๐Ÿƒ Flashcards ยท Topic 2.2

Change in Linear &
Exponential Functions

10 cards across 2 print sheets. Print double-sided, cut along dashed lines.

๐Ÿ“‹ Notes ๐Ÿƒ Flashcards ๐Ÿง  Quiz
Print double-sided (flip on long edge) โ†’ cut along dashed lines.
10 cards ยท 2 print sheets
๐Ÿ“„ Page 1 โ€” Questions FRONT ยท Sheet 1/2
1/10
Topic 2.2 ยท Linear Test
How do you identify a LINEAR function from a table?
Equal ฮ”x intervals
Think: what stays constant?
2/10
Topic 2.2 ยท Exponential Test
How do you identify an EXPONENTIAL function from a table?
Equal ฮ”x intervals
Think: what stays constant?
3/10
Topic 2.2 ยท Key Distinction
How does a linear function change vs. an exponential function?
Linear: f(x) = b + mx Expo: f(x) = aยทbหฃ
Addition vs multiplication
4/10
Topic 2.2 ยท Sequences โ†” Functions
Arithmetic sequence โ†” which function type? Geometric sequence โ†” which function type?
Both have parallel formulas
5/10
Topic 2.2 ยท Point-Slope โ€” Linear
Write the point-slope form for an arithmetic sequence / linear function given a known point (k, aโ‚–).
Like slope-intercept but anchored at k
6/10
Topic 2.2 ยท Point-Slope โ€” Exponential
Write the point-slope form for a geometric sequence / exponential function given a known point (k, aโ‚–).
Like standard form but anchored at k
7/10
Topic 2.2 ยท Finding r (Example 2)
P(3)=43 and P(6)=140 model a geometric sequence. Write the equation for r.
P(n) = 43ยทr^(nโˆ’3)
Substitute n=6, then solve for r
8/10
Topic 2.2 ยท Finding m (Example 3)
Theater: row 5 has 31 seats, row 11 has 49 seats. Find the common difference m.
s(n) = 31 + mยท(nโˆ’5)
Substitute n=11, solve for m
๐Ÿ“„ Page 2 โ€” Answers BACK ยท columns swapped
2/10
โœ“ Exponential Test
Constant RATIO between outputs
Divide consecutive outputs. If ratio is always the same โ†’ Exponential.
Change is based on multiplication.
1/10
โœ“ Linear Test
Constant DIFFERENCE between outputs
Subtract consecutive outputs. If difference is always the same โ†’ Linear.
Change is based on addition.
4/10
โœ“ Sequences โ†” Functions
Arithmetic โ†” Linear ยท Geometric โ†” Exponential
Arithmetic (add d each term) behaves like linear.
Geometric (multiply by r each term) behaves like exponential.
3/10
โœ“ Key Distinction
Linear: add same amount ยท Exponential: multiply same ratio
Linear: change d based on addition โ†’ constant rate.
Exponential: change r based on multiplication โ†’ proportional change.
6/10
โœ“ Point-Slope โ€” Exponential
aโ‚™ = aโ‚– ยท r^(n โˆ’ k)
Anchor at your known point (k, aโ‚–). The exponent is (nโˆ’k) โ€” the number of steps from k.
Equivalent: f(x) = f(xโ‚€)ยทb^(xโˆ’xโ‚€)
5/10
โœ“ Point-Slope โ€” Linear
aโ‚™ = aโ‚– + dยท(n โˆ’ k)
Anchor at your known point (k, aโ‚–). Add d for each step away from k.
Equivalent: f(x) = f(xโ‚€) + mยท(xโˆ’xโ‚€)
8/10
โœ“ Common Difference m
m = 3
s(11) = 31 + mยท(6) = 49 โ†’ 6m = 18 โ†’ m = 3.
s(25) = 31 + 3ยท(20) = 91 seats.
7/10
โœ“ Finding r (Example 2)
r = (140/43)^(1/3)
43ยทrยณ = 140 โ†’ rยณ = 140/43 โ†’ r = (140/43)^(1/3).
Cube root because the span is 3 days (6โˆ’3=3).
๐Ÿ“„ Page 3 โ€” Questions FRONT ยท Sheet 2/2
9/10
Topic 2.2 ยท Neither
g(x): outputs 0, 1, 4, 9, 16 over equal intervals. Is it linear, exponential, or neither? Why?
Check differences then ratios
10/10
Topic 2.2 ยท Two-Point Rule
If you are given any two points of a function, what can you always write?
Four possible answers
๐Ÿ“„ Page 4 โ€” Answers BACK ยท columns swapped
10/10
โœ“ Two-Point Rule
Linear, exponential, arithmetic, or geometric
Two points are always enough to write a linear function, exponential function, arithmetic sequence, or geometric sequence using point-slope form.
9/10
โœ“ Neither
Neither โ€” differences AND ratios are not constant
Differences: 1, 3, 5, 7 โ†’ not constant โ†’ not linear.
Ratios: undef, 4, 9/4... โ†’ not constant โ†’ not exponential.
โ†’ Neither.