still fits: Y = 2X

Equation: Y = 2X

All points work.

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Table 4 (bottom left):

X | Y
-2 | 0
-1 | 2
0 | 4
1 | 6
2 | 8

From X=0 to 1: Y from 4 to 6 → +2 → slope = 2
At X=0, Y=4 → b=4

Equation: Y = 2X + 4

Check:
X=-2 → 2*(-2)+4 = -4+4=0 ✔
X=-1 → 2*(-1)+4 = -2+4=2 ✔
X=2 → 4+4=8 ✔
Good.

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Table 5 (bottom middle):

X | Y
-2 | -8
-1 | -5
0 | -2
1 | 1
4 | 10

Let’s find slope between consecutive points.

From X=0 to 1: Y from -2 to 1 → +3
From X=1 to 4: Y from 1 to 10 → over 3 steps in X, up 9 in Y → 9/3=3 → slope=3

At X=0, Y=-2 → b=-2

Equation: Y = 3X - 2

Check:
X=-2 → 3*(-2)-2 = -6-2=-8 ✔
X=-1 → -3-2=-5 ✔
X=4 → 12-2=10 ✔
Perfect.

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Table 6 (bottom right):

X | Y
0 | 5
1 | 9
2 | 13
3 | 17
4 | 13 ← Wait! This breaks the pattern.

First, look at X=0 to 3:

X=0 → Y=5
X=1 → Y=9 → +4
X=2 → Y=13 → +4
X=3 → Y=17 → +4
So slope = 4, and at X=0, Y=5 → should be Y = 4X + 5

But then X=4 → Y=13? According to formula: 4*4 + 5 = 16+5=21, but given Y=13 → not matching.

Wait — maybe typo? Or perhaps it's not linear? But all others are linear.

Looking again: X=4, Y=13 — that’s less than previous. Maybe mistake in problem?

But let’s double-check the pattern:

From X=0 to 3: every time X increases by 1, Y increases by 4 → consistent.

Then X=4 should be 17+4=21, but it says 13.

Possibility: maybe it’s a trick? Or misprint?

Alternatively, could it be quadratic? Let’s test.

Assume Y = aX² + bX + c

Use three points:

At X=0, Y=5 → c=5

At X=1, Y=9 → a(1) + b(1) + 5 = 9 → a + b = 4 → eq1

At X=2, Y=13 → 4a + 2b + 5 = 13 → 4a + 2b = 8 → divide by 2: 2a + b = 4 → eq2

Now subtract eq1 from eq2:

(2a + b) - (a + b) = 4 - 4 → a = 0

Then from eq1: 0 + b = 4 → b=4

So Y = 0*X² + 4X + 5 → Y=4X+5 → same as before.

Then at X=3: 4*3+5=17 ✔
At X=4: 4*4+5=21, but table says 13 → contradiction.

Unless... maybe the last point is wrong? Or perhaps it’s intentional to test attention?

But since all other tables are perfectly linear, and this one almost is except last point, likely a typo. In educational contexts, sometimes they include an outlier to see if you notice — but here, since the question asks for "Y = ___", implying a single equation, and 4 out of 5 points fit Y=4X+5, probably that’s intended.

Perhaps the last entry is meant to be X=4, Y=21? Or maybe X= something else?

Wait — what if it’s not X=4, but X= -1 or something? No, listed as 4.

Another idea: maybe it’s absolute value or piecewise? Unlikely for this level.

Given context, I think we go with the clear pattern: Y = 4X + 5, and assume the last point is a mistake.

But let’s verify with another approach.

Compute differences:

X: 0,1,2,3,4
Y:5,9,13,17,13

First differences (ΔY): 4,4,4,-4 → not constant → not linear.

Second differences: 0,0,-8 → not constant → not quadratic either.

Hmm.

Wait — perhaps it’s Y = 4X + 5 for X ≤ 3, and something else? But no indication.

Maybe the last row is X= -1? But written as 4.

Looking back at image description — user said “image containing task”, and in bottom right table, last row is X=4, Y=13.

But in many such worksheets, sometimes there’s a trick — like after X=3, it decreases? But why?

Alternative: maybe it’s Y = |something|? Not obvious.

Or perhaps it’s a different function.

Let me try fitting with X=4,Y=13 included.

Suppose we ignore X=0 for a moment.

Points: (1,9), (2,13), (3,17), (4,13)

From 1 to 2: +4
2 to 3: +4
3 to 4: -4 → symmetric? Like a parabola opening down.

Vertex at X=3? Since it peaks at 17.

Assume vertex form: Y = a(X-h)^2 + k, with h=3, k=17

Then Y = a(X-3)^2 + 17

At X=2: Y=13 → a(2-3)^2 +17 = a(1) +17 =13 → a= -4

Then equation: Y = -4(X-3)^2 + 17

Check X=1: -4(1-3)^2 +17 = -4(4) +17 = -16+17=1 ≠9 ✘

Not working.

At X=0: -4(0-3)^2 +17 = -4*9 +17= -36+17=-19 ≠5 ✘

No.

Try using (0,5), (1,9), (2,13), (3,17) — these are perfect for Y=4X+5

And (4,13) is outlier.

In standard curriculum, if 4 points fit a line and one doesn't, and no other pattern, we assume the line is intended, and the last point might be error.

Perhaps it's X= -1 for last? But written as 4.

Another thought: maybe the last X is -1? But in table it's listed as 4.

Looking at the sequence: X=0,1,2,3,4 — increasing.

Y=5,9,13,17,13 — so after 17, drops to 13.

Difference of -4.

If it were continuing, should be 21.

Perhaps it's a typo, and it's supposed to be Y=21.

I think for the purpose of this exercise, since all other tables are straightforward linear, and this one has a clear linear trend for first four points, we'll go with Y=4X+5.

To confirm, let's see if any other interpretation works.

Suppose it's Y = 4X + 5 for X<4, and at X=4, something else — but no rule given.

I think safest is to state the equation based on majority pattern.

So for Table 6: Y = 4X + 5

Even though last point doesn't fit, it's likely an error.

In some contexts, they might expect you to notice and say it's not linear, but the format suggests filling in an equation.

Given that, I'll proceed.

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Final Equations:

Table 1: Y = -3X
Table 2: Y = X + 6
Table 3: Y = 2X
Table 4: Y = 2X + 4
Table 5: Y = 3X - 2
Table 6: Y = 4X + 5 (assuming last point is typo)

But to be precise, let's list them as per order in image.

Image has two rows of three tables each.

Top row: left, middle, right
Bottom row: left, middle, right

So:

Top Left: Y = -3X
Top Middle: Y = X + 6
Top Right: Y = 2X
Bottom Left: Y = 2X + 4
Bottom Middle: Y = 3X - 2
Bottom Right: Y = 4X + 5

For bottom right, if we must account for X=4,Y=13, perhaps it's not linear, but I think for this level, it's expected to be linear.

Another idea: maybe the last point is X= -1? But written as 4.

Or perhaps it's X=4, Y=21, and 13 is misread.

I recall in some fonts, 21 looks like 13? Unlikely.

Perhaps it's Y = 4X + 5 for X≤3, and for X>3, different, but no.

I think we have to go with Y=4X+5.

So final answers:

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Final Answer:
Top Left: Y = -3X
Top Middle: Y = X + 6
Top Right: Y = 2X
Bottom Left: Y = 2X + 4
Bottom Middle: Y = 3X - 2
Bottom Right: Y = 4X + 5