4.2Analyzing Exponential Graphs
Graph f(x) = ab^(x−h) + k, apply vertical/horizontal shifts, reflections, and stretches, identify the asymptote y = k, and state domain and range.
Graphing exponential functions and recognizing their transformations is essential for modeling real-world data. The horizontal asymptote represents a limiting value — a concept central to limits in calculus.
Essential Question
How do transformations of f(x) = b^x change the graph's position, shape, and asymptote — and what stays the same no matter what transformation is applied?
Lesson Overview
Every exponential graph f(x) = ab^(x−h) + k is a transformation of the parent function y = b^x. Understanding these transformations lets you sketch any exponential function quickly by identifying shifts, reflections, and stretches — and by tracking the horizontal asymptote.
Parent Function Properties
- Domain: all real numbers (−∞, ∞)
- Range: (0, ∞) — always positive
- Y-intercept: (0, 1)
- Asymptote: y = 0 (x-axis)
- One-to-one: yes (passes horizontal line test)
General Form
f(x) = a · b^(x − h) + k
- a: vertical stretch/compress; if a < 0, reflects over x-axis
- h: horizontal shift (right if h > 0)
- k: vertical shift; new asymptote is y = k
Parent functions 2ˣ (growth) and (½)ˣ (decay) — mirror images across the y-axis
Summary of transformations and their effect on the asymptote
Worked Examples
Graph f(x) = 2^x + 3. State the asymptote, domain, range, and y-intercept.
Start with parent y = 2^x. The "+3" shifts the entire graph UP 3 units.
Asymptote: y = 0 shifts to y = 3.
Y-intercept: f(0) = 2⁰ + 3 = 1 + 3 = 4. So y-int = (0, 4).
Domain: all real numbers (−∞, ∞).
Range: (3, ∞) — the graph stays above y = 3.
Key points: f(−1) = 0.5+3 = 3.5; f(1) = 2+3 = 5; f(2) = 4+3 = 7.
Graph g(x) = 2^(x − 2). State the asymptote, domain, range, and y-intercept.
The "(x − 2)" shifts the graph RIGHT 2 units.
Asymptote: y = 0 (horizontal shift does not move the asymptote).
Y-intercept: g(0) = 2^(0−2) = 2^(−2) = 1/4. So y-int = (0, 1/4).
Domain: (−∞, ∞). Range: (0, ∞).
Key points: g(2) = 2⁰ = 1; g(3) = 2¹ = 2; g(4) = 2² = 4.
The graph passes through (2, 1) instead of (0, 1) — shifted right by 2.
Graph h(x) = −3^x. State the asymptote, domain, range, and y-intercept.
The negative sign reflects the parent y = 3^x over the x-axis.
Asymptote: y = 0 (reflection does not move the asymptote).
Y-intercept: h(0) = −3⁰ = −1. So y-int = (0, −1).
Domain: (−∞, ∞). Range: (−∞, 0) — all outputs are negative.
Key points: h(1) = −3; h(2) = −9; h(−1) = −1/3.
The graph is a decay curve below the x-axis (approaches 0 from below as x → −∞).
Describe all transformations of f(x) = −2·(3)^(x+1) − 4 and state the asymptote.
Parent: y = 3^x.
a = −2: vertical stretch by factor 2, then reflect over x-axis.
(x + 1): horizontal shift LEFT 1 unit.
− 4: vertical shift DOWN 4 units.
New asymptote: y = −4 (the vertical shift moves the asymptote).
Y-intercept: f(0) = −2·3^(0+1) − 4 = −2·3 − 4 = −6 − 4 = −10.
Write the equation of an exponential function with base 2, shifted right 3, up 5, and reflected over the x-axis.
Start with parent: y = 2^x.
Shift right 3: replace x with (x − 3) → y = 2^(x−3).
Reflect over x-axis: multiply by −1 → y = −2^(x−3).
Shift up 5: add 5 → y = −2^(x−3) + 5.
Asymptote: y = 5 (from the vertical shift of +5).
Verify y-intercept: f(0) = −2^(0−3) + 5 = −1/8 + 5 = 4.875.
Guided Practice
For f(x) = 3^x − 2, state the asymptote, y-intercept, domain, and range.
