Exponential Functions

Exponential Functions: AP Pre-Calculus Study Guide

Prepare to dive into the exhilarating world of exponential functions, where numbers grow and decay faster than your favorite TikTok dances become trends! 🌟 Whether you’re solving problems about population growth, radioactive decay, or even the rise and fall of your savings account, understanding exponential functions is your golden ticket. So grab your graphing calculator and let’s get exponential! 📈

An exponential function is like the rollercoaster of mathematical functions—its variable, x, sits high in the exponent seat, not down in the base area as usual. The general form of an exponential function looks like this:

[ f(x) = ab^x ]

where:

• ( a ) is the initial value (the y-intercept), think of it as the starting point of your financial savings jar.
• ( b ) is the base, a positive number other than 1 (because math decided 1’s too boring for this ride 🌟).

The magic behind exponential functions lies in the base ( b ).

• If ( b > 1 ), you’ve got exponential growth, kind of like how your social media likes skyrocket! As x increases, ( f(x) ) increases faster than you can say “viral video”. 📈
• If ( 0 < b < 1 ), it’s exponential decay, similar to how fast-touch screen phones get outdated. As x increases, ( f(x) ) decreases rapidly.

The domain of an exponential function is all real numbers, meaning you can plug in any real number you like for x. It’s like an all-you-can-eat buffet, but for math!

When we plug in natural numbers (1, 2, 3, ...) into ( f(x) ), each input tells us how many times to multiply the base ( b ).

For instance:

• When ( x = 1 ): ( f(1) = ab^1 = ab )
• When ( x = 2 ): ( f(2) = ab^2 = ab \times b = ab^2 )

Flip the scenario and consider if your savings are growing at 5% per year. Your exponential function would look like this:

[ P(1.05)^n ]

where P is your initial savings. Here, the base ( 1.05 ) represents a 5% growth rate, and the exponent ( n ) is the number of years your investment grows. Cha-ching! 💰

Exponential functions tend to have pretty consistent graph styles:

• If ( b > 1 ), the graph is concave up and looks like the steepest uphill climb you’ve seen.
• If ( 0 < b < 1 ), you’ll get a graph that’s concave down, like a skateboard ramp on Decay Street.

By now, you might have noticed that exponential graphs always trend up or down and never change direction—no inflection points here! So no matter how much your math teacher wants to find one, they won’t!

Exponential functions don’t have highest or lowest points (extrema) in open intervals, just like you won’t find a top or bottom while zooming through space. That’s because these graphs are always increasing or decreasing forever and ever (and ever). 🚀

Now, if you want to give your function ( f(x) ) a little nudge up or down, you can create an additive transformation with:

[ g(x) = f(x) + k ]

Here, ( k ) is a constant that vertically shifts the entire graph. If ( f(x) ) represents exponential growth, then ( f(x) + k ) will also follow the growth but start from a different place on your graph paper. 🖍️

Let's talk limits, but no, not the ones your parents set! In the world of exponential functions, we have three scenarios:

• If ( b > 1 ), as ( x ) approaches infinity, ( f(x) ) also shoots up to infinity. Imagine your bank account growing with infinite interest rates! 🏦
• If ( 0 < b < 1 ), as ( x ) approaches infinity, ( f(x) ) gets closer and closer to zero, just like the melting ice cream you left out too long. 🍦
• When ( x ) heads to negative infinity:
  • ( b > 1 ): ( f(x) ) plunges down to negative infinity.
  • ( 0 < b < 1 ): ( f(x) ) rises up toward zero.

Exponential functions are the rock stars of real-world applications. From the rapid growth of bacteria in a petri dish 🍽️ to the depreciating value of your new car the minute you drive it off the lot 🚗, exponential functions are everywhere!

For example: Imagine a population of rabbits growing by 10% every year. If you start with 100 rabbits, after ( n ) years the population will be:

[ 100 \times (1.10)^n ]

Just hope you’ve got enough carrots! 🥕

Now that you're well-versed in the world of exponential functions, you're ready to tackle any problem that comes your way. Remember, whether it’s growth or decay, these functions are like the highs and lows of life: inevitable and ever-changing.

As you conquer your AP Pre-Calculus exam, let the exponential rise of your knowledge be your guide. May your grades grow as exponentially as the stars in the sky! 🌠

Good luck, and remember: Math is just another rollercoaster; all you need to enjoy the ride is a little bit of courage and a firm grip on the bar of knowledge. 🎢