Parent Functions and Their Properties

Vocabulary review flashcards covering parent functions, including their equations, symmetry, domains, ranges, end behaviors, critical points, and asymptotes.

Constant Function

The parent function $y = C$ (e.g., $y = 2$) with Even symmetry, Domain: $(-\infty, \infty)$, Range: $\{y: y = C\}$, End Behavior: as $x \rightarrow -\infty, y \rightarrow C$ and as $x \rightarrow \infty, y \rightarrow C$, and Critical points: $(-1, C)$, $(0, C)$, $(1, C)$.

Linear Function

The parent function $y = x$ with Odd symmetry, Domain: $(-\infty, \infty)$, Range: $(-\infty, \infty)$, End Behavior: as $x \rightarrow -\infty, y \rightarrow -\infty$ and as $x \rightarrow \infty, y \rightarrow \infty$, and Critical points: $(-1, -1)$, $(0, 0)$, $(1, 1)$.

Absolute Value Function

The parent function $y = |x|$ with Even symmetry, Domain: $(-\infty, \infty)$, Range: $[0, \infty)$, End Behavior: as $x \rightarrow -\infty, y \rightarrow \infty$ and as $x \rightarrow \infty, y \rightarrow \infty$, and Critical points: $(-1, 1)$, $(0, 0)$, $(1, 1)$.

Quadratic Function

The parent function $y = x^2$ with Even symmetry, Domain: $(-\infty, \infty)$, Range: $[0, \infty)$, End Behavior: as $x \rightarrow -\infty, y \rightarrow \infty$ and as $x \rightarrow \infty, y \rightarrow \infty$, and Critical points: $(-1, 1)$, $(0, 0)$, $(1, 1)$.

Square Root Function

The radical parent function $y = \sqrt{x}$ with Neither symmetry, Domain: $[0, \infty)$, Range: $[0, \infty)$, End Behavior: as $x \rightarrow 0, y \rightarrow 0$ and as $x \rightarrow \infty, y \rightarrow \infty$, and Critical points: $(0, 0)$, $(1, 1)$, $(4, 2)$.

Cubic Function

The parent function $y = x^3$ with Odd symmetry, Domain: $(-\infty, \infty)$, Range: $(-\infty, \infty)$, End Behavior: as $x \rightarrow -\infty, y \rightarrow -\infty$ and as $x \rightarrow \infty, y \rightarrow \infty$, and Critical points: $(-1, -1)$, $(0, 0)$, $(1, 1)$.

Cube Root Function

The parent function $y = \sqrt[3]{x}$ with Odd symmetry, Domain: $(-\infty, \infty)$, Range: $(-\infty, \infty)$, End Behavior: as $x \rightarrow -\infty, y \rightarrow -\infty$ and as $x \rightarrow \infty, y \rightarrow \infty$, and Critical points: $(-1, -1)$, $(0, 0)$, $(1, 1)$.

Exponential Function

The parent function $y = b^x, b > 1$ (e.g., $y = 2^x$) with Neither symmetry, Domain: $(-\infty, \infty)$, Range: $(0, \infty)$, End Behavior: as $x \rightarrow -\infty, y \rightarrow 0$ and as $x \rightarrow \infty, y \rightarrow \infty$, Asymptote: $y = 0$, and Critical points: $(-1, \frac{1}{b})$, $(0, 1)$, $(1, b)$.

Logarithmic Function

The parent function $y = \log_b(x), b > 1$ (e.g., $y = \log_2(x)$) with Neither symmetry, Domain: $(0, \infty)$, Range: $(-\infty, \infty)$, End Behavior: as $x \rightarrow 0, y \rightarrow -\infty$ and as $x \rightarrow \infty, y \rightarrow \infty$, Asymptote: $x = 0$, and Critical points: $(\frac{1}{b}, -1)$, $(1, 0)$, $(b, 1)$.

Rational Function (Inverse)

The parent function $y = \frac{1}{x}$ with Odd symmetry, Domain: $(-\infty, 0) \cup (0, \infty)$, Range: $(-\infty, 0) \cup (0, \infty)$, End Behavior: as $x \rightarrow -\infty, y \rightarrow 0$ and as $x \rightarrow \infty, y \rightarrow 0$, Asymptotes: $y = 0$ and $x = 0$, and Critical points: $(-1, -1)$, $(1, 1)$.

Greatest Integer Function

The parent function $y = \text{int}(x) = [x]$ with Neither symmetry.