Regents & Unit Test Master Study Guide: Functions, Transformations & High-Yield Pitfalls

A comprehensive vocabulary review of function transformations, dilations, reflections, coordinate shifts, and high-yield Regents pitfalls.

Horizontal Shift

A transformation that occurs when a constant is added to or subtracted from the input inside the function notation: $f(x \pm h)$.

Vertical Shift

A transformation that occurs when a constant is added to or subtracted from the output outside the function notation: $f(x) \pm k$.

Function Notation $f(x \pm h)$

The notation indicating a horizontal shift where a constant is added to or subtracted from the input inside parentheses.

Function Notation $f(x) \pm k$

The notation indicating a vertical shift where a constant is added to or subtracted from the output outside the function notation.

Horizontal Shift Right Notation $f(x - h)$

The algebraic notation that shifts the graph of a function to the right by $h$ units.

Horizontal Shift Left Notation $f(x + h)$

The algebraic notation that shifts the graph of a function to the left by $h$ units.

Vertical Shift Up Notation $f(x) + k$

The algebraic notation that shifts the graph of a function up by $k$ units.

Vertical Shift Down Notation $f(x) - k$

The algebraic notation that shifts the graph of a function down by $k$ units.

Regents Pitfall for Horizontal Shifts

The high-yield trap warning students to always reverse the sign inside the parentheses for horizontal shifts.

Input inside the Function Notation

The position within the function parentheses where adding or subtracting a constant results in a horizontal shift.

Output outside the Function Notation

The position outside the function parentheses where adding or subtracting a constant results in a vertical shift.

Right Shift of $h$ Units

The graphical displacement caused by subtracting a positive constant $h$ inside the input: $f(x - h)$.

Left Shift of $h$ Units

The graphical displacement caused by adding a positive constant $h$ inside the input: $f(x + h)$.

Upward Shift of $k$ Units

The graphical displacement caused by adding a constant $k$ to the output outside the function: $f(x) + k$.

Downward Shift of $k$ Units

The graphical displacement caused by subtracting a constant $k$ from the output outside the function: $f(x) - k$.

Vertical Shift Coordinate Effect on $x$

In a vertical shift $y = f(x) + k$, the $x$-coordinates remain unchanged.

Vertical Shift Coordinate Effect on $y$

In a vertical shift $y = f(x) + k$, $k$ is added directly to the $y$-coordinates.

Original Minimum Point in Study Guide Example

The point $(-2, -4)$ given as the minimum point on the original curve $y = f(x)$.

Transformed Minimum Point in Study Guide Example

The point $(-2, 3)$ obtained by shifting the minimum point $(-2, -4)$ of $y = f(x)$ by $+7$ units vertically.

Example Transformation $y = f(x) + 7$

The vertical shift example that adds $7$ to the $y$-coordinates of $y = f(x)$, moving $(-2, -4)$ to $(-2, 3)$.

Dilations

Transformations of a function that cause stretches and compressions either vertically or horizontally.

Vertical Stretches/Compressions Equation

The formula $g(x) = a \cdot f(x)$ where the function output is multiplied by a constant $a$.

Vertical Stretch Condition

The condition $a > 1$ in $g(x) = a \cdot f(x)$, which causes the function to undergo a vertical stretch by a factor of $a$.

Vertical Stretch Factor

The scaling factor $a$ by which a function stretches vertically when $a > 1$ in $g(x) = a \cdot f(x)$.

Vertical Compression Condition

The condition $0 < a < 1$ in $g(x) = a \cdot f(x)$, which causes the function to undergo a vertical compression (shrink) by a factor of $a$.

Vertical Compression (Shrink)

The transformation that scales the vertical height of a function down by a factor of $a$ when $0 < a < 1$ in $g(x) = a \cdot f(x)$.

Vertical Compression Factor

The scaling factor equal to $a$ by which a function compresses vertically when $0 < a < 1$ in $g(x) = a \cdot f(x)$.

Horizontal Stretches/Compressions Equation

The formula $g(x) = f(b \cdot x)$ where the input variable is multiplied by a constant $b$.

