Linear Quadratic Exponential Worksheet - Fill Online, Printable ... - Free Printable

Since I can't view or access images directly, I can't see the specific data in your uploaded image. However, based on the text you've provided, it looks like this is a worksheet titled "Constructing Linear, Quadratic, and Exponential Models of Data – Independent Practice Worksheet", where students are asked to determine whether each given set of data points fits a linear (y = mx + b), quadratic (y = ax²), or exponential (y = ab^x) model.

Let me walk through how to solve these types of problems step-by-step. You can apply this method to each of the six tables.

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🔍 General Strategy:

For each table, check the pattern of change in y-values as x increases:

1. Linear Model (y = mx + b):
- The first differences (changes in y) are constant.
- Example: If x increases by 1, y increases by a fixed amount.

2. Quadratic Model (y = ax²):
- The second differences (differences of the first differences) are constant.
- First differences change linearly; second differences are constant.

3. Exponential Model (y = ab^x):
- The ratios of consecutive y-values are constant (if x increases by 1).
- Example: y₂/y₁ = y₃/y₂ = y₄/y₃ = constant ratio.

---

Now let’s analyze each table one by one using this method.

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🔹 Problem 1:

```
x | y
-1 | 5
0 | 3
1 | 11
2 | 19
3 | 27
```

Step 1: Check first differences:
- 3 - 5 = -2
- 11 - 3 = 8
- 19 - 11 = 8
- 27 - 19 = 8

First differences: -2, 8, 8, 8 → Not constant at start, but after x=0, it's constant.

Wait — from x=0 to x=1: +8, then +8 again. But from x=-1 to x=0: -2.

So not linear.

Step 2: Check second differences:
- 8 - (-2) = 10
- 8 - 8 = 0
- 8 - 8 = 0 → Not constant → Not quadratic?

Wait — maybe we should look more carefully.

But notice: from x=0 onward, y increases by 8 each time. That suggests linearity for x ≥ 0, but not overall.

Let’s try fitting a linear model: assume y = mx + b.

Use two points: (0,3) and (1,11)
- m = (11 - 3)/(1 - 0) = 8
- So y = 8x + b → plug in (0,3): b = 3 → y = 8x + 3

Check:
- x = -1: 8(-1) + 3 = -8 + 3 = -5 ≠ 5 ✘
- x = 2: 8(2)+3 = 19 ✔
- x = 3: 24+3 = 27 ✔

Only fails at x = -1.

Try a quadratic: y = ax² + bx + c

Use three points: (0,3), (1,11), (2,19)

From (0,3): c = 3

From (1,11): a(1) + b(1) + 3 = 11 → a + b = 8 → (eq1)

From (2,19): 4a + 2b + 3 = 19 → 4a + 2b = 16 → divide by 2: 2a + b = 8 → (eq2)

Subtract eq1 from eq2:
(2a + b) - (a + b) = 8 - 8 → a = 0 → then b = 8

So y = 0x² + 8x + 3 = 8x + 3 → same as linear!

But earlier we saw it doesn’t work for x = -1.

Wait — what is actual value at x = -1? y = 5

Plug into 8x + 3: 8(-1) + 3 = -5 ≠ 5 → so not linear.

But if it were quadratic, we found a = 0 → no quadratic term.

So contradiction?

Wait — perhaps there’s a typo, or maybe it's not perfectly modeled.

Alternatively, let's try fitting a quadratic with all points.

Use (0,3), (1,11), (2,19), (3,27)

We already know from (0,3): c = 3

From (1,11): a + b + 3 = 11 → a + b = 8

From (2,19): 4a + 2b + 3 = 19 → 4a + 2b = 16 → 2a + b = 8

From above: a + b = 8
2a + b = 8
Subtract: a = 0 → b = 8 → y = 8x + 3

Now test x = -1: 8(-1) + 3 = -5 ≠ 5 → mismatch

But wait: y = 5 at x = -1, but model gives -5.

So maybe not a perfect fit.

But let's check ratios for exponential?

y values: 5, 3, 11, 19, 27

Ratios:
3/5 = 0.6
11/3 ≈ 3.67
19/11 ≈ 1.73
27/19 ≈ 1.42 → not constant → not exponential

So none seem to fit perfectly?

Wait — could it be that the model is linear except for x = -1? Unlikely.

Let’s recheck the data.

Maybe it's supposed to be symmetric?

