Adding the same amount each step is linear. Multiplying by the same factor is exponential. The SAT hides the difference inside one word.
Which equation represents this situation?
A gym opened with 240 members. The number of members increases by 15% each year. Which equation gives the number of members, M, in terms of the number of years after opening, t?
A percent change is a repeated multiplication: keep 100% and add 15%, so multiply by 1.15 every year and M = 240(1.15)t. After one year that is 240 × 1.15 = 276, a gain of 36 members. A adds 15 members a year — the linear reading of the same sentence, and the trap this lesson is about. B uses 0.15, the part that is added rather than the whole multiplier, which would shrink the gym to 36 members in a year. D uses 15 instead of 1.15, multiplying the membership fifteenfold each year.
1
One adds the same amount; the other multiplies by the same factor.
Linear means a constant difference. Exponential means a constant ratio. Everything else in this lesson follows from those two sentences.
x
A(x) — add 32
B(x) — multiply by 1.5
0
64
64
1
96
96
2
128
144
3
160
216
4
192
324
Both columns start at 64 and both reach 96 after one step, because 64 + 32 and 64 × 1.5 happen to agree there. One pair of values never decides the model; from the second step on these two part company for good.
2
Given a table, run both tests.
Subtract down the column first. If the differences are not equal, divide down the column instead.
A: 96 − 64 = 32, 128 − 96 = 32, 160 − 128 = 32
same difference every time → linear
A(x) = 64 + 32x
the start, then the difference as the slope
B: 96 − 64 = 32, 144 − 96 = 48, 216 − 144 = 72
differences grow, so try dividing
B: 9664 = 1.5, 14496 = 1.5, 216144 = 1.5
same ratio every time → exponential
B(x) = 64(1.5)x
the start, then the ratio as the base
constant difference → y = a + dxconstant ratio → y = a(b)xa is the value at x = 0
Only equal steps in x may be compared
Both tests assume x rises by the same amount each row. If a table jumps from x = 1 to x = 3, that row covers two steps: a linear model adds the difference twice, an exponential model multiplies by the base twice.
3
“By 40 each year” and “by 40% each year” are different worlds.
A store sold 250 lamps in its first year, x = 0. Here are the two versions of the next sentence.
up 40 lamps a year
L(x) = 250 + 40x
the same 40 lamps forever
up 40% a year
E(x) = 250(1.4)x
40% of a number that keeps growing
100% + 40% = 140% = 1.4
keep all of it, then add 40% more
L(3) = 250 + 40(3) = 250 + 120 = 370
three equal additions
1.43 = 1.4 × 1.4 × 1.4 = 2.744
three multiplications, not three additions
E(3) = 250 × 2.744 = 686
or 250 → 350 → 490 → 686 the long way
370 lamps vs 686 lamps — the sentences differ by one percent sign
The multiplier is 1.4, not 0.4 and not 40
0.4 is only the part that gets added, so 250(0.4)x would model a 60% drop each year. 40 is the percent number with the decimal point left in the wrong place, and 250(40)x reaches 16 million lamps in three years. When a stem says “increases by r percent”, write 1 + r100 before anything else.
4
The base alone tells you growth or decay.
In y = a(b)x with a positive a: b > 1 grows, 0 < b < 1 decays, b = 1 never moves.
1.05 → up 5%1.4 → up 40%0.92 → down 8%0.75 → down 25%0.5 → halved
For decay the multiplier is what you keep: a 25% loss leaves 75%, so b = 1 − 0.25 = 0.75.
V(x) = 1200(0.75)x
a $1,200 laptop losing 25% of its value a year
0.753 = 0.421875
three years of keeping three quarters
V(3) = 1200 × 0.421875 = $506.25
1200 → 900 → 675 → 506.25, the same trip
Decay never reaches 0 and growth never turns around. A quantity that falls and then rises is neither — that shape belongs to a quadratic.
5
Exponential is patient: it trails for years, then wins.
Channel A starts at 100 thousand subscribers and gains 20 thousand a year. Channel B starts at 20 thousand and grows 40% a year.
x = 4: A = 180, B = 20(1.4)4 ≈ 76.8
the line is still more than double the curve
x = 7: A = 240, B ≈ 210.8
closing fast, but the line still leads
x = 8: A = 260, B ≈ 295.2
overtaken, and the gap only widens
Where the calculator earns its keep
Type y = 100 + 20x and y = 20(1.4)^x into Desmos and click where the graphs meet: the intersection is reported at about (7.51, 250.2). Reading an overtaking point off the graph is far quicker than testing years one at a time, and you can drag the window out to see how far ahead the curve is by year 12.
6
Say what each number means in the situation's own words.
Interpretation items hand you the model and ask what one piece of it represents.
B(x) = 20(1.4)x
Linear a + dx
Exponential a(b)x
a
value at x = 0
value at x = 0
the other number
d is added each step
b multiplies each step
in words
“increases by d per year”
“increases by (b − 1) × 100% per year”
graph
straight line
a curve that steepens or flattens
So 20 means “the channel had 20 thousand subscribers at launch”, and 1.4 means “the number of subscribers increases by 40% each year”. The base is not the increase itself — subtract 1 first, then name the percent.
Choices that swap the two roles
“The number of subscribers increases by 1.4 thousand each year” and “… increases by 140% each year” are both built from the right number and the wrong meaning. A base of 1.4 is a 40% increase, applied once per year.
✓
Check: does the model behave like the sentence?
the gym240 × 1.15 = 276up 36 in year 1, more later ✓
the lamps250 → 350 → 490 → 686×1.4 three times ✓
the laptop1200 → 900 → 675 → 506.25each year keeps 75% ✓
Remember the flow
Difference or ratio · Multiplier = 1 ± r · Exponent counts the periods
A percent in the stem becomes a base, never a slope.
Now you try
Practice problem
Which function represents this situation?
A cell culture contains 8,000 cells. The number of cells decreases by 30% every 4 hours. Which function gives the number of cells, C, remaining h hours later?
Two decisions. The multiplier is what remains, 1 − 0.30 = 0.7, and it is applied once every 4 hours, so the exponent is h4:C(h) = 8,000(0.7)h/4. Check at h = 4: 8,000 × 0.7 = 5,600 cells, exactly a 30% loss. A uses 0.3, the part that disappears, leaving 2,400 after 4 hours instead of 5,600. C multiplies by 4 instead of dividing, so it applies 16 decays in the first 4 hours and all but wipes the culture out. D decays every hour and reaches 8,000(0.7)4 ≈ 1,920 after 4 hours.
Hint: ask the two questions in order — what is the multiplier, and how many times is it applied? The percent answers the first; the period named in the sentence answers the second.