Horizontal Compression Condition

The condition $b > 1$ in $g(x) = f(b \cdot x)$, which causes the function to undergo a horizontal compression by a factor of $\frac{1}{b}$.

Horizontal Compression Factor

The factor equal to $\frac{1}{b}$ by which a graph is compressed horizontally when $b > 1$ in $g(x) = f(b \cdot x)$.

Horizontal Stretch Condition

The condition $0 < b < 1$ in $g(x) = f(b \cdot x)$, which causes the function to undergo a horizontal stretch by a factor of $\frac{1}{b}$.

Horizontal Stretch Factor

The factor equal to $\frac{1}{b}$ by which a graph is stretched horizontally when $0 < b < 1$ in $g(x) = f(b \cdot x)$.

Reflection across the $x$-axis

A vertical reflection of the function across the horizontal axis, defined by $g(x) = -f(x)$.

Vertical Reflection

A reflection across the $x$-axis produced algebraically by placing a negative sign outside the function: $g(x) = -f(x)$.

Vertical Reflection Equation

The algebraic equation $g(x) = -f(x)$.

Reflection across the $y$-axis

A horizontal reflection of the function across the vertical axis, defined by $g(x) = f(-x)$.

Horizontal Reflection

A reflection across the $y$-axis produced algebraically by placing a negative sign inside the function input: $g(x) = f(-x)$.

Horizontal Reflection Equation

The algebraic equation $g(x) = f(-x)$.

Parameter $a$ in $g(x) = a \cdot f(x)$

The coefficient multiplying $f(x)$ vertically that determines whether a stretch ($a > 1$) or compression ($0 < a < 1$) occurs.

Parameter $b$ in $g(x) = f(b \cdot x)$

The coefficient multiplying the input $x$ horizontally that determines whether a compression ($b > 1$) or stretch ($0 < b < 1$) occurs.

The Transformation Order-of-Operations Trap

A major Regents topic highlighted in the guide involving expressions such as $y_1 = k \cdot f(x) + c$.

Expression $y_1 = k \cdot f(x) + c$

The transformed function equation examined under the Transformation Order-of-Operations Trap.

Parameter $c$ in $y_1 = k \cdot f(x) + c$

The constant added outside the function notation representing a vertical shift.

Multiplier $k$ in $y_1 = k \cdot f(x) + c$

The vertical scaling multiplier positioned outside the function notation multiplying $f(x)$.

Constant $h$

The value added to or subtracted from the input variable inside $f(x \pm h)$ to shift the curve horizontally.

Constant $k$ in $f(x) \pm k$

The value added to or subtracted from the function output to shift the curve vertically.

Sign Reversal Rule for Horizontal Shifts

The specific instruction to reverse the operation sign inside the parentheses when evaluating horizontal movements.

Function Input

The argument inside the parentheses of $f(x)$, where modifications create horizontal transformations.

Function Output

The value of $f(x)$, where outside modifications create vertical transformations.

Horizontal Compression

The dilation that shrinks a graph horizontally by a factor of $\frac{1}{b}$ when $b > 1$ in $g(x) = f(b \cdot x)$.

Horizontal Stretch

The dilation that widens a graph horizontally by a factor of $\frac{1}{b}$ when $0 < b < 1$ in $g(x) = f(b \cdot x)$.

Vertical Stretch

The dilation that elongates a graph vertically by a factor of $a$ when $a > 1$ in $g(x) = a \cdot f(x)$.

Vertical Shrink

An alternate term for vertical compression, occurring when $0 < a < 1$ in $g(x) = a \cdot f(x)$.

Negative Coefficient Outside Function

The algebraic placement of a negative sign in front of $f(x)$ resulting in a vertical reflection across the $x$-axis.

Negative Coefficient Inside Function Input

The algebraic placement of a negative sign in front of $x$ inside $f(x)$ resulting in a horizontal reflection across the $y$-axis.

Minimum Point

A coordinate point representing a lowest value on a function, such as $(-2, -4)$ on $y = f(x)$.

Condition $b > 1$

The mathematical requirement for an input multiplier $b$ to induce a horizontal compression by $\frac{1}{b}$.

