Parent Functions Graphic Organizer

Vocabulary flashcards detailing key parent functions, their algebraic definitions, domains, ranges, intercepts, symmetry, extrema, and end behaviors.

Function Family

A group of functions that share similar characteristics.

Parent Function

The simplest function in a function family.

Linear Parent Function

The parent function defined by $f(x) = x$, having domain and range of $(-\infty, \infty)$, an intercept at $(0,0)$, no extrema, odd symmetry, and end behavior where $f(x) \to -\infty$ as $x \to -\infty$ and $f(x) \to \infty$ as $x \to \infty$.

Absolute Value Parent Function

The parent function defined by $f(x) = |x|$, having domain $(-\infty, \infty)$, range $[0, \infty)$, intercept at $(0,0)$, a minimum at $x = 0$, even symmetry, decreasing on $(-\infty, 0]$, and increasing on $[0, \infty)$.

Quadratic Parent Function

The parent function defined by $f(x) = x^2$, having domain $(-\infty, \infty)$, range $[0, \infty)$, intercept at $(0,0)$, a minimum at $x = 0$, even symmetry, decreasing on $(-\infty, 0]$, and increasing on $[0, \infty)$.

Cubic Parent Function

The parent function defined by $f(x) = x^3$, having domain and range of $(-\infty, \infty)$, intercept at $(0,0)$, no extrema, odd symmetry, and increasing on the interval $(-\infty, \infty)$.

Square Root Parent Function

The parent function defined by $f(x) = \sqrt{x}$, having domain $[0, \infty)$, range $[0, \infty)$, intercept at $(0,0)$, a minimum value of $0$ at $x = 0$, increasing on $[0, \infty)$, and neither even nor odd symmetry.

Cube Root Parent Function

The parent function defined by $f(x) = \sqrt[3]{x}$, having domain and range of $(-\infty, \infty)$, intercept at $(0,0)$, no extrema, odd symmetry, and increasing on the interval $(-\infty, \infty)$.

Reciprocal Parent Function

The hyperbola parent function defined by $f(x) = \frac{1}{x}$, having domain and range of $(-\infty, 0) \cup (0, \infty)$, no intercepts, no extrema, decreasing on $(-\infty, 0) \cup (0, \infty)$, odd symmetry, a horizontal asymptote at $y = 0$, and a vertical asymptote at $x = 0$.

Exponential Parent Function

The parent function defined by $f(x) = e^x$, having domain $(-\infty, \infty)$, range $(0, \infty)$, no $x$-intercept, a $y$-intercept at $(0, 1)$, increasing on $(-\infty, \infty)$, no extrema, and end behavior where $f(x) \to 0$ as $x \to -\infty$ and $f(x) \to \infty$ as $x \to \infty$.

Logarithmic Parent Function

The parent function defined by $f(x) = \ln(x)$, having domain $(0, \infty)$, range $(-\infty, \infty)$, an $x$-intercept at $(1, 0)$, no $y$-intercept, increasing on $(0, \infty)$, no extrema, and end behavior where $f(x) \to -\infty$ as $x \to 0$ and $f(x) \to \infty$ as $x \to \infty$.

Greatest Integer Parent Function

The step function defined by $f(x) = [x]$ (or $f(x) = \text{int}(x)$, $f(x) = \text{floor}(x)$), having domain $(-\infty, \infty)$, range consisting of all integers, $x$-intercepts for all $x$ such that $0 \le x < 1$, $y$-intercept at $(0,0)$, and jump discontinuities at every integer.

Sine Parent Function

The periodic parent function defined by $f(x) = \sin(x)$, having domain $(-\infty, \infty)$, range $[-1, 1]$, $y$-intercept at $(0,0)$, maxima of $1$ at $x = -\frac{3\pi}{2}, \frac{\pi}{2}$, minima of $-1$ at $x = -\frac{\pi}{2}, \frac{3\pi}{2}$, odd symmetry, and end behavior oscillating between $-1$ and $1$ as $x \to \pm\infty$.

Cosine Parent Function

The periodic parent function defined by $f(x) = \cos(x)$, having domain $(-\infty, \infty)$, range $[-1, 1]$, $y$-intercept at $(0, 1)$, maxima of $1$ at $x = -2\pi, 0, 2\pi$, minima of $-1$ at $x = -\pi, \pi$, even symmetry, and end behavior oscillating between $-1$ and $1$ as $x \to \pm\infty$.

Tangent Parent Function

The periodic parent function defined by $f(x) = \tan(x)$, having domain of all real numbers except odd multiples of $\frac{\pi}{2}$, range $(-\infty, \infty)$, intercept at $(0,0)$, odd symmetry, and vertical discontinuities at $x = \frac{(2k+1)\pi}{2}$ where $k \in \mathbb{Z}$.

Logistic Parent Function

The parent function defined by $f(x) = \frac{1}{1 + e^{-x}}$, having domain $(-\infty, \infty)$, range $(0, 1)$, no $x$-intercept, a $y$-intercept at $(0, 0.5)$, increasing on $(-\infty, \infty)$, neither even nor odd symmetry, and end behavior where $f(x) \to 0$ as $x \to -\infty$ and $f(x) \to 1$ as $x \to \infty$.