MAT137 - Week 5 - Exponential Functions and Rules
Welcome to Week 5 of MAT137! This week, we focused on sections 4.1 - 4.3 of the textbook, dedicated to understanding exponential functions and their governing rules. Let's recap the key concepts we covered.
Key Concepts Covered:
- Definition of Exponential Functions: An exponential function is defined as $f(x) = a^x$, where $a > 0$ and $a \neq 1$. The base, $a$, determines whether the function represents exponential growth or decay.
- Rules of Exponents: We explored the fundamental rules that govern how exponents behave. These rules are essential for simplifying expressions and solving equations involving exponents. Here's a quick reminder:
- Product of Powers: $a^m \cdot a^n = a^{m+n}$
- Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
- Power of a Power: $(a^m)^n = a^{m \cdot n}$
- Power of a Product: $(ab)^n = a^n \cdot b^n$
- Power of a Quotient: $(\frac{a}{b})^n = \frac{a^n}{b^n}$
- Zero Exponent: $a^0 = 1$ (where $a \neq 0$)
- Negative Exponent: $a^{-n} = \frac{1}{a^n}$
Understanding and applying these rules is crucial for success in this chapter. Remember to practice using these rules with various examples to solidify your understanding.
- Exponential Growth and Decay: We discussed how the value of the base, $a$, influences the behavior of the exponential function. If $a > 1$, the function represents exponential growth, meaning the function's value increases rapidly as $x$ increases. If $0 < a < 1$, the function represents exponential decay, meaning the function's value decreases as $x$ increases.
- Transformations of Exponential Functions: We looked at how transformations such as shifts, stretches, and reflections affect the graph of an exponential function. For example, $f(x) = a^{x} + k$ shifts the graph vertically by $k$ units, and $f(x) = a^{x-h}$ shifts the graph horizontally by $h$ units.
Examples
Let's look at a couple of examples:
- Simplify: $\frac{(x^2y^3)^4}{x^3y^5}$
Solution: $\frac{(x^2y^3)^4}{x^3y^5} = \frac{x^{2*4}y^{3*4}}{x^3y^5} = \frac{x^8y^{12}}{x^3y^5} = x^{8-3}y^{12-5} = x^5y^7$ - Solve for $x$: $2^{x+1} = 8$
Solution: Since $8 = 2^3$, we have $2^{x+1} = 2^3$. Therefore, $x+1 = 3$, so $x=2$.
Remember to practice lots of problems! Exponential functions are a fundamental concept in mathematics, and a solid understanding of their properties will be beneficial in your future studies.
Good luck with your studies, and don't hesitate to reach out if you have any questions! You've got this!
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