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Precalculus Unit 4: Transformations & function modeling

New functions come from transforming, combining and inverting known ones. Choosing the right model for data is the applied side of the same skill.

Covers: Shifts/dilations, composition, inverse, choosing a model from data

Study guide

Transformations change a function's graph predictably. Additive changes shift: f(x) + k moves the graph up k, and f(x − h) moves it right h. Multiplicative changes dilate: a·f(x) stretches vertically by |a| (and reflects over the x-axis if a < 0), while f(bx) compresses horizontally by a factor of 1/|b| (and reflects over the y-axis if b < 0). For example, f(2x) squeezes the graph toward the y-axis by half.

Composition combines functions: (f ∘ g)(x) = f(g(x)) feeds the output of g into f. Order matters, and the domain of the composition includes only inputs where g is defined and g(x) is in the domain of f. Decomposing a complicated function into simpler pieces is also a useful skill.

An inverse function reverses inputs and outputs. To find it, swap x and y and solve for y; its graph is the reflection of the original across y = x. Only one-to-one functions have inverse functions, though you can restrict a domain to make one.

Function modeling means choosing a function type that fits data or a context. Constant differences in outputs suggest a linear model, constant second differences a quadratic, and constant ratios an exponential. Check a model by examining residuals (actual minus predicted): a good model has residuals that are small and show no pattern.

A function is even if f(−x) = f(x), so its graph is symmetric about the y-axis. It is odd if f(−x) = −f(x), so it is symmetric about the origin. To test, substitute −x and simplify. To fit y = ax² + bx + c through three given points, substitute each point to get three linear equations in a, b and c, then solve the system (by elimination or with a calculator). For a model with two unknowns, two points are enough.

A model's domain should be limited to inputs that make sense, for example ending when the quantity reaches zero or a maximum. Predictions far outside the range of the data used to build the model are less reliable, because the real pattern may change.

Key ideas

• g(x) = a·f(b(x − h)) + k: a stretches vertically, b compresses horizontally by 1/b, h and k shift.
• Composition chains functions; the inside function runs first.
• Inverses undo a function and reflect its graph across y = x.
• Pick models by behavior: constant change → linear; constant second differences → quadratic; constant ratio → exponential.
• Residuals (actual − predicted) show whether a model fits.

Formulas and rules

Common mistakes

• Reading f(2x) as a horizontal stretch instead of a compression.
• Choosing a model only by how the scatterplot looks without checking residuals.

Words to know

Effect of f(2x)

Horizontal compression by a factor of 1/2.

Exponential model

Fits outputs that change by a constant ratio.

Residual

Actual value minus predicted value.

Effect of f(x + 3)

Shift left 3.

Graph of an inverse function

Reflection of the original across y = x.

Vertical shift

f(x) + k moves the graph up k.

Horizontal shift

f(x − h) moves the graph right h.

Vertical dilation

a·f(x) stretches vertically by |a|.

Horizontal dilation

f(bx) compresses horizontally by a factor of 1/|b|.

Composition

(f ∘ g)(x) = f(g(x)).

One-to-one function

Each output comes from exactly one input; it has an inverse.

Function model

Choosing linear, quadratic or exponential from patterns in the data.

Sample practice questions

23 more questions like these, from level 1 to 5, are in the practice set.