Resources For Teachers For Tutors For Students & Parents Pricing
Create Free Account
Algebra 2 Relations and functions (advanced)

Key attributes (domain, range, intercepts, asymptotes, end behavior)

20 practice questions 0 video lessons Theory + worked examples

Key Attributes of Functions

Texas Algebra II (TEKS) • Standard 2A.2(A) • Relations & Functions

Key Attributes of Functions is a topic in Relations & Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.2(A), which requires students to graph and analyze the key attributes of functions, including domain, range, intercepts, symmetries, and asymptotic behavior.

The key attributes of a function are its domain and range, intercepts and zeros, extrema, intervals of increase/decrease, asymptotes, and end behavior.

Texas Algebra II (TEKS) › Relations & Functions › Key Attributes of Functions  —  Standard 2A.2(A)

Practice 20 questions
Practice questions

Every question with a fully worked solution.

Start practising
Create a free accountTrack your progress and save your work as you go.
Create free account

Theory

The key attributes summarize a function's behavior:

  • Domain / range: allowable inputs / outputs.
  • Intercepts: where the graph meets the axes; the \(x\)-intercepts are the zeros.
  • Extrema: maximum and minimum values.
  • Intervals of increase and decrease.
  • Asymptotes and end behavior as \(x\to\pm\infty\).
Read attributes from the graph or the equation — both should agree.
Key attributes of a graph A graph's zeros, turning points, and end behavior are read directly from its shape. x y end ↓ end ↑ zeros
Zeros, turning points, and end behavior of a graph.
Key attributes Key attributes Key attributes domain / range x- and y-intercepts (zeros) maxima / minima increasing / decreasing intervals asymptotes; end behavior
The key attributes to identify.

Finding intercepts:

\[\text{y-intercept: } f(0),\qquad \text{x-intercepts: solve } f(x)=0\]
the y-intercept is f of 0; the x-intercepts solve f of x equals 0
End behavior of a polynomial is set by its leading term.

How to analyze a function

  1. Find the domain (exclude values that break the function).
  2. Find intercepts: \(f(0)\) and the solutions of \(f(x)=0\).
  3. Locate extrema and intervals of increase/decrease.
  4. Describe asymptotes and end behavior.
Example 1 — Find the intercepts
Find the intercepts of \(f(x)=x^2-4\).
Solution

Set each variable to zero.

\(x=0:\ f(0)\)\(=\)\(-4\ \text{(y-intercept)}\)
\(f(x)=0:\ x^2-4\)\(=\)\(0\)
\(x\)\(=\)\(\pm 2\ \text{(x-intercepts)}\)
y-intercept negative 4, x-intercepts plus and minus 2
Example 2 — State the domain and range
Give the domain and range of \(f(x)=x^2-4\).
Solution

A parabola opening up has a minimum at its vertex \((0,-4)\).

\(\text{domain}\)\(=\)\((-\infty,\infty)\)
\(\text{range}\)\(=\)\([-4,\infty)\)
domain all reals, range y at least negative 4
Example 3 — End behavior
Describe the end behavior of \(f(x)=x^2-4\).
Solution

The leading term \(x^2\) dominates for large \(|x|\).

\(x\to\pm\infty\)\(\Rightarrow\)\(f(x)\to +\infty\)
as x goes to plus or minus infinity, f goes to positive infinity
Example 4 — Increasing and decreasing
On what interval is \(f(x)=x^2-4\) increasing?
Solution

A upward parabola decreases then increases at its vertex \(x=0\).

\(\text{increasing}\)\(\text{on}\)\((0,\infty)\)
increasing for x greater than 0

Common pitfalls

Zeros are \(x\)-intercepts, found by solving \(f(x)=0\) — not the \(y\)-intercept.
Range depends on the shape; don't assume all reals.
End behavior follows the leading term for polynomials.

Frequently asked questions

What are the key attributes of a function?

Domain, range, intercepts, extrema, intervals of increase/decrease, asymptotes, and end behavior.

What is a zero of a function?

An \(x\)-value where \(f(x)=0\) — an \(x\)-intercept of the graph.

How do you find the y-intercept?

Evaluate \(f(0)\).

What is end behavior?

How the outputs behave as \(x\to\pm\infty\); for polynomials the leading term decides it.

More in Relations and functions (advanced)
Inverse functions
20 questions
Function composition
20 questions
Transformations of parent functions
20 questions
Interval and set notation for domain and range
20 questions