What happens when change is proportional to what's already there? Start with sequences and exponentials, learn composition and inverses, and use them to meet logarithms. You'll finish by modeling data and choosing the best model.
Episode Guide — Season 2
A sequence is a function on the whole numbers. Arithmetic sequences grow by a common difference and geometric sequences by a common ratio, and each has a general term you can write.
Linear functions are the continuous cousins of arithmetic sequences, and exponential functions are the cousins of geometric ones. One adds a constant each step, the other multiplies by one.
Meet f(x) = abˣ: initial value, base, and the difference between growth and decay. Exponential graphs are always increasing or always decreasing, with a horizontal asymptote that you can describe with limits.
The properties of exponents have graphical meaning. A horizontal translation of bˣ is the same as a vertical dilation, and a horizontal dilation is the same as a change of base.
Build exponential models from a ratio and initial value, from two input-output pairs, or with exponential regression on your calculator. Then interpret the base as a growth factor and as a percent change.
Linear, quadratic, and exponential models can all fit the same data. Use residuals and residual plots to justify which model is most appropriate, and consider where each one over- or underestimates.
Feed the output of one function into another. Composition relates quantities that have no direct formula, and it isn't commutative. Evaluate, build, and decompose composite functions.
An inverse function reverses a mapping, but only on a domain where each output comes from a unique input. Find inverses from tables, graphs, and equations, and check them by composing.
A logarithm is an exponent: log_b c is the power you raise b to in order to get c. Evaluate logarithmic expressions, and see how a logarithmic scale turns multiplication into equal steps.
Logarithmic functions are the inverses of exponential functions. Exponential growth multiplies as inputs add, and logarithmic growth adds as inputs multiply. The graphs are mirror images over y = x.
Describe what log graphs do: domain x > 0, all real numbers as range, always increasing or always decreasing, and a vertical asymptote at x = 0.
The product, power, and change-of-base properties let you rewrite logarithmic expressions. Each one also has a graphical meaning, and ln x is the log with natural base e.
Use exponent properties, log properties, and the inverse relationship to solve equations and inequalities. Check every solution for extraneous results, and construct inverses of transformed exponential and log functions.
When inputs change proportionally over equal-length output intervals, a logarithmic model fits. Construct one from a proportion and a zero, from two points, or with logarithmic regression.
On a semi-log plot, exponential data looks linear. Use that to decide whether an exponential model is appropriate, and linearize the data to find the model's parameters.