Algebra 1 · Unit 6
From a picture pattern, write a function and connect each part of the formula to the picture.
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Count first
For figure n = 1, 2, 3, count the dots, tiles, or sticks. Put the counts in a table.
Look at the change
If the count goes up by the same amount each time, the pattern is linear: f(n) = mn + b.
Connect parts to the picture
The slope is the new pieces added each figure. The intercept is what is there when n = 0, or a fixed core that never changes.
Check figure 4
Use the formula to predict, then count if you can.
A constant increase. The picture is in the question.
Each new figure adds 2 dots, so the slope is 2.
Figure 1 has 2 dots: f(1) = 2.
f(n) = 2n
The 2 is “two dots per figure.” There is no extra core.
Answer: f(n) = 2n
Adds 2 each time, so m = 2.
f(1) = 3, so 2(1) + b = 3, b = 1.
f(n) = 2n + 1
The +1 is one extra dot that stays in every figure.
Answer: f(n) = 2n + 1
Adds 1 each time.
f(n) = n
Answer: f(n) = n
m = 3 new tiles each figure.
f(1) = 4 = 3(1) + 1
f(n) = 3n + 1
3 is the tiles added. 1 is a tile that is already there in every figure.
Answer: f(n) = 3n + 1
Nothing is added. f(n) = 5
A constant pattern.
Answer: f(n) = 5
m = 4
f(n) = 4n
Answer: f(n) = 4n
Predict figure 5, or a pattern that starts with a core.
m = 4
f(1) = 8 = 4(1) + 4, so b = 4
f(n) = 4n + 4
The +4 is a four-tile core. Each new figure adds 4 around the outside.
f(5) = 20 + 4 = 24
Answer: f(n) = 4n + 4; f(5) = 24
f(n) = 3n + 1
Each new square shares a side, so you add 3 sticks, not 4.
Answer: f(n) = 3n + 1
m = 3
f(n) = 3n − 1
f(10) = 29
Answer: 29
f(1) = 4, which matches.
2n + 2 is correct. Also 2(n + 1).
Answer: f(n) = 2n + 2
f(n) = 5n + 2
The core is 2 tiles that are not part of the +5.
Answer: f(n) = 5n + 2
3n + 1 ≥ 40
n ≥ 13
Figure 13 has 40.
Answer: figure 13
Two possible formulas, or a growing group plus a fixed group.
2n + 3
Or 2(n + 1) + 1 = 2n + 3
The 2n is the rectangle. The +3 is the end cap.
Answer: f(n) = 2n + 3
The first differences are not constant (3, then 5).
1, 4, 9 are squares: f(n) = n².
A linear function cannot fit three terms with changing differences.
Answer: f(n) = n², not linear
f(n) = 2n + 1
Each arm has n squares, plus one corner square counted once.
Answer: f(n) = 2n + 1
A single square needs 4 sticks. Each new square shares one stick and adds 3.
The +1 is the extra stick that started the first square (4 = 3 + 1).
Answer: add 3 sticks each square; +1 from the first square
A is linear. B doubles, so exponential, not 2n + 1.
Answer: only A
f(n) = 4n + 2
4 is the new column. 2 is a starter pair that is always there.
Answer: f(n) = 4n + 2
Figure 1 has 5 pieces: a 2-piece core plus 3 new.
Each later figure adds 3 pieces (one group of three).
Answer: core 2, add 3 each time
f(1) = 2, but the picture has 3.
m = 2 still, so f(n) = 2n + 1.
Answer: f(n) = 2n + 1
Each figure number n gives exactly one count.
n is the figure number, the input. 2n is the count of dots.
Answer: n is the figure number
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