Transformations of Exponential Functions Made Easy

Welcome to our exploration of exponential functions! We'll start by understanding the parent function.The parent exponential function is written as f of x equals b to the x power, where b is our base.Every exponential function passes through the point (0,1). This is because any number raised to the zero power equals one.The value of b determines whether our function shows growth or decay. Let's understand what this means.Let's start with b equals 2, which gives us growth. As x increases, the function grows exponentially.Notice how quickly the values grow. Each time x increases by 1, we multiply our previous value by 2.Now let's look at decay, where b is one-half. This means we're repeatedly multiplying by a fraction less than 1.With decay, our values get smaller and smaller, approaching but never reaching zero.Let's compare different bases. A larger base like 3 or 4 creates an even steeper growth curve.Notice how the steepness increases with larger bases. This shows how the base affects the rate of growth.Now we'll explore how to transform exponential functions vertically.When we add a constant k, like 3, to our function, the entire graph shifts up by that amount.Notice how the y-intercept has shifted from 1 to 4.Subtracting a constant shifts the graph down. Here we subtract 1.Multiplying by a constant a greater than 1 stretches the graph vertically. Let's multiply by 3.Each y-value is now three times its original value.Multiplying by a constant between 0 and 1 compresses the graph vertically. Here we multiply by one-half.We can combine vertical stretches and shifts. Here's three times two to the x plus two.The y-intercept is now at 5, showing both the stretch and shift effects.