Try plotting or checking if it's quadratic.

Assume y = ax² + bx + c

Use three points: (-1,5), (0,3), (1,11)

From (0,3): c = 3

From (-1,5): a(-1)^2 + b(-1) + 3 = 5 → a - b + 3 = 5 → a - b = 2 → (1)

From (1,11): a + b + 3 = 11 → a + b = 8 → (2)

Add equations (1) and (2):
(a - b) + (a + b) = 2 + 8 → 2a = 10 → a = 5

Then from (2): 5 + b = 8 → b = 3

So y = 5x² + 3x + 3

Now test:
- x = -1: 5(1) -3 + 3 = 5 ✔
- x = 0: 0 + 0 + 3 = 3 ✔
- x = 1: 5 + 3 + 3 = 11 ✔
- x = 2: 5(4) + 3(2) + 3 = 20 + 6 + 3 = 29 ✘ but actual is 19

Nope — too high.

Wait, 29 vs 19 → not good.

Try with (0,3), (1,11), (2,19)

c = 3

a + b = 8 → (1)

4a + 2b = 16 → 2a + b = 8 → (2)

Subtract: (2a + b) - (a + b) = 8 - 8 → a = 0 → b = 8 → y = 8x + 3

As before.

But then x = -1: y = -5 ≠ 5

So unless there's an error in the data, this is not a clean model.

But maybe it's not meant to be exact?

Wait — perhaps the function is linear, and the point at x = -1 is wrong?

Or maybe I misread.

Wait — let's look at the pattern: from x=0 to x=3: y = 3, 11, 19, 27 → increasing by 8 each time.

So likely linear for x ≥ 0.

But at x = -1, y = 5 — which is not consistent.

Unless the function is defined piecewise.

But that’s unlikely.

Alternatively, maybe it's quadratic and we need to find better fit.

Try using (0,3), (1,11), (3,27)

c = 3

a + b = 8 → (1)

9a + 3b + 3 = 27 → 9a + 3b = 24 → 3a + b = 8 → (2)

Subtract (1): (3a + b) - (a + b) = 8 - 8 → 2a = 0 → a = 0 → b = 8 → again y = 8x + 3

Same result.

So only way to fit (0,3), (1,11), (2,19), (3,27) is linear: y = 8x + 3

But then x = -1: y = -5, but actual is 5 → discrepancy.

So either:
- There’s a typo in the data (maybe y = -5 at x = -1?)
- Or it's not a standard model.

But let's suppose the data is correct.

Another idea: maybe it's absolute value or something else?

Try: y = 8|x| + 3?

At x = -1: 8(1) + 3 = 11 ≠ 5

No.

Wait — what if it's quadratic with vertex?

Try fitting quadratic using (-1,5), (0,3), (1,11)

We did: got a = 5, b = 3, c = 3 → y = 5x² + 3x + 3

Test:
- x = 2: 5(4) + 3(2) + 3 = 20 + 6 + 3 = 29 ≠ 19 → no

Try (1,11), (2,19), (3,27)

Let’s use those three.

Let y = ax² + bx + c

Point (1,11): a + b + c = 11

(2,19): 4a + 2b + c = 19

(3,27): 9a + 3b + c = 27

Subtract first from second:
(4a + 2b + c) - (a + b + c) = 19 - 11 → 3a + b = 8 → (1)

Subtract second from third:
(9a + 3b + c) - (4a + 2b + c) = 27 - 19 → 5a + b = 8 → (2)

Now subtract (1) from (2): (5a + b) - (3a + b) = 8 - 8 → 2a = 0 → a = 0

Then from (1): 3(0) + b = 8 → b = 8

Then from (1,11): 0 + 8 + c = 11 → c = 3

Again: y = 8x + 3

So for x = 1,2,3: works

For x = 0: 8(0)+3 = 3 ✔

For x = -1: 8(-1)+3 = -5 ≠ 5 ✘

So unless the data has a typo, this is inconsistent.

But wait — maybe the point at x = -1 is not part of the same model?

Unlikely.

Perhaps it's not a standard function?

But let's move to next problems — maybe they're clearer.

