ACT Math Functions: notation, linear and quadratic functions, exponentials and logs, transformations and sequences

What the Functions area covers

Functions, about 12 to 15 percent of the ACT Math test, is one of its largest areas. It runs from reading function notation to graphing parabolas, exponentials, transformations and sequences. This guide ties together the dot points: function notation and evaluation, linear functions and slope, quadratic functions and graphs, exponential and logarithmic functions, transformations of functions, and sequences and series.

Function notation and evaluation

$f(a)$ means substitute $x = a$ into the rule. Composition $f(g(x))$ applies $g$ first, then $f$ (inside out). The domain is the allowed inputs (exclude zero denominators and negative even roots); the range is the possible outputs. The vertical-line test decides whether a graph is a function.

Linear functions and slope

In $f(x) = mx + b$, the slope $m$ is the rate of change and $b$ is the starting value. Build a model from a constant rate (slope) and an initial amount (intercept), using a negative slope for a decreasing quantity. Slope from points is $\frac{y_2 - y_1}{x_2 - x_1}$; from a table, a constant difference over a constant input step.

Quadratic functions and graphs

A quadratic graphs as a parabola: up if $a > 0$ (minimum), down if $a < 0$ (maximum). Standard form shows the $y$-intercept $c$; factored shows the zeros; vertex form $a(x - h)^{2} + k$ shows the vertex $(h, k)$. The axis of symmetry is $x = -\frac{b}{2a}$.

Exponential and logarithmic functions

$f(x) = a \cdot b^{x}$ has initial value $a$ and growth factor $b$ ($b > 1$ grows, $0 < b < 1$ decays); a factor of $1.05$ is $+5\%$. Linear change adds a constant; exponential change multiplies. Compound growth is $A = P(1 + r)^{t}$. A logarithm $\log_b x = y$ means $b^{y} = x$, the inverse of an exponent.

Transformations of functions

From $y = f(x)$: $f(x) + k$ shifts up; $f(x - h)$ shifts right (subtract inside moves right); $-f(x)$ reflects across the $x$-axis; $a \cdot f(x)$ stretches ($a > 1$) or compresses ($0 < a < 1$). Changes outside act vertically as written; changes inside act horizontally and opposite to the sign.

Sequences and series

Arithmetic sequences add a common difference $d$: $a_n = a_1 + (n - 1)d$. Geometric sequences multiply by a common ratio $r$: $a_n = a_1 \cdot r^{n-1}$. Classify by a constant difference (arithmetic) or constant ratio (geometric). Watch the $(n - 1)$: the $n$th term uses one fewer step than $n$.

Check your knowledge

Try these, then read the solutions.

1. If $f(x) = 3x - 2$, find $f(5)$. [1 point]
2. Find the vertex of $f(x) = (x - 4)^{2} + 2$. [1 point]
3. In $P(t) = 200(1.06)^{t}$, what is the growth rate? [1 point]
4. The graph of $g(x) = f(x) - 4$ is $f$ shifted how? [1 point]
5. Find the 6th term of the arithmetic sequence $4, 9, 14, \ldots$ [2 points]