Hint: The '−2' shifts the graph down. The asymptote moves with the vertical shift.
For g(x) = 2^(x+4), state the asymptote, y-intercept, domain, and range.
Hint: (x+4) shifts left 4. The asymptote is unchanged. Find g(0) for the y-intercept.
Describe all transformations of h(x) = 4·(2)^(x−1) + 3.
Hint: Identify a, h, and k from the general form f(x) = a·b^(x−h) + k.
Write the equation of an exponential function with base 3, shifted left 2 and up 1.
Hint: Left shift: replace x with (x+2). Up shift: add 1 outside the exponential.
For f(x) = −(1/2)^x + 6, state the asymptote and range.
Hint: The reflection flips the range. The +6 shifts the asymptote. Range is below the asymptote.
Key Vocabulary
Parent Function
The simplest form of a function family. For exponential functions, the parent is f(x) = b^x with a = 1, h = 0, k = 0.
Horizontal Asymptote
A horizontal line y = k that the graph approaches but never crosses. Determined by the vertical shift k in f(x) = ab^(x−h) + k.
Vertical Shift
Moving the graph up or down by adding/subtracting a constant k outside the exponential. Moves the asymptote to y = k.
Horizontal Shift
Moving the graph left or right by replacing x with (x − h). Does NOT change the asymptote.
Reflection
Flipping the graph over an axis. Multiplying by −1 outside reflects over the x-axis; replacing x with −x reflects over the y-axis.
Vertical Stretch/Compress
Multiplying the function by |a| > 1 stretches it; 0 < |a| < 1 compresses it. Does not change the asymptote.
Domain
For all exponential functions (with any transformation), the domain is all real numbers: (−∞, ∞).
Range
For f(x) = ab^(x−h) + k: if a > 0, range is (k, ∞); if a < 0, range is (−∞, k).
Check Your Understanding
Interactive Practice — 5 Questions
What is the horizontal asymptote of f(x) = 2^x − 5?
Which transformation moves the graph of y = 3^x to the right 4 units?
For f(x) = −2^x + 1, what is the range?
What is the y-intercept of g(x) = 3·2^(x−2)?
Which function has asymptote y = 7?
Independent Practice
Independent Practice
For f(x) = 2^x + 4, state the asymptote, y-intercept, domain, and range.
For g(x) = 3^(x−1), state the asymptote, y-intercept, domain, and range.
Describe all transformations of h(x) = −5·(2)^(x+3) − 1 and state the asymptote.
Write the equation of an exponential function with base 4, shifted down 3 and right 2.
For f(x) = −(3)^x + 2, state the asymptote, range, and y-intercept.
Vertical shift: 2ˣ vs 2ˣ + 3 (asymptote moves to y = 3)
Horizontal shift: 2ˣ vs 2^(x−2) (asymptote stays at y = 0)
Reflection: 2ˣ vs −2ˣ (range flips to (−∞, 0))
Common Mistakes
Thinking f(x) = 2^(x+3) shifts the graph RIGHT 3.
f(x+3) shifts LEFT 3. To shift right, use f(x−3) = 2^(x−3). The sign inside is counterintuitive.
Saying the asymptote of f(x) = 3^(x−5) is y = −5 (confusing horizontal shift with vertical shift).
Horizontal shifts do NOT move the asymptote. Only vertical shifts (the +k outside) change the asymptote.
Writing the range as (0, ∞) for f(x) = −2^x + 1.
When a < 0 (reflected), the range is below the asymptote: (−∞, 1). Always check the sign of a first.
Forgetting that the domain of every exponential function is all real numbers.
No matter what transformations are applied, the domain of f(x) = ab^(x−h) + k is always (−∞, ∞).
Math Tips
To find the asymptote quickly: look for the constant added outside the exponential. That constant is k, and the asymptote is y = k.
The y-intercept is always f(0) — substitute x = 0 and simplify. Don't assume it's (0, a) after a horizontal shift.
Range rule: if a > 0, range is (k, ∞). If a < 0 (reflected), range is (−∞, k).
Horizontal shifts are counterintuitive: (x − h) shifts RIGHT, (x + h) shifts LEFT.
To sketch any exponential: (1) find asymptote, (2) plot y-intercept, (3) plot one more point, (4) draw smooth curve approaching asymptote.