Exponential Functions
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Domain and Range of Exponential Functions
Standard Form:
\[ f(x) = a \cdot b^x + k \]
where:
• \( a \) = initial value/vertical stretch factor
• \( b \) = base (growth/decay factor), \( b > 0, b \neq 1 \)
• \( x \) = exponent/independent variable
• \( k \) = vertical shift (horizontal asymptote)
Domain:
\[ \text{Domain: } (-\infty, \infty) \text{ or all real numbers} \]
Exponential functions are defined for ALL real numbers
Range:
Depends on the vertical shift (k) and direction:
• If \( a > 0 \): Range is \( (k, \infty) \) or \( y > k \)
• If \( a < 0 \): Range is \( (-\infty, k) \) or \( y < k \)
Key Point:
The horizontal asymptote \( y = k \) determines the boundary of the range
Examples:
\( f(x) = 2^x \)
Domain: \( (-\infty, \infty) \)
Range: \( (0, \infty) \) (horizontal asymptote at y = 0)
\( g(x) = 3^x - 2 \)
Domain: \( (-\infty, \infty) \)
Range: \( (-2, \infty) \) (horizontal asymptote at y = -2)
2. Match Exponential Functions and Graphs
Key Characteristics:
Growth vs Decay:
• Exponential Growth: \( b > 1 \) → graph rises from left to right
• Exponential Decay: \( 0 < b < 1 \) → graph falls from left to right
Y-intercept:
When \( x = 0 \): \( f(0) = a \cdot b^0 + k = a + k \)
The y-intercept is at point \( (0, a + k) \)
Horizontal Asymptote:
The line \( y = k \)
The graph approaches but never touches or crosses this line
Direction:
• If \( a > 0 \): Graph is above the asymptote
• If \( a < 0 \): Graph is below the asymptote (reflected over x-axis)
Common Forms:
| Function | Type | Y-intercept | Asymptote |
|---|---|---|---|
| \( f(x) = 2^x \) | Growth | (0, 1) | y = 0 |
| \( f(x) = (0.5)^x \) | Decay | (0, 1) | y = 0 |
| \( f(x) = 3^x + 2 \) | Growth | (0, 3) | y = 2 |
| \( f(x) = -2^x \) | Reflected growth | (0, -1) | y = 0 |
3. Linear and Exponential Functions Over Unit Intervals
Key Differences:
Linear Functions: \( f(x) = mx + b \)
• Constant rate of change
• Equal differences in consecutive y-values
• When x increases by 1, y increases by m (the slope)
• Graph is a straight line
Exponential Functions: \( f(x) = a \cdot b^x \)
• Constant percent change (common ratio)
• Equal ratios of consecutive y-values
• When x increases by 1, y is multiplied by b
• Graph is a curved line
Comparison Table:
| x | Linear: f(x) = 2x + 1 | Difference | Exponential: g(x) = 2ˣ | Ratio |
|---|---|---|---|---|
| 0 | 1 | — | 1 | — |
| 1 | 3 | +2 | 2 | ×2 |
| 2 | 5 | +2 | 4 | ×2 |
| 3 | 7 | +2 | 8 | ×2 |
| 4 | 9 | +2 | 16 | ×2 |
4. Identify Linear and Exponential Functions
From a Table:
Test for Linear Function:
1. Check if x-values have equal intervals
2. Calculate differences between consecutive y-values
3. If differences are constant → LINEAR
Test for Exponential Function:
1. Check if x-values have equal intervals
2. Calculate ratios of consecutive y-values
3. If ratios are constant → EXPONENTIAL
From an Equation:
Linear:
Variable x is the BASE: \( f(x) = mx + b \)
Exponential:
Variable x is the EXPONENT: \( f(x) = a \cdot b^x \)
From a Graph:
Linear:
Straight line
Exponential:
Curved line (J-shaped or reverse J-shaped)
Has a horizontal asymptote
5. Describe Linear and Exponential Growth and Decay
Exponential Growth:
\[ f(x) = a(1 + r)^t \quad \text{or} \quad f(x) = a \cdot b^x \text{ where } b > 1 \]
Characteristics:
• \( a \) = initial amount
• \( r \) = growth rate (as a decimal)
• \( t \) = time
• Growth factor: \( b = 1 + r > 1 \)
• Function increases as x increases
Example:
A population of 1000 grows at 5% per year:
\( P(t) = 1000(1.05)^t \)
Exponential Decay:
\[ f(x) = a(1 - r)^t \quad \text{or} \quad f(x) = a \cdot b^x \text{ where } 0 < b < 1 \]
Characteristics:
• \( a \) = initial amount
• \( r \) = decay rate (as a decimal)
• \( t \) = time
• Decay factor: \( b = 1 - r \), where \( 0 < b < 1 \)
• Function decreases as x increases
Example:
A car worth $20,000 depreciates 15% per year:
\( V(t) = 20000(0.85)^t \)
Linear Growth and Decay:
\[ f(x) = mx + b \]
Characteristics:
• \( m \) = slope (constant rate of change)
• \( b \) = y-intercept (initial value)
• If \( m > 0 \): Linear growth
• If \( m < 0 \): Linear decay
• Changes by the same amount over equal intervals
Comparison Summary:
| Feature | Linear | Exponential |
|---|---|---|
| Rate of change | Constant | Changing (proportional) |
| Pattern | Add/subtract same amount | Multiply/divide by same factor |
| Graph shape | Straight line | Curved (J-shape) |
| Long-term behavior | Steady increase/decrease | Rapid increase/decrease |
6. Quick Reference Summary
Key Formulas:
Standard Form: \( f(x) = a \cdot b^x + k \)
Domain: Always \( (-\infty, \infty) \)
Range: \( (k, \infty) \) if \( a > 0 \); \( (-\infty, k) \) if \( a < 0 \)
Growth: \( b > 1 \) or \( f(x) = a(1 + r)^t \)
Decay: \( 0 < b < 1 \) or \( f(x) = a(1 - r)^t \)
Horizontal Asymptote: \( y = k \)
Y-intercept: \( (0, a + k) \)
📚 Study Tips
✓ Domain is always all real numbers for exponential functions
✓ Check b value: b > 1 is growth, 0 < b < 1 is decay
✓ Linear has constant differences; exponential has constant ratios
✓ Exponential functions grow/decay much faster than linear
✓ Horizontal asymptote y = k determines the range boundary