Square Root Functions and Transformations

A set of vocabulary flashcards covering key definitions, properties, and transformations of square root functions based on Lesson 5.1.1.

Square Root Function

A function defined as $y = \sqrt{x}$, which gives the positive number whose square is $x$ for each non-negative input $x$.

Parent Square Root Function

The function $y = \sqrt{x}$, which is the simplest form and base model of a square root function.

Domain of $y = \sqrt{x}$

The set of non-negative real numbers, $x \ge 0$ or $[0, +\infty)$, because square roots of negative numbers are not defined in the real number system.

Range of $y = \sqrt{x}$

The set of non-negative real numbers, $y \ge 0$ or $[0, +\infty)$.

Intercepts of $y = \sqrt{x}$

The $x$-intercept and $y$-intercept are both at the origin $(0, 0)$.

Graph Shape of $y = \sqrt{x}$

A curve that starts at the origin $(0, 0)$ and curves upward to the right on its domain.

End Behavior of $y = \sqrt{x}$

As $x \rightarrow \infty$, $y \rightarrow \infty$, and as $x \rightarrow 0^+$, $y \rightarrow 0$; the rate of increase decreases as $x$ increases.

Symmetry of Square Root Functions

The function $y = \sqrt{x}$ is neither even nor odd and is not symmetric about the y-axis or origin.

General Form of Transformation

The equation $y = a\sqrt{b(x - h)} + k$, where $a$, $b$, $h$, and $k$ transform the parent function $y = \sqrt{x}$.

Vertical Shift Parameter ($k$)

A constant added or subtracted outside the radical in $y = \sqrt{x} + k$ that shifts the graph up $k$ units if $k > 0$ or down $|k|$ units if $k < 0$.

Horizontal Shift Parameter ($h$)

A constant changed inside the radical in $y = \sqrt{x - h}$ that shifts the graph right $h$ units if $h > 0$ or left $|h|$ units if $h < 0$.

Vertical Stretch and Compression Parameter ($a$)

A multiplier $a > 0$ outside the radical in $y = a\sqrt{x}$ that stretches the graph vertically if $a > 1$ or compresses it vertically if $0 < a < 1$.

Reflection Across the x-axis

A transformation of $y = \sqrt{x}$ given by $y = -\sqrt{x}$ that flips the graph upside down, changing the range to $y \le 0$ or $(-\infty, 0]$.

Reflection Across the y-axis

A transformation achieved by replacing $x$ with $-x$ inside the radical sign in $y = \sqrt{-x}$, which reflects the graph over the y-axis and changes the domain to $x \le 0$ or $(-\infty, 0]$.

Transformations of $y = \sqrt{x} + 2$

A vertical shift of $y = \sqrt{x}$ upward by $2$ units, having domain $[0, +\infty)$ and range $[2, +\infty)$.

Transformations of $y = \sqrt{x} - 4$

A vertical shift of $y = \sqrt{x}$ downward by $4$ units, having domain $[0, +\infty)$ and range $[-4, +\infty)$.

Transformations of $y = \sqrt{x - 2}$

A horizontal shift of $y = \sqrt{x}$ to the right by $2$ units, having domain $[2, +\infty)$ and range $[0, +\infty)$.

Transformations of $y = \sqrt{x + 4}$

A horizontal shift of $y = \sqrt{x}$ to the left by $4$ units, having domain $[-4, +\infty)$ and range $[0, +\infty)$.

Transformations of $y = 2\sqrt{x}$

A vertical stretch of $y = \sqrt{x}$ by a factor of $2$, having domain $[0, +\infty)$ and range $[0, +\infty)$.

Transformations of $y = \frac{1}{2}\sqrt{x}$

A vertical compression of $y = \sqrt{x}$ by a factor of $\frac{1}{2}$, having domain $[0, +\infty)$ and range $[0, +\infty)$.

Combined Transformation: $y = \sqrt{x - 2} + 3$

A square root function shifted right by $2$ units and up by $3$ units, having starting point $(2, 3)$, domain $[2, +\infty)$, and range $[3, +\infty)$.

Combined Transformation: $y = -2\sqrt{x + 4} - 1$

A square root function shifted left by $4$ units, vertically stretched by $2$, reflected across the x-axis, and shifted down by $1$ unit, with starting point $(-4, -1)$, domain $[-4, +\infty)$, and range $(-\infty, -1]$.