Precalculus Functions and Graph Transformations

Vocabulary flashcards reviewing basic precalculus functions, properties, continuity, symmetry, and graph transformations based on lecture notes.

Continuous Function

A function whose entire graph can be drawn without lifting your pencil.

Increasing Function

A function whose graph rises from left to right.

Decreasing Function

A function whose graph falls from left to right.

Identity Function

The basic function defined by $f(x) = x$.

Squaring Function

The basic function defined by $f(x) = x^2$.

Cubing Function

The basic function defined by $f(x) = x^3$.

Square Root Function

The basic function defined by $f(x) = \text{sqrt}(x)$ or $f(x) = \frac{1}{2}$ power.

Cube Root Function

The basic function defined by $y = \text{sqrt}[3]{x}$.

Absolute Value Function

The basic function defined by $y = |x|$.

Symmetry (of a Graph)

A property of a graph where folding it results in two halves that coincide exactly.

Even Function

A function where $f(-x) = f(x)$ for all $x$, whose graph is symmetric to the y-axis, and all term exponents are even.

Odd Function

A function where $f(-x) = -f(x)$ for all $x$, whose graph is symmetric to the origin, and all term exponents are odd.

Vertical Stretch

A transformation of $y = c \times f(x)$ that stretches the graph of $f(x)$ vertically when $c > 1$.

Vertical Shrink

A transformation of $y = c \times f(x)$ that shrinks the graph of $f(x)$ vertically when $0 < c < 1$.

Vertical Shift Downward

A transformation caused by subtracting a value outside the function expression, such as $-3$ in $y = x^2 - 3$, which moves the graph down by that many units.

Horizontal Shift Right

A transformation caused by subtracting a value inside the parenthesis/function argument, such as $-3$ in $y = (x - 3)^2$, which moves the graph to the right by that many units.

Reflection Across the x-axis

A transformation created by placing a negative sign outside the square root or absolute value symbol, such as $y = -\text{sqrt}(x)$, flipping the graph upside down across the horizontal axis.

Reflection Across the y-axis

A transformation created by placing a negative sign inside the input variable, such as $y = \text{sqrt}(-x)$, flipping the graph horizontally across the vertical axis.

Symmetry to the Origin

A graphical property where rotating the graph $180^\text{deg}$ around $(0,0)$ yields the same graph, characteristic of odd functions.

Symmetry to the line y = x

A graphical property where folding the graph across the line $y = x$ reflects the two halves perfectly onto each other.

Transformations of y = -\frac{1}{4}|x + 2| - 3

A composite transformation resulting in shifting two units to the left, shrinking vertically by a factor of $\frac{1}{4}$, reflecting across the x-axis, and shifting vertically three units downward.