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Amy Ferguson Moncure

Algebra 1 · Unit 8

Quadratic Patterns as Functions

From a picture pattern, write a quadratic function and connect each part of the formula to the picture.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Count, then table

Figure n = 1, 2, 3, … . Count tiles or dots. Look at first and second differences.

Constant second differences

A quadratic f(n) = an² + bn + c. The picture’s growing square or growing L often supplies the n² term.

Connect parts to the picture

n² can be an n-by-n square. A linear term can be a border. A constant can be a core that never changes.

Check the next figure

Use the formula to predict figure 4, then count if you can.

Level 1

Easy

A growing square. The picture is in the question.

Example
Figures show 1, 4, 9 dots. Write f(n) and connect it to the picture.

1, 4, 9 are 1², 2², 3².

f(n) = n²

Figure n is an n-by-n square of dots.

Answer: f(n) = n²

1
Figures show 2, 5, 10 dots (one extra besides a square). Write f(n).
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Squares 1, 4, 9 plus 1.

f(n) = n² + 1

The extra 1 is a dot that sits beside the square in every figure.

Answer: f(n) = n² + 1

2
Tiles: 3, 6, 11. First differences 3, 5. Second difference 2. Which family?
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Second differences constant (so far). Quadratic is the family to try.

Answer: quadratic

3
A student writes f(n) = 2n + 1 for 1, 4, 9. Show it fails at n = 2.
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f(2) = 5, but the picture has 4 dots.

The first differences are not constant.

Answer: not linear

4
An n-by-n square of tiles plus 1 extra tile. Write f(n) and check figure 3.
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f(n) = n² + 1

Figure 3 has 9 + 1 = 10 tiles.

Answer: f(n) = n² + 1

5
Counts 4, 9, 16. Write f(n) if these are figures n = 2, 3, 4 of a square pattern.
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f(n) = n²

Figure 2 has 4 dots.

Answer: f(n) = n²

Level 2

Medium

A square plus a border, or predict figure 5.

Example
A growing square of n-by-n tiles plus a row of n tiles along the bottom. Write f(n) and find f(5).

n² + n

f(n) = n(n + 1)

The n² is the square. The +n is the extra row.

f(5) = 25 + 5 = 30

Answer: f(n) = n² + n; f(5) = 30

1
Counts: 2, 6, 12, 20. Write f(n) = n² + n and check n = 1 to 4.
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1+1=2, 4+2=6, 9+3=12, 16+4=20. Matches.

Each figure is a triangle of dots that makes n(n + 1) — an n by (n+1) rectangle half, but here the formula is n² + n, a square plus a side.

Answer: f(n) = n² + n

2
An L-shape has 1, 3, 5, 7 tiles. A student says quadratic because it is an L. Correct them.
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First differences 2, 2, 2. Linear: f(n) = 2n − 1.

An L can still grow by a constant amount.

Answer: f(n) = 2n − 1

3
Square frames: figure n is the border of an n-by-n square (hollow). Write f(n) for n ≥ 2.
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n² − (n − 2)² = 4n − 4

Four sides of length n, minus 4 corners double-counted: 4(n − 1).

Answer: f(n) = 4n − 4

4
Counts 5, 8, 13, 20. Second differences 2, 2. Fit f(n) = n² + 4.
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1+4=5, 4+4=8, 9+4=13, 16+4=20. Yes.

The +4 is a 4-tile core beside a growing square.

Answer: f(n) = n² + 4

5
Figure n has an n-by-n square of blue tiles and n red tiles. Write f for the total, and say what n² and n mean.
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f(n) = n² + n

n² blue, n red.

Answer: n² + n

Level 3

Hard

Justify each term, or reject a wrong family.

Example
A picture is an n-by-n square plus two extra n-strips and 3 corner tiles. Write f(n) two ways and show they match.

n² + 2n + 3

Or (n + 1)² + (n + 2) = n² + 2n + 1 + n + 2 = n² + 3n + 3, which is a different picture.

Stick to the description: square n², two strips 2n, three corners +3.

f(n) = n² + 2n + 3

f(1) = 6, f(2) = 11, f(3) = 18.

Answer: f(n) = n² + 2n + 3

1
Counts 3, 8, 15, 24. Show second differences are 2, and write f(n) = n² + 2n.
Show Answer

First: 5, 7, 9. Second: 2, 2.

1+2=3, 4+4=8, 9+6=15, 16+8=24. Yes, n² + 2n.

Answer: f(n) = n² + 2n

2
Why is f(n) = 2n² a reasonable model for two separate n-by-n squares in each figure?
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Each square contributes n². Two squares: 2n².

Figure 1 has 2 tiles, figure 2 has 8, figure 3 has 18.

Answer: f(n) = 2n²

3
A pattern 2, 4, 8, 16 is labeled quadratic because it curves up. Write the right family and why second differences fail.
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Ratios 2, 2, 2. Exponential f(n) = 2ⁿ.

Second differences 2, 4, 8 are not constant.

Answer: exponential

4
Hollow square plus a diagonal: 4n − 4 + n for n ≥ 2, but the corner of the diagonal was already on the border. Fix the formula.
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Border 4n − 4. Diagonal of n tiles shares 2 corners with the border.

Add n − 2 interior diagonal tiles.

f(n) = 4n − 4 + n − 2 = 5n − 6.

Answer: f(n) = 5n − 6

5
Match f(n) = n² − n to a picture: an n-by-n square with one row of n tiles removed.
Show Answer

n² − n = n(n − 1), an n by (n − 1) rectangle.

Either description is the same count.

Answer: n(n − 1)

Super Challenge

Super Challenge

1
Create a picture for f(n) = n² + n + 1. Describe figure 1 and what is added from figure n to n + 1.
Show Answer

Figure 1 has 3 tiles: a 1-by-1 square, one extra, and a core of 1.

Going to n + 1 adds 2n + 1 tiles (the usual square growth) plus 1 more for the extra side: 2n + 2.

Check: f(n+1) − f(n) = (n+1)² + (n+1) + 1 − n² − n − 1 = 2n + 1 + 1 = 2n + 2.

Answer: core 1, square, extra side

2
A student fits f(n) = 3n to 1, 4, 9. Use second differences to show a linear model is wrong, then write the quadratic.
Show Answer

Differences 3, 5, not constant.

f(n) = n²

Answer: f(n) = n²

3
Why does connecting n² in the formula to an n-by-n block in the picture justify the symbolic rule?
Show Answer

Each figure’s square grows with n in both directions, which is n × n.

If the picture really is that square (plus whatever the other terms name), the formula is counting the same pieces.

Answer: n² counts the square

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