TNReady Algebra I: a complete guide to functions

What this category demands

This guide covers the Functions reporting category (TN domains A1.F.IF, A1.F.BF, A1.F.LE), the largest on the TNReady Algebra I test at about 32 to 40 percent. It runs from the definition of a function through reading graphs, rate of change, writing linear functions, sequences, and exponential models. Each dot-point page has its own practice: function notation, domain, and range, interpreting key features, average rate of change, writing linear functions, sequences, exponential functions, and comparing function families.

Function basics

A function assigns each input exactly one output; the vertical line test checks a graph. Function notation $f(x)$ names the output at input $x$, so $f(3)$ means "evaluate at $x = 3$." The domain is the inputs, the range is the outputs. These ideas make every later topic readable.

Reading graphs and rates

Key features (A1.F.IF.C.4), the intercepts, intervals of increase or decrease, maxima or minima, and end behavior, are interpreted in context: the $y$-intercept is a starting value, an $x$-intercept is a break-even or landing point. The average rate of change over $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$, the slope of the segment between the endpoints, constant for a line and varying for a curve.

Writing functions and sequences

Linear functions come from the slope formula plus slope-intercept ($y = mx + b$) or point-slope ($y - y_1 = m(x - x_1)$) form. Sequences are functions on the integers: arithmetic adds a common difference ($a_n = a_1 + (n-1)d$, like a line) and geometric multiplies by a common ratio ($a_n = a_1 r^{n-1}$, like an exponential). Both sequence formulas are on the reference sheet.

Exponential models and comparison

Exponential growth is $y = a(1 + r)^t$ and decay is $y = a(1 - r)^t$ (these are not on the reference sheet). The graph passes through $(0, a)$ with a horizontal asymptote at $y = 0$. A situation is exponential when it changes by a constant percent, linear when by a constant amount. A1.F.LE.A.3: exponential growth eventually exceeds linear and quadratic growth.

How this category is examined

• Numeric response. Evaluate functions, find a rate of change, a sequence term, or an exponential value.
• Multiple choice and multiple select. Identify a function, a key feature's meaning, a family, or the correct model.
• Graphing / inline choice. Identify intercepts, asymptotes, or whether data is linear, quadratic, or exponential.

Check your knowledge

Work these as you would for credit on the EOC.

1. If $f(x) = 2x^2 - 3$, find $f(-3)$. (1 point)
2. Is $\{(1, 2), (2, 4), (1, 6)\}$ a function? (1 point)
3. For $f(x) = x^2$, find the average rate of change from $x = 2$ to $x = 5$. (2 points)
4. Write the line through $(0, 4)$ and $(2, 10)$. (2 points)
5. Find the $12$th term of $3, 7, 11, \dots$. (1 point)
6. Write the explicit rule for $2, 6, 18, 54, \dots$. (1 point)
7. A \1000$investment grows$6%$per year. Write the model and find its value after$2$ years. (2 points)
8. A table has outputs $1, 3, 9, 27$ for inputs $0,1,2,3$. Which family? (1 point)