Exponential Functions
1️⃣ Understanding Exponential Functions
An exponential function is a function where the variable appears as an exponent. It is written in the form:
\[ f(x)=a\cdot b^x \]
- ✔ \( a \) is the initial value (the value when \( x=0 \)).
- ✔ \( b \) is the base (the constant growth or decay factor).
- ✔ \( x \) is the exponent.
📌 If \( b>1 \), the function represents exponential growth (values increase).
📌 If \( 0
Examples:
- Growth: \( f(x)=3\cdot2^x \) – values double each time.
- Decay: \( g(x)=100\cdot(0.5)^x \) – values halve each time.
Not Exponential: \( f(x)=x^2 \) (quadratic) and \( f(x)=3x+5 \) (linear).
2️⃣ Key Features of Exponential Functions
- Initial Value \( a \): The starting value when \( x=0 \).
- Growth Factor \( b \): Determines how fast the function grows or decays.
- Horizontal Asymptote: The graph approaches \( y=0 \) but never touches it.
- Domain and Range:
- Domain: \( (-\infty, \infty) \).
- Range: For growth functions, \( (0,\infty) \).
Example: For \( f(x)=5\cdot3^x \), the initial value is 5, the growth factor is 3, and the horizontal asymptote is \( y=0 \).
3️⃣ Exponential Growth and Decay
Exponential functions model growth and decay as follows:
- Exponential Growth: \( f(x)=a\cdot (1+r)^x \)
- Exponential Decay: \( f(x)=a\cdot (1-r)^x \)
Growth Example: A population of 1000 growing at 8% per year: \[ P(t)=1000(1.08)^t \] After 5 years, \( P(5)\approx1469 \).
Decay Example: A radioactive substance decaying at 10% per year from 500g: \[ A(t)=500(0.9)^t \] After 3 years, \( A(3)\approx364.5 \) g.
4️⃣ Graphing Exponential Functions
The graph of an exponential function typically has:
- A horizontal asymptote at \( y=0 \).
- A curve that rises rapidly (growth) or falls rapidly (decay).
- A y-intercept at \( (0,a) \).
Example: Graph \( f(x)=2^x \) using the table below:
| \( x \) | \( f(x)=2^x \) |
|---|---|
| -2 | 0.25 |
| -1 | 0.5 |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
The graph rises quickly for \( x>0 \) and never goes negative.
5️⃣ Real-Life Applications of Exponential Functions
- Population Growth: A country's population growing at 2% per year.
- Radioactive Decay: The half-life of substances like carbon-14.
- Bank Interest: Compound interest follows an exponential model.
- Virus Spread: Infections can increase exponentially.
Compound Interest Example:
\[ A = P\left(1+\frac{r}{n}\right)^{nt} \]
For a \$1000 investment at 5% annual interest compounded monthly for 10 years,
A = 1000\(\left(1+\frac{0.05}{12}\right)^{12\cdot10}\)
which is approximately \$1647.
🎯 Practice Questions – Fundamental
- Identify whether \( f(x)=3^x \), \( g(x)=(0.7)^x \), and \( h(x)=5\cdot2^x \) represent exponential growth or decay.
- Find the y-intercept and horizontal asymptote of \( f(x)=4\cdot3^x \).
- Solve for \( x \) in \( 2^x=16 \).
- A bacteria culture starts with 500 cells and doubles every 3 hours. Find the population after 9 hours.
- A car depreciates by 15% per year. If its initial value is \$20,000, what is its value after 5 years?
- Graph \( f(x)=5(0.8)^x \).
- Calculate the compound interest on a \$2000 investment at 6% annual interest for 8 years, compounded quarterly.
- If \( f(x)=10\cdot2^x \), find \( f(3) \).
- The half-life of a drug is 6 hours. If 200 mg is taken, how much remains after 18 hours?
- Solve: \( 3^{x+1}=81 \).
🔥 Challenging Problems – Push Your Limits!
- Solve for \( x \) without logarithms: \( 5^x=125 \).
- The population of a city follows \( P(t)=5000(1.04)^t \). Find the time when the population doubles.
- A radioactive substance decays according to \( A(t)=100(0.88)^t \). Determine its half-life.
- A scientist models virus spread as \( N(t)=10e^{0.3t} \). Find \( N(10) \).
- Find the equation of an exponential function passing through \( (0,3) \) and \( (2,12) \).
7️⃣ Summary
- ✅ Exponential functions are written as \( f(x)=a\cdot b^x \) and represent growth or decay at a constant rate.
- ✅ Key features include the initial value, growth/decay factor, and a horizontal asymptote at \( y=0 \).
- ✅ They model real-life phenomena such as population growth, radioactive decay, and compound interest.
8️⃣ Fantastic Job!
Excellent work! Next up: Logarithmic Functions! 🚀