Exponential functions over unit intervals

Learning Objectives Define an exponential function and identify its initial value and base. Identify the common ratio of an exponential function from a table of values over unit intervals. Explain how the output of an exponential function changes when the input increases by one unit. Write the equation of an exponential function in the form y = a * b^x using a table of values. Distinguish between exponential growth and exponential decay based on the value of the common ratio. Create a table of values for a given exponential function over a sequence of unit intervals. Compare the constant multiplicative change of exponential functions to the constant additive change of linear functions over unit intervals. Have you ever seen a video go viral? 📈 How does one share become mi...

TermDefinitionExample Exponential FunctionA function where the input variable 'x' is an exponent. Its general form is y = a * b^x.y = 3 * 2^x is an exponential function where the initial amount is 3 and it doubles at each step. Unit IntervalA change of exactly 1 in the input variable (x). We study the function's behavior as x goes from 0 to 1, 1 to 2, 2 to 3, and so on.The interval between x=4 and x=5 is a unit interval. Initial Value (a)The starting amount of the function. It is the value of y when x = 0.In the function y = 50 * (1.1)^x, the initial value is 50. Common Ratio (b)The constant factor you multiply by to get the next output value when the input 'x' increases by 1. It is also called the base or the growth/decay factor.In the sequence 4, 12, 36, 108...,...

General Form of an Exponential Function y = a \cdot b^x This is the standard formula for an exponential function. 'a' is the initial value (the y-intercept), 'b' is the common ratio (the growth/decay factor), and 'x' is the input. Finding the Common Ratio (b) over a Unit Interval b = \frac{f(x+1)}{f(x)} To find the base 'b', divide any output value by the output value that comes directly before it (where the inputs are consecutive integers like x and x+1). Conditions for Growth and Decay \text{If } b > 1 \rightarrow \text{Growth} \\ \text{If } 0 < b < 1 \rightarrow \text{Decay} The value of the base 'b' tells you the function's behavior. A base greater than 1 means the values are growing. A base between...