B.E.S.T. Algebra 1 EOC: a complete guide to functions

What this category demands

This guide covers the Functions strand (MA.912.F), the heart of the Functions and Modeling reporting category, about 40 percent of the B.E.S.T. Algebra 1 EOC. The skills are function notation and domain and range, reading key features of graphs, average rate of change, transformations, and comparing functions across representations. Because almost every later topic, quadratics, exponentials, modeling, is stated in function language, fluency here lifts the whole test. Each dot-point page has its own practice: function notation, domain, and range, key features of graphs, average rate of change, transformations of functions, and comparing functions.

Function notation, domain, and range

$f(x)$ names the output for input $x$; $f(4)$ means substitute, not multiply. A relation is a function when every input has exactly one output, confirmed by the vertical line test. The domain is the inputs ($x$, left-to-right extent), and the range is the outputs ($y$, bottom-to-top extent), both restricted to the context in a word problem. Counting situations are discrete (dots); measuring situations are usually continuous (connected curves).

Key features of graphs

Read the $y$-intercept (where $x = 0$, often a starting value), the $x$-intercepts or zeros (where $y = 0$), the increasing and decreasing intervals, the maximum and minimum (report the $y$-value), and the end behavior. In context, always interpret the feature: the $y$-intercept is the start, an $x$-intercept is when something hits zero, the maximum is the most. Because the on-screen calculator is scientific, build these from the equation: $f(0)$ for the $y$-intercept, $f(x) = 0$ for zeros, $\frac{-b}{2a}$ for a vertex.

Average rate of change

The average rate of change from $x = a$ to $x = b$ is $\frac{f(b) - f(a)}{b - a}$, the slope of the line through the endpoints. For a linear function it is constant; for a curve it varies. Attach units (output per input) in context.

Transformations

Outside the function shifts vertically: $f(x) + k$ moves up for $k > 0$. Inside the function shifts horizontally opposite to the sign: $f(x - h)$ moves right $h$. Multiplying, $a \cdot f(x)$, gives a stretch ($|a| > 1$), a compression ($0 < |a| < 1$), and a reflection across the $x$-axis ($a < 0$). This is exactly why vertex form $a(x - h)^2 + k$ has its vertex at $(h, k)$.

Comparing functions across representations

To compare two functions in different forms, extract the same feature from each and compare the numbers. Rate of change is the coefficient of $x$, the table's constant difference, or the graph's slope. The $y$-intercept is the constant term, the output at $x = 0$, or the $y$-axis crossing. Name the feature, read it from both, then compare.

How this category is examined

• Multiple choice and multiselect. Evaluate functions, interpret features, or compare two functions.
• GRID and hot-spot. Plot or click intercepts, maximums, or transformed vertices; identify increasing intervals.
• Equation editor. Compute average rate of change or a function value.
• Matching and editing task. Pair equations with graphs or describe a transformation.

Check your knowledge

Work these as you would for credit on the computer-based test.

1. If $f(x) = 2x + 9$, find $f(-3)$. (1 point)
2. State the domain and range of the segment from $(-1, 2)$ to $(5, 8)$, endpoints included. (2 points)
3. A downward parabola has vertex $(1, 10)$. State the maximum value and the increasing interval. (2 points)
4. For $f(x) = x^2 + 2$, find the average rate of change from $x = 0$ to $x = 4$. (2 points)
5. How is $g(x) = f(x) - 6$ related to $f$? (1 point)
6. Starting from $y = x^2$, give the vertex of $y = (x + 2)^2 + 5$. (1 point)
7. Function A: $f(x) = 5x + 1$. Function B: table $(0, 4), (2, 10)$. Which has the greater rate of change? (1 point)
8. Is a function giving the number of buses needed for $n$ students discrete or continuous? (1 point)