SAT Math – Advanced Math
Exponential Graphs
Understanding growth, decay, and transformations of exponential functions
Exponential graphs represent functions of the form \(f(x) = a \cdot b^x\) where the variable is in the exponent. On the SAT, you'll identify growth versus decay, find y-intercepts and horizontal asymptotes, analyze transformations, compare rates of change, and interpret graphs modeling population growth, compound interest, and radioactive decay.
Success requires understanding how base b determines growth or decay, recognizing that exponential functions never reach zero (horizontal asymptote), identifying transformations from equation parameters, and comparing exponential versus linear growth rates. These graphs aren't just mathematical curves—they model bacterial populations, investment returns, viral spread, carbon dating, and any process involving repeated multiplication or percentage change.
Understanding Exponential Function Features
Standard Exponential Form
The general form \(f(x) = a \cdot b^x\) reveals key information about the graph.
b (base): Growth factor (if b > 1) or decay factor (if 0 < b < 1)
Growth: \(b > 1\) → function increases as x increases
Decay: \(0 < b < 1\) → function decreases as x increases
Domain: All real numbers; Range: \(y > 0\) (if a > 0)
Horizontal Asymptote
Exponential functions approach but never reach a horizontal line.
Transformed: \(f(x) = a \cdot b^x + k\) has asymptote y = k
Growth graphs: Approach asymptote as \(x \to -\infty\)
Decay graphs: Approach asymptote as \(x \to +\infty\)
Never crosses: Function never equals asymptote value
Y-Intercept
Where the graph crosses the y-axis (when x = 0).
Y-intercept: Always (0, a)
No x-intercept: Exponential graphs never cross x-axis
Why: \(b^x > 0\) for all x when b > 0
Growth vs. Decay Recognition
Visual and algebraic ways to distinguish growth from decay.
Decay (0 < b < 1): Falls to the right, decreases toward asymptote
Visual test: Which direction does graph rise?
Algebraic test: Is base greater than 1 or between 0 and 1?
Essential Forms and Properties
Standard Exponential Function
\(f(x) = a \cdot b^x\)
a = initial value (y-intercept)
b = base (growth/decay factor)
x = exponent (input variable)
Percentage Growth/Decay Form
Growth: \(f(x) = a(1 + r)^x\) where r is growth rate
Decay: \(f(x) = a(1 - r)^x\) where r is decay rate
Example: 5% growth → \(b = 1.05\)
Example: 15% decay → \(b = 0.85\)
Transformations
\(f(x) = a \cdot b^{x-h} + k\)
• Horizontal shift: h units right (if h > 0)
• Vertical shift: k units up (if k > 0)
• Horizontal asymptote becomes y = k
• Reflection: negative a flips over x-axis
Key Properties
Domain: All real numbers \((-\infty, \infty)\)
Range: \((0, \infty)\) for standard form (a > 0)
Y-intercept: (0, a)
No x-intercepts: Never crosses x-axis
Common Pitfalls & Expert Tips
❌ Confusing growth rate with growth factor
8% growth means multiply by 1.08 (not 0.08). Growth factor = 1 + rate. Don't forget the 1!
❌ Thinking exponential graphs cross x-axis
Exponential functions (with positive a and b) never equal zero. They approach the asymptote but never reach it!
❌ Mixing up domain and range
Domain is ALL x-values (unlimited). Range is limited by horizontal asymptote. Don't reverse these!
❌ Misidentifying base from percentage
Decreases by 25% means multiply by 0.75 (not 0.25). Remaining amount = 1 - 0.25 = 0.75.
✓ Expert Tip: Check direction for growth/decay
If graph rises to the right, it's growth (b > 1). If falls to the right, it's decay (0 < b < 1). Simple visual test!
✓ Expert Tip: Y-intercept is coefficient a
In \(f(x) = a \cdot b^x\), plug in x = 0 to get f(0) = a. The y-intercept is always (0, a) immediately!
✓ Expert Tip: Exponential eventually dominates linear
Exponential growth always overtakes linear growth given enough time. Test by comparing values at large x!
Fully Worked SAT-Style Examples
Which of the following represents exponential decay?
A) \(f(x) = 3(1.2)^x\)
B) \(g(x) = 5(0.8)^x\)
C) \(h(x) = 2x + 7\)
D) \(j(x) = x^2 + 3\)
Solution:
Check each option:
A) Base = 1.2 > 1 → Growth
B) Base = 0.8 (between 0 and 1) → Decay ✓
C) Linear function, not exponential
D) Quadratic function, not exponential
Key insight:
Decay requires exponential form with 0 < b < 1
0.8 = 80% means decreasing by 20% each time
Answer: B
What is the y-intercept of \(f(x) = 7(2)^x\)?
