NC Math 1: a complete guide to functions and exponential models

What this strand demands

This guide covers functions and exponential models on the NC Math 1 EOC, drawing on Interpreting Functions (NC.M1.F-IF), Building Functions (NC.M1.F-BF), and Linear, Quadratic, and Exponential Models (NC.M1.F-LE). The Functions block is about 32 to 36 percent of the test, the second-largest reporting block, so this strand is essential to a strong score. Each dot-point page has its own practice: function notation, domain, and range, interpreting key features, average rate of change, arithmetic and geometric sequences, exponential functions, growth, and decay, comparing function families, and solving quadratic equations.

Function notation, domain, and range

A function assigns each input exactly one output, and its graph passes the vertical line test. Function notation $f(x)$ names the output: $f(4)$ means evaluate at $x = 4$, and $f(4) = 7$ places $(4, 7)$ on the graph. The domain is the valid inputs and the range is the outputs; in context, restrict the domain to values that make sense (non-negative time, whole-number counts).

Key features and rate of change

Key features (F-IF.4) are the intercepts, intervals of increase and decrease, maximum or minimum (the vertex for a parabola), and end behavior. The average rate of change over $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$, the slope of the secant line; it is constant for a line but changes by interval for quadratics and exponentials.

Sequences

Arithmetic sequences add a common difference $d$ ($a_n = a_1 + (n - 1)d$); geometric sequences multiply by a common ratio $r$ ($a_n = a_1 r^{\,n-1}$). Both can be written recursively (from the previous term) or explicitly (from $n$). Arithmetic sequences are linear; geometric sequences are exponential.

Exponential growth and decay

An exponential function $f(x) = ab^x$ has initial value $a$ and growth/decay factor $b$: growth if $b > 1$, decay if $0 < b < 1$. Writing $b = 1 \pm r$ shows the rate. The fingerprint of an exponential situation is a constant percent change, not a constant amount.

Comparing function families and solving quadratics

The families differ by their pattern of change: constant first difference (linear), constant second difference (quadratic), constant ratio (exponential). An increasing exponential eventually exceeds any linear or quadratic function. To solve a quadratic, set it to zero, then factor and use the zero-product property, take square roots (keeping $\pm$), or apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

How this strand is examined

• Gridded response. Evaluate a function, compute an average rate of change, find a term of a sequence, or enter a zero. Exact-match scoring.
• Multiple choice and multiple select. Classify a family, choose an exponential model, or identify a key feature.
• Technology-enhanced. Match graphs, tables, and equations, or plot features.

Check your knowledge

Work these as you would for credit on the EOC.

1. If $f(x) = 2x^2 - 3$, find $f(4)$. (1 point)
2. State the domain and range of $y = x^2$ (vertex at the origin, opens up). (2 points)
3. For $y = (x - 1)(x + 5)$, find the x-intercepts. (1 point)
4. Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 3$. (2 points)
5. Find the $6$th term of the arithmetic sequence $2, 7, 12, \ldots$. (1 point)
6. Write a function for \400$growing$5%$ per year. (1 point)
7. Outputs $3, 6, 12, 24$ for inputs $0, 1, 2, 3$: which family? (1 point)
8. Solve $x^2 - 5x + 6 = 0$. (2 points)