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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

Common Core Algebra 2 • Standard F-IF.4 • Relations & Functions

Key Attributes of Functions is a topic in Relations & Functions in the Common Core State Standards. It is aligned to Standard F-IF.4, which requires students to interpret the key features of a function from its graph and its table or equation.

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

Common Core Algebra 2 › Relations & Functions › Key Attributes of Functions  —  Standard F-IF.4

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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.

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