Summary of Common Function Properties
A comparative reference table of common function properties consolidates domain, range, extrema, axis of symmetry, and end behavior for quadratic, absolute value, square root, and exponential parent functions. In Grade 11 enVision Algebra 1 (Chapter 10: Working With Functions), students use this summary to quickly identify key properties: the quadratic f(x) = x² has domain all reals, range [0, ∞), minimum at (0, 0), axis of symmetry x = 0, and both ends rising; absolute value f(x) = |x| has the same domain, range, and minimum; square root and exponential have half-line domains.
Key Concepts
Property A summary table of key graphical features for common parent functions. This table provides a quick reference for the domain, range, extrema, axis of symmetry, and end behavior of quadratic, absolute value, square root, and exponential functions.
|| Function Type | Domain | Range | Extrema | Axis of Symmetry | End Behavior | | | | | | | | | Quadratic $f(x)=x^2$ | $( \infty, \infty)$ | $[0, \infty)$ | Minimum at $(0, 0)$ | $x=0$ | As $x \to \pm\infty$, $f(x) \to \infty$ | | Absolute Value function $f(x)=\lvert x\rvert$ | $( \infty, \infty)$ | $[0, \infty)$ | Minimum at $(0, 0)$ | $x=0$ | As $x \to \pm\infty$, $f(x) \to \infty$ | | Square Root $f(x)=\sqrt{x}$ | $[0, \infty)$ | $[0, \infty)$ | Minimum at $(0, 0)$ | None | As $x \to \infty$, $f(x) \to \infty$ | | Exponential $f(x)=b^x, b 1$ | $( \infty, \infty)$ | $(0, \infty)$ | None | None | As $x \to \infty$, $f(x) \to \infty$ As $x \to \infty$, $f(x) \to 0$ |.
Examples Quadratic Function $f(x) = x^2 + 2$: This function has a maximum value at $(0, 2)$, a domain of $( \infty, \infty)$, a range of $( \infty, 2]$, and an axis of symmetry at $x=0$. Square Root Function $g(x) = \sqrt{x 3}$: This function has a minimum at $(3, 0)$, a domain of $[3, \infty)$, and a range of $[0, \infty)$. It does not have an axis of symmetry.
Common Questions
What are the domain and range of f(x) = x² (quadratic parent function)?
Domain: all real numbers (−∞, ∞). Range: [0, ∞), since x² is never negative.
What are the domain and range of f(x) = √x?
Domain: [0, ∞), since the square root requires non-negative input. Range: [0, ∞).
What is the end behavior of f(x) = |x|?
As x → ∞, f(x) → ∞, and as x → −∞, f(x) → ∞. Both ends of the V-shape rise upward.
What is the minimum of the absolute value parent function?
The minimum is at (0, 0) — the vertex of the V-shape.
What is the end behavior of a basic exponential growth function f(x) = 2ˣ?
As x → ∞, f(x) → ∞. As x → −∞, f(x) → 0 (horizontal asymptote at y = 0).
Which parent functions have a minimum but no maximum?
The quadratic f(x) = x² and the absolute value f(x) = |x| both have a global minimum at (0, 0) but no maximum (they rise to infinity).