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Unit 2
25–40% of examThis unit compares growth that adds the same amount each step with growth that multiplies by the same factor. You'll work with sequences, exponential and logarithmic functions, inverses and composition, and use these functions to model data like population, interest and sound levels. It's the math behind anything that grows or shrinks by a percent.
Longer videos that cover the whole unit. Good for a first pass or a final review.
An arithmetic sequence adds the same number (the common difference) each step, while a geometric sequence multiplies by the same number (the common ratio). You'll write a formula for the nth term of each and compare how fast they grow.
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Linear functions are like arithmetic sequences: equal input steps give equal output differences. Exponential functions are like geometric sequences: equal input steps give equal output ratios. You'll write each from two points or from a table.
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In f(x) = a·bˣ with a > 0, a is the starting value and b is the base (b > 0, b ≠ 1). If b > 1 the function grows, and if 0 < b < 1 it decays. The graph has a horizontal asymptote at y = 0 and never changes concavity.
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Exponent rules let you rewrite exponential expressions, which can show a different way to see the same function. For example, 2^(x+3) = 8·2ˣ shows a horizontal shift is the same as a vertical stretch, and 4^(x/2) = 2ˣ.
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You'll build exponential models from a context, like money growing at a percent rate or a substance decaying with a half-life, or from data using exponential regression. The base e ≈ 2.718 comes up naturally in continuous growth.
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When several models could fit a data set, a residual plot helps you choose: a residual is the actual value minus the predicted value. A good model leaves residuals scattered with no pattern, and you can say where a model over- or underestimates.
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Composing f(g(x)) means using the output of g as the input of f; order matters, so f(g(x)) usually isn't the same as g(f(x)). You'll compose functions from equations, tables and graphs, and break a complicated function into simpler pieces.
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An inverse function undoes the original, swapping inputs and outputs, so its graph is the reflection across the line y = x. A function only has an inverse on a domain where it's one-to-one, so sometimes you restrict the domain first.
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A logarithm answers the question 'what power?': log_b(c) = a means bᵃ = c. You'll evaluate logs, estimate them, and read the common log (base 10) and natural log (base e).
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The log function log_b(x) is the inverse of the exponential function bˣ, so input-output pairs swap and the graphs reflect across y = x. The exponential's horizontal asymptote becomes the log's vertical asymptote.
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A basic log function like log₂ x only accepts positive inputs, has a vertical asymptote at x = 0, and grows without bound, but very slowly. Over equal-ratio input steps, a log function's outputs change by equal amounts.
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The product, quotient and power rules for logs let you rewrite expressions, such as log(xy) = log x + log y and log(xⁿ) = n·log x. The change-of-base rule lets you rewrite any log in base 10 or base e.
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You'll solve exponential equations by taking logs, and solve log equations by rewriting them in exponential form, then check for answers outside the domain. You'll also find inverses of exponential and log functions with transformations.
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Logs model situations where a quantity grows quickly at first and then much more slowly. You'll build log models from context or with logarithmic regression, and interpret what they mean.
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A semi-log plot uses a log scale on the y-axis. Exponential data looks like a straight line there, which makes it easy to spot and to find a linear model for log(y), which you can convert back into an exponential model.
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