Virginia SOL Algebra I: a complete guide to functions (A.F)

What this category demands

This guide covers the Functions reporting category, the A.F standards, part of the 20-item Functions and Statistics block on the Virginia Algebra I SOL. The skills are function notation, domain and range, slope as rate of change, writing linear equations, key features, and quadratic functions. Reading graphs and tables fluently, and connecting them to equations, is what decides the higher scores. Each dot-point page has its own practice: function notation and evaluation, domain and range, linear functions and rate of change, writing linear equations, zeros and key features, and quadratic functions.

Function notation and evaluation

A function gives one output per input. Use the vertical line test on graphs, and check that no input repeats with different outputs in a table. Function notation $f(x)$ names the output; to evaluate $f(a)$, substitute $a$ for every $x$ (use parentheses) and simplify. The statement $f(3) = 7$ is the point $(3, 7)$.

Domain and range

The domain is the set of inputs ($x$); the range is the set of outputs ($y$). Read a graph left-to-right for domain and bottom-to-top for range. A discrete domain is separate counted values; a continuous domain is an unbroken interval. In context, choose a reasonable domain (lengths and counts are nonnegative).

Linear functions and rate of change

Slope is the constant rate of change, $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ (rise over run). In context, the slope is a per-unit rate and the $y$-intercept is the starting value at input $0$. A constant rate of change is the signature of a linear function.

Writing linear equations

Use slope-intercept $y = mx + b$ and point-slope $y - y_1 = m(x - x_1)$. Find the slope, then $b$. Parallel lines share a slope; perpendicular lines have opposite-reciprocal slopes (product $-1$).

Zeros and key features

Zeros ($x$-intercepts) are where $y = 0$; the $y$-intercept is $f(0)$. The maximum (downward parabola) or minimum (upward parabola) is at the vertex. A function is increasing where the graph rises left to right and decreasing where it falls.

Quadratic functions

A quadratic graphs as a parabola. The sign of $a$ sets the direction (up for $a > 0$, down for $a < 0$). The axis of symmetry is $x = \dfrac{-b}{2a}$ (memorize, not on the sheet), and the vertex is on it. Three forms each show a feature: standard (the $y$-intercept), vertex (the vertex), factored (the zeros).

How this category is examined

• Multiple choice. Identify functions, interpret slope and intercepts, find a vertex or zero.
• Fill-in-the-blank. Evaluate $f(a)$, compute a slope, write a line, or find an axis of symmetry.
• Coordinate-plane and hot-spot. Plot points and vertices, read key features, graph lines.

Check your knowledge

Work these as you would for credit on the online test.

1. For $f(x) = 3x^2 - 2x + 1$, find $f(-2)$. (2 points)
2. Is $\{(1, 2), (1, 3), (2, 4)\}$ a function? (1 point)
3. State the domain of $\{(-2, 3), (0, 5), (4, 9)\}$. (1 point)
4. Find the slope through $(1, 4)$ and $(5, 16)$. (2 points)
5. Write the line with slope $4$ through $(0, -3)$. (1 point)
6. What slope is perpendicular to $y = \frac{1}{2}x + 6$? (1 point)
7. Find the zeros of $f(x) = x^2 - 4$. (2 points)
8. What is the $y$-intercept of $f(x) = 2x^2 + 3x - 5$? (1 point)
9. Find the axis of symmetry of $f(x) = x^2 - 6x + 5$. (2 points)
10. Does $f(x) = -2x^2 + 4x + 1$ open up or down? (1 point)