ACT Math Study Notes: Understanding and Using Functions

Function

A rule that assigns exactly one output to each allowed input.

Function notation

Notation like f(x) that means the output of function f when the input is x; it does not mean f times x.

Domain

The set of all allowed inputs of a function.

Range

The set of all possible outputs of a function.

Vertical line test

A graph represents a function if every vertical line intersects it at most once.

Linear function

A function whose graph is a straight line and whose rate of change is constant.

Slope

The rate of change of a line, found by $\frac{y_2 - y_1}{x_2 - x_1}$.

Slope-intercept form

The form y = mx + b, where m is the slope and b is the y-intercept.

Point-slope form

The form $y - y_1 = m(x - x_1)$, used when a slope and one point on the line are known.

Polynomial function

A function made from powers of x with nonnegative integer exponents and real coefficients.

Degree

The highest exponent with a nonzero coefficient in a polynomial.

End behavior

How a graph behaves as x becomes very large positive or very large negative, determined mainly by the leading term.

Zero (root)

An input value that makes $f(x) = 0$; it corresponds to an x-intercept.

Discriminant

The expression $b^2 - 4ac$ in the quadratic formula, which tells how many real solutions a quadratic has.

Radical function

A function that contains a root of the variable, such as a square root.

Piecewise function

A function defined by different rules on different parts of its domain.

Open and closed dots

Graph markers showing whether a boundary point is excluded with an open dot or included with a closed dot.

Exponential function

A function of the form $a \times b^x$, where the variable is in the exponent; it models growth if $b > 1$ and decay if $0 < b < 1$.

Logarithmic function

The inverse of an exponential function; $y = \log_b(x)$ asks what exponent on base $b$ gives $x$.

Function transformation

A change to a parent function by shifting, stretching, compressing, or reflecting its graph.

Horizontal shift

A transformation where $f(x - h)$ moves a graph right $h$ units and $f(x + h)$ moves it left $h$ units.

Vertical shift

A transformation where f(x) + k moves a graph up k units and f(x) - k moves it down k units.

Reflection

A flip of a graph, where -f(x) reflects across the x-axis and f(-x) reflects across the y-axis.

Intercepts

x-intercepts occur where y = 0, and the y-intercept occurs where x = 0, or at f(0).

Average rate of change

The quantity $\frac{f(b) - f(a)}{b - a}$, which is the slope of the secant line between two points on a function.