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Algebra 2 Unit 1: Functions & transformations

Algebra 2 treats functions as objects you can shift, stretch, combine and invert.

Covers: Function notation, domain and range, composition and inverses, piecewise functions, transformations

Study guide

A function assigns exactly one output to each input. In function notation, f(3) means the output when the input is 3. The domain is the set of allowed inputs and the range is the set of outputs. Restrictions come from denominators (can't be zero), even roots (radicand can't be negative) and logarithms (input must be positive).

Composition (f ∘ g)(x) = f(g(x)) means apply g first and then f. An inverse function undoes the original: to find it, swap x and y and solve for y. The graph of an inverse is the reflection of the original across y = x, and a function has an inverse function only if it passes the horizontal line test (it's one-to-one).

Piecewise functions use different rules on different intervals of x. To evaluate, find which interval the input belongs to and use that rule. Absolute value is a common piecewise function.

Transformations change a parent graph in predictable ways. f(x) + k shifts up k, f(x − h) shifts right h (the sign looks backward), −f(x) reflects over the x-axis, f(−x) reflects over the y-axis, a·f(x) stretches vertically by a, and f(bx) compresses horizontally by a factor of 1/b. Apply stretches and reflections before shifts when combining them.

A function is even if f(−x) = f(x) for every x, so its graph is symmetric about the y-axis (like x² or cos x). It is odd if f(−x) = −f(x), so its graph is symmetric about the origin (like x³ or sin x). A vertical shift or a phase shift usually destroys odd symmetry, so y = a sin(bx) is odd but y = sin x + 2 is not. To solve an equation whose two sides are different kinds of functions, graph y = f(x) and y = g(x) and read the x-coordinates of their intersection points. Those x-values are the solutions, and a graphing calculator's intersect feature makes them exact to several decimals.

The average rate of change of f on [a, b] is [f(b) − f(a)]/(b − a). In context it has units of output per unit of input, such as dollars per year.

Key ideas

• f(x − h) + k shifts right h and up k.
• Composition f(g(x)) feeds g's output into f.
• Inverses swap inputs and outputs; graphs reflect over y = x.
• Piecewise functions use different rules on different intervals.
• Domain excludes division by zero and even roots of negatives.

Formulas and rules

Common mistakes

• Shifting the wrong direction for f(x − h).
• Evaluating a piecewise function with the wrong piece.

Words to know

Effect of f(x + 2)

Shift left 2.

Composition (f ∘ g)(x)

f(g(x)): apply g first, then f.

Graph of an inverse

Reflection over y = x.

Domain of a rational function

All real numbers except where the denominator is 0.

Piecewise function

A function with different rules on different intervals of x.

Function

A rule assigning exactly one output to each input.

Domain

The set of allowed inputs.

Range

The set of outputs.

Inverse function

Undoes the original; swap x and y and solve for y.

Horizontal line test

A function has an inverse if no horizontal line hits its graph twice.

Vertical stretch

a·f(x) stretches by a factor of a.

Reflection over the x-axis

−f(x)

Transformations

f(x) + k shifts up, f(x − h) shifts right, −f(x) reflects, a·f(x) stretches.

Sample practice questions

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