Functions, Logs & Trigonometry – Unit 2 Review

Vocabulary flashcards covering vertical & horizontal transformations, discriminant cases, exponential & logarithmic properties, rational equations, and trigonometric area and law rules.

Vertical Transformation

Any change applied to the y-coordinate of a function; x values stay the same.

Vertical Reflection

Multiply y by −1 so the graph flips across the x-axis.

Vertical Shift Up

Add a constant to y; the graph moves upward.

Vertical Shift Down

Subtract a constant from y; the graph moves downward.

Vertical Stretch

Multiply y by a factor >1; the graph becomes narrower (stretches vertically).

Vertical Compression

Multiply y by a factor between 0 and 1 (e.g., ½); the graph widens.

Horizontal Transformation

Any change applied to the x-coordinate of a function; y values stay the same and operations occur ‘backwards.’

Horizontal Reflection

Replace x with −x; flips the graph across the y-axis.

Horizontal Shift Right

Subtract inside brackets, which actually adds to x, moving the graph right.

Horizontal Shift Left

Add inside brackets, which actually subtracts from x, moving the graph left.

Horizontal Compression

Multiply x inside brackets by a factor >1 (do the opposite and divide), making the graph squeeze horizontally.

Horizontal Stretch

Divide x inside brackets by a factor >1 (do the opposite and multiply), making the graph stretch horizontally.

Discriminant

The value b² − 4ac under a quadratic’s radical; tells how many real roots exist.

Discriminant > 0

Two distinct real roots and two x-intercepts.

Discriminant = 0

One repeated real root and one x-intercept.

Discriminant < 0

No real roots and no x-intercepts.

Exponential Function

A function where the variable is in the exponent, showing growth or decay by a factor.

Logarithm

The inverse of an exponential; logₐx answers ‘what exponent y satisfies aʸ = x?’

Change of Form Rule

logₐ1 = 0 and logₐa = 1; logarithms of 0 or negatives are undefined.

Log Power Rule (#1)

logₐbᶜ = c·logₐb and conversely c·logₐb = logₐbᶜ.

Log Product Rule (#2)

logₐM + logₐN = logₐ(M·N).

Log Quotient Rule (#3)

logₐM − logₐN = logₐ(M⁄N).

Change of Base Rule

logₐb = (logk b)⁄(logk a) for any positive base k ≠ 1.

Steps to Solve Rational Equations

1) Find LCD, 2) Multiply every term by LCD, 3) Eliminate denominators, 4) Solve.

Inverse of a Rational Function

Swap x and y, cross-multiply to clear denominator, collect x and y on one side, and factor y.

Area of a Non-right Triangle

A = ½ab sin C; use ½ only for triangles, not rectangles.

Sine Rule

a⁄sin A = b⁄sin B = c⁄sin C; relates sides and opposite angles.

Cosine Rule

Used when two sides and the included angle are known to find the third side: c² = a² + b² − 2ab cos C.