Solution:
Y-intercept when x = 0:
\(f(0) = 7(2)^0 = 7 \times 1 = 7\)
Shortcut:
In \(f(x) = a \cdot b^x\), y-intercept is always a
Here a = 7, so y-intercept is (0, 7)
Answer: (0, 7) or y = 7
What is the horizontal asymptote of \(g(x) = 4(0.5)^x + 3\)?
Solution:
Standard form with vertical shift:
\(f(x) = a \cdot b^x + k\)
Here k = 3
Horizontal asymptote:
The +3 shifts graph up 3 units
Asymptote moves from y = 0 to y = 3
Behavior:
This is decay (b = 0.5 < 1)
As \(x \to \infty\), function approaches 3 from above
Answer: y = 3
What is the range of \(f(x) = 2(3)^x\)?
Solution:
Identify key features:
Coefficient a = 2 > 0
Base b = 3 > 1 (growth)
No vertical shift (k = 0)
Determine range:
Horizontal asymptote: y = 0
Function approaches 0 but never reaches it
Since growth, extends upward to infinity
Answer: \((0, \infty)\) or \(y > 0\)
Which function grows faster: \(f(x) = 100 + 10x\) (linear) or \(g(x) = 2(1.5)^x\) (exponential)?
Solution:
Test at several x-values:
At x = 0: f(0) = 100, g(0) = 2
At x = 10: f(10) = 200, g(10) = 2(1.5)^{10} ≈ 115
At x = 20: f(20) = 300, g(20) = 2(1.5)^{20} ≈ 6{,}536
Key principle:
Exponential growth eventually overtakes linear
Linear may start higher, but exponential wins long-term
Answer: Exponential \(g(x)\) grows faster eventually
A quantity increases by 12% each year. If it starts at 50, write the exponential function.
Solution:
Identify components:
Initial value: a = 50
Growth rate: r = 0.12 (12% as decimal)
Calculate growth factor:
Growth factor = 1 + r = 1 + 0.12 = 1.12
Write function:
\(f(x) = 50(1.12)^x\)
Answer: \(f(x) = 50(1.12)^x\)
An exponential graph passes through (0, 3) and (1, 12). What is the function?
Solution:
Use y-intercept:
Point (0, 3) means a = 3
So far: \(f(x) = 3 \cdot b^x\)
Use second point to find b:
Point (1, 12): \(12 = 3 \cdot b^1\)
\(12 = 3b\)
\(b = 4\)
Answer: \(f(x) = 3(4)^x\)
Quick Recognition Guide
Growth Indicators
• Base b > 1
• Graph rises to the right
• Increases without bound
• Example: \(2^x\), \(1.5^x\)
Decay Indicators
• Base 0 < b < 1
• Graph falls to the right
• Approaches asymptote
• Example: \(0.5^x\), \(0.9^x\)
Exponential Graphs: Understanding Rapid Change
Exponential graphs capture the mathematics of compounding change—processes where quantities multiply repeatedly rather than adding incrementally. The SAT tests these functions because they model phenomena central to modern life: bacterial populations doubling every hour, investments compounding annually, viral videos spreading geometrically, radioactive isotopes halving predictably, and pandemics accelerating exponentially before interventions flatten curves. Understanding that \(f(x) = a \cdot b^x\) creates fundamentally different behavior than linear \(f(x) = mx + b\) distinguishes students who grasp mathematical structure from those memorizing formulas—exponential growth accelerates relentlessly while linear growth plods steadily, explaining why small percentage differences compound into enormous long-term disparities and why early intervention matters immensely in exponential processes. The horizontal asymptote represents a fundamental limit: exponential decay approaches but never reaches zero (half-lives continue forever, investments never completely vanish), while exponential growth has no upper bound unless external factors intervene. Recognizing growth versus decay from the base—greater than one means multiplying by increasing factors, between zero and one means multiplying by shrinking factors—connects algebraic parameters to graphical behavior instantly. The y-intercept reveals initial conditions (starting population, principal investment, initial dose), while the base encodes the rate of change (1.05 means 5% growth, 0.92 means 8% decay). Master the counterintuitive nature of exponential growth: doubling seems harmless initially but produces astronomical values surprisingly quickly, explaining compound interest magic, population explosion warnings, and why technological advances accelerate. These graphs transcend academic exercises, equipping you to evaluate retirement savings projections, understand epidemiological models, question claims about resource depletion or economic growth sustainability, and recognize when linear thinking fails for exponential realities—skills essential for navigating a world where compounding effects dominate from climate change to technological disruption.