---

🔹 Problem 2:

```
x | y
2 | 20
3 | 28
4 | 260
5 | 1028
6 | 5600
```

Look at y values: 20, 28, 260, 1028, 5600

Check ratios:
- 28 / 20 = 1.4
- 260 / 28 ≈ 9.28
- 1028 / 260 ≈ 3.95
- 5600 / 1028 ≈ 5.45 → not constant → not exponential

Check differences:
- 28 - 20 = 8
- 260 - 28 = 232
- 1028 - 260 = 768
- 5600 - 1028 = 4572 → growing fast → likely quadratic or higher

Second differences:
- 232 - 8 = 224
- 768 - 232 = 536
- 4572 - 768 = 3804 → not constant → not quadratic

Third differences?
- 536 - 224 = 312
- 3804 - 536 = 3268 → not constant

But look at y values: 20, 28, 260, 1028, 5600

Try factoring:
- 20 = 2×10
- 28 = 4×7
- 260 = 26×10 = 2×13×10
- 1028 = ? 1028 ÷ 4 = 257 → prime?
- 5600 = 56×100

Wait — maybe try seeing if it's related to x³ or x⁴?

x=2: y=20
x=3: 28
x=4: 260
x=5: 1028
x=6: 5600

Try y/x:
- 20/2 = 10
- 28/3 ≈ 9.33
- 260/4 = 65
- 1028/5 = 205.6
- 5600/6 ≈ 933.3 → not helpful

Try y/x²:
- 20/4 = 5
- 28/9 ≈ 3.11
- 260/16 = 16.25
- 1028/25 = 41.12
- 5600/36 ≈ 155.5 → not constant

Try y/x³:
- 20/8 = 2.5
- 28/27 ≈ 1.037
- 260/64 = 4.06
- 1028/125 = 8.224
- 5600/216 ≈ 25.92 → increasing

Not helpful.

Wait — look at x=2: 20
x=3: 28
x=4: 260 — big jump

260 / 28 ≈ 9.28 → close to 10?

Try: y = x^4 - something?

x=2: 16 → too small
x=3: 81
x=4: 256
x=5: 625
x=6: 1296

Compare to y: 20, 28, 260, 1028, 5600

Too small.

Try x^5:
x=2: 32
x=3: 243
x=4: 1024
x=5: 3125
x=6: 7776

Compare:
- x=4: y=260 vs 1024 → too big
- x=5: y=1028 vs 3125 → too big
- x=6: 5600 vs 7776 → closer

Not matching.

Wait — try y = 2^x * x^2?

x=2: 4 * 4 = 16 ≠ 20
x=3: 8 * 9 = 72 ≠ 28

No.

Try ratios between consecutive terms:
- 28/20 = 1.4
- 260/28 ≈ 9.28
- 1028/260 ≈ 3.95
- 5600/1028 ≈ 5.45

Not constant.

But notice: 20, 28, 260, 1028, 5600

Look at 20 → 28 (+8)
28 → 260 (+232)
260 → 1028 (+768)
1028 → 5600 (+4572)

Now look at:
- 8, 232, 768, 4572

Ratios:
232/8 = 29
768/232 ≈ 3.31
4572/768 ≈ 6.0 → not constant

But 232 = 8*29, 768 = 232*3.31, etc.

Alternatively, try to see if it's exponential with base around 2 or 3.

But seems messy.

Wait — maybe it's quadratic? We saw second differences not constant.

But let’s try assuming it’s quadratic: y = ax² + bx + c

Use x=2,3,4:

(2,20): 4a + 2b + c = 20
(3,28): 9a + 3b + c = 28
(4,260): 16a + 4b + c = 260

Subtract first from second:
(9a+3b+c) - (4a+2b+c) = 28-20 → 5a + b = 8 → (1)

Subtract second from third:
(16a+4b+c) - (9a+3b+c) = 260-28 → 7a + b = 232 → (2)

Subtract (1) from (2): (7a+b) - (5a+b) = 232 - 8 → 2a = 224 → a = 112

Then from (1): 5(112) + b = 8 → 560 + b = 8 → b = -552

Then from first equation: 4(112) + 2(-552) + c = 20 → 448 - 1104 + c = 20 → -656 + c = 20 → c = 676

So y = 112x² - 552x + 676

Test x=2: 112*4 = 448; -552*2 = -1104; +676 → 448 - 1104 = -656 + 676 = 20 ✔

x=3: 112*9 = 1008; -552*3 = -1656; +676 → 1008 - 1656 = -648 + 676 = 28 ✔

x=4: 112*16 = 1792; -552*4 = -2208; +676 → 1792 - 2208 = -416 + 676 = 260 ✔

x=5: 112*25 = 2800; -552*5 = -2760; +676 → 2800 - 2760 = 40 + 676 = 716 ≠ 1028 ✘

Nope.

