Digital SAT · Math · Unit 2 · Advanced Math

Lesson 2.4 · Exponential Functions

On the test: 2–4 questions every test — almost always a percent-growth or doubling story

Big idea. Linear functions ADD the same amount each step; exponential functions MULTIPLY by the same factor each step. Every model is f(x) = a·b^x: a is where you start, b is what you multiply by each step. Growing by r% means b = 1 + r; shrinking by r% means b = 1 − r.

🎛 Move it · build the curve

Slide the start, the factor and the shift. Then press ‘compare’ to see why multiplying beats adding.

f (x) = 2 (1.5)^{x}

start a2factor b1.5shift k0

b = 1.5 = 1 + 0.50 → GROWS 50% each step

y-intercept = a + k = 2. The curve hugs y = 0 (the asymptote) but never touches it.

🧑‍🏫 Rule 1 · Percent change → growth factorstep 1 of 5

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📋 The question · SAT · June 2023 · Math 800 Book Q6
Population 1,340 in 2009, increasing 3% per year. Which equation for $t$ years after 2009?
1. A) $N = 0.03 (1, 340)^{t}$
2. B) $N = 1, 340 (0.03)^{t}$
3. C) $N = 1, 340 (1.03)^{t}$
4. D) $N = 1.03 (1, 340)^{t}$
the question
$1340, + 3% each year$
June 2023: a population of 1,340 grows by 3% of the previous year, every year.

🧑‍🏫 Rule 2 · Factor → percent change (the reverse)step 1 of 4

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📋 The question · SAT · March 2024 · Math 800 Book Q19
$f (x) = 42 (0.24)^{x}$ . For any positive integer $n$ , $f (n)$ is $p %$ less than $f (n - 1)$ . What is $p$ ?
1. A)76
2. B)58
3. C)42
4. D)24
the question
$f (x) = 42 (0.24)^{x}, f (n) is p % less than f (n - 1)$
March 2024.

🧑‍🏫 Rule 3 · Find a and b from datastep 1 of 4

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📋 The question · SAT · May US 2024 · Math 800 Book Q33
Table $h (0) = 1.46$ , $h (2) = 2.32$ , $h (4) = 3.68$ (exponential). Best function?
1. A) $h (x) = 1.26 (0.46)^{x}$
2. B) $h (x) = 1.26 (1.46)^{x}$
3. C) $h (x) = 1.46 (0.26)^{x}$
4. D) $h (x) = 1.46 (1.26)^{x}$
a = the value at 0
$h (0) = 1.46$
May US 2024 table: at $x = 0$ the factor part is $b^{0} = 1$ , so $h (0) = a$ .

Trap · the leftover percent

0.24

means “keep 24%”, not “lose 24%”. Always ask: is this the part that STAYS?

🧑‍🏫 Rule 4 · “Every k units” → exponent t/kstep 1 of 4

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📋 The question · SAT · March US 2023 · Math 800 Book Q3
Ulmus trees doubled every 664 years; $n$ at the start. Which gives the number $x$ years later?
1. A) $n (2)^{664 x}$
2. B) $n (2)^{\frac{664}{x}}$
3. C) $n (2)^{\frac{x}{664}}$
4. D) $n (2)^{664 + x}$
the question
$doubles every 664 years$
March US 2023: $n$ trees at the start.

🧑‍🏫 Rule 5 · Change per a DIFFERENT stepstep 1 of 5

1
📋 The question · SAT · June 2024 · Math 800 Book Q42
$f (x) = 24 (1.30)^{x /3}$ increases by $p %$ per increase of 6. What is $p$ ?
1. A)30
2. B)41
3. C)60
4. D)69
the question
$f (x) = 24 (1.30)^{x /3}, per 6$
June 2024: by what percent per increase of 6?

Real question 1SAT · March US 2023 · Math 800 Book Q3

The number of Ulmus trees doubled every 664 years. There were $n$ trees in the earliest known sample. Which expression gives the number of trees $x$ years later?

🧑‍🏫 Rule 6 · y-intercept and asymptotestep 1 of 4

1
y-intercept
$f (0) = a \cdot b^{0} + k = a + k$
Anything to the power 0 is 1.

🧑‍🏫 Rule 7 · Shiftsstep 1 of 4

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📋 The question · SAT · March US 2023 · Math 800 Book Q1
$figure from the test$
The graph of $y = f (x) - 1$ is shown. Which equation could define $f$ ?
1. A) $f (x) = 2^{x}$
2. B) $f (x) = 2^{x} - 1$
3. C) $f (x) = 2^{x} + 1$
4. D) $f (x) = 2^{x} + 2$
graph of f − 1
$f - 1 = 2^{x} + 1 \Rightarrow f = 2^{x} + 2$
March US 2023: the graph shows $f (x) - 1$ with intercept 2 and asymptote 1. Add 1 back.

Real question 2SAT · March 2024 · Math 800 Book Q21

$figure from the test$

The graph of $y = 2^{x} + k$ is shown, where $k$ is a constant. What is the value of $k$ ?

🧑‍🏫 Rule 8 · Interpreting f(input) = outputstep 1 of 4

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📋 The question · SAT · November 2023 · Math 800 Book Q16
$f (t) = 6000 (1.01)^{2 t}$ . Best interpretation of " $f (5) \approx 6627.73$ "?
1. A)5 years after the initial deposit, the amount of money in the account is 6,627.73
2. B)Every 5 years it increases by 6,627.73
3. C)5 years after, it has increased by 6,627.73
4. D)Every 5 years it decreases by 6,627.73
the question
$f (t) = 6000 (1.01)^{2 t}, f (5) \approx 6627.73$
November 2023.

🧑‍🏫 Rule 9 · Which form displays a value?step 1 of 4

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📋 The question · SAT · August US 2024 · Math 800 Book Q47
For the exponential $f$ , $f (1) = k$ . Which form shows $k$ as the coefficient or base?
1. A) $f (x) = 50 (1.4)^{x + 1}$
2. B) $f (x) = 70 (1.4)^{x}$
3. C) $f (x) = 98 (1.4)^{x - 1}$
4. D) $f (x) = 137.2 (1.4)^{x - 2}$
the question
$f (1) = k$
August US 2024: which form shows $k$ as the coefficient or base?

Desmos move · regression

With a table, type it into Desmos and fit y₁ ~ a·b^x₁. Desmos gives

a

and

b

; match the options.

🧑‍🏫 Rule · Interest compounded n times a yearstep 1 of 3

1
the formula
$A = P (1 + \frac{r}{n})^{n t}$
Each period adds $\frac{r}{n}$ of the balance; there are $n t$ periods in $t$ years.

Take with you

▸ $f (x) = a \cdot b^{x}$ : $a$ = start, $b$ = factor per step.
▸+r% → $b = 1 + r$ ; −r% → $b = 1 - r$ . $b = 0.24$ means 76% lost each step.
▸“Every $k$ units” → exponent $\frac{t}{k}$ ; per a different step, raise $b$ to the new exponent change.
▸ $a \cdot b^{x} + k$ : $y$ -intercept $a + k$ , asymptote $y = k$ .
▸ $f (x - h)$ = shift right; tidy $b^{x - h} = b^{- h} b^{x}$ .
▸The coefficient equals $f$ at the input that makes the exponent 0.

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