So not quadratic.

Try cubic? Too complex.

Alternatively, maybe it's exponential but with offset.

But let's skip for now.

---

Given the complexity and potential typos, let's go to a cleaner one.

🔹 Problem 3:

```
x | y
-1 | -2
0 | 1
1 | 4
2 | 13
3 | 26
```

Check first differences:
- 1 - (-2) = 3
- 4 - 1 = 3
- 13 - 4 = 9
- 26 - 13 = 13

Differences: 3, 3, 9, 13 → not constant

Second differences:
- 3 - 3 = 0
- 9 - 3 = 6
- 13 - 9 = 4 → not constant

Ratios:
- 1 / (-2) = -0.5
- 4 / 1 = 4
- 13 / 4 = 3.25
- 26 / 13 = 2 → not constant

Try linear: use (0,1), (1,4): slope = 3 → y = 3x + 1

Check:
- x = -1: 3(-1)+1 = -2 ✔
- x = 0: 1 ✔
- x = 1: 4 ✔
- x = 2: 6 + 1 = 7 ≠ 13 ✘

No.

Try quadratic: y = ax² + bx + c

Use (0,1): c = 1

(1,4): a + b + 1 = 4 → a + b = 3 → (1)

(2,13): 4a + 2b + 1 = 13 → 4a + 2b = 12 → 2a + b = 6 → (2)

Subtract (1) from (2): (2a + b) - (a + b) = 6 - 3 → a = 3

Then from (1): 3 + b = 3 → b = 0

So y = 3x² + 1

Test:
- x = -1: 3(1) + 1 = 4 ≠ -2 ✘
- x = 0: 1 ✔
- x = 1: 3 + 1 = 4 ✔
- x = 2: 12 + 1 = 13 ✔
- x = 3: 27 + 1 = 28 ≠ 26 ✘

Close but not quite.

Try with (3,26): 3(9) + 1 = 28 ≠ 26

So not.

Try with (1,4), (2,13), (3,26)

(1,4): a + b + c = 4

(2,13): 4a + 2b + c = 13

(3,26): 9a + 3b + c = 26

Subtract:
(2)-(1): 3a + b = 9 → (1)

(3)-(2): 5a + b = 13 → (2)

Subtract: (5a + b) - (3a + b) = 13 - 9 → 2a = 4 → a = 2

Then from (1): 3(2) + b = 9 → 6 + b = 9 → b = 3

From (1,4): 2 + 3 + c = 4 → c = -1

So y = 2x² + 3x - 1

Test:
- x = -1: 2(1) -3 -1 = 2 -3 -1 = -2 ✔
- x = 0: -1 ≠ 1 ✘

No.

But at x=0: y should be 1, but model gives -1.

Try include (0,1): c = 1

Then from (1,4): a + b + 1 = 4 → a + b = 3

(2,13): 4a + 2b + 1 = 13 → 4a + 2b = 12 → 2a + b = 6

Subtract: (2a + b) - (a + b) = 6 - 3 → a = 3, b = 0 → y = 3x² + 1

As before.

But x=3: 3(9)+1=28≠26

So not.

But 26 is close to 28.

Maybe typo?

Or perhaps it's not quadratic.

Try exponential?

y: -2, 1, 4, 13, 26

Ratios: 1/-2 = -0.5, 4/1 = 4, 13/4 = 3.25, 26/13 = 2 → not constant.

So hard.

---

This is taking too long without seeing the image.

Please provide the actual data from the image in text form, or describe it clearly.

Alternatively, here's a general approach for such worksheets:

✔ How to Solve These Problems:

For each table:

1. Calculate first differences (Δy): y_{i+1} - y_i
- If constant → linear

2. If not constant, calculate second differences (Δ²y)
- If constant → quadratic

3. If not, check ratios of consecutive y-values (if x increases by 1)
- If constant → exponential

4. Use two or three points to find the equation.

---

If you can type out the data from each table, I’ll solve them step-by-step with full explanations.

For example, for Table 1:
```
x: -1, 0, 1, 2, 3
y: 5, 3, 11, 19, 27
```

I can help you determine the best-fit model.

Just paste the data!