Condition $0 < b < 1$

The mathematical requirement for an input multiplier $b$ to induce a horizontal stretch by $\frac{1}{b}$.

Condition $a > 1$

The mathematical requirement for an output multiplier $a$ to induce a vertical stretch by $a$.

Condition $0 < a < 1$

The mathematical requirement for an output multiplier $a$ to induce a vertical compression by $a$.

Horizontal Dilation Factor

The scaling factor $\frac{1}{b}$ applied to the horizontal dimensions of the graph in $g(x) = f(b \cdot x)$.

Vertical Dilation Factor

The scaling factor $a$ applied directly to the vertical dimensions of the graph in $g(x) = a \cdot f(x)$.

Unchanged Coordinate in Vertical Shift

The $x$-coordinate of any point on $y = f(x)$, which stays constant when forming $y = f(x) + k$.

Modified Coordinate in Vertical Shift

The $y$-coordinate of any point on $y = f(x)$, to which $k$ is directly added when forming $y = f(x) + k$.

Initial Coordinate $-2$

The $x$-coordinate of the minimum point on $y = f(x)$, which remains $-2$ on $y = f(x) + 7$.

Initial Coordinate $-4$

The original $y$-coordinate of the minimum point on $y = f(x)$ before the vertical shift of $+7$.

Resulting Coordinate $3$

The new $y$-coordinate of the minimum point on $y = f(x) + 7$, calculated from $-4 + 7$.

Shift Value $+7$ in Example

The constant $k = 7$ added to $f(x)$ that moves the minimum point $(-2, -4)$ up to $(-2, 3)$.

Horizontal Axis of Reflection

The $x$-axis, across which a function is flipped during a vertical reflection $g(x) = -f(x)$.

Vertical Axis of Reflection

The $y$-axis, across which a function is flipped during a horizontal reflection $g(x) = f(-x)$.

Upward Shift Transformation

A translation produced by adding a positive constant to the function output: $f(x) + k$.

Downward Shift Transformation

A translation produced by subtracting a positive constant from the function output: $f(x) - k$.

Leftward Shift Transformation

A translation produced by adding a positive constant to the function input: $f(x + h)$.

Rightward Shift Transformation

A translation produced by subtracting a positive constant from the function input: $f(x - h)$.

Direct Coordinate Addition

The arithmetic process of adding $k$ straight to the $y$-value for points under $y = f(x) + k$.

Coordinate Invariance

The property where the $x$-coordinates are unaffected during purely vertical shifts of a function.

Vertical Dilation Factor for $a > 1$

The factor $a$ representing the extent of vertical stretch applied to $f(x)$.

Vertical Dilation Factor for $0 < a < 1$

The factor $a$ representing the extent of vertical compression (shrink) applied to $f(x)$.

Horizontal Dilation Factor for $b > 1$

The factor $\frac{1}{b}$ representing the extent of horizontal compression applied to $f(x)$.

Horizontal Dilation Factor for $0 < b < 1$

The factor $\frac{1}{b}$ representing the extent of horizontal stretch applied to $f(x)$.

Formula $g(x) = -f(x)$

The function definition indicating a reflection across the $x$-axis.

Formula $g(x) = f(-x)$

The function definition indicating a reflection across the $y$-axis.

High-Yield Regents Traps

Critical transformation concepts, such as sign changes and order of operations, where students commonly make errors.

Sign Reversal in $f(x - h)$

The counterintuitive transformation rule where a minus sign produces movement to the positive (right) direction.

Sign Reversal in $f(x + h)$

The counterintuitive transformation rule where a plus sign produces movement to the negative (left) direction.

Vertical Shift Sign Preservation

The property that vertical shifts maintain their sign: $+k$ shifts up and $-k$ shifts down.

Horizontal Shift Sign Inversion

The property that horizontal shifts invert their sign: $-h$ shifts right and $+h$ shifts left.

Dilation Property for $b > 1$

The counterintuitive horizontal compression behavior that occurs despite multiplying the input by a number greater than $1$.

Dilation Property for $0 < b < 1$

The counterintuitive horizontal stretch behavior that occurs when multiplying the input by a number between $0$ and $1$.

Dilation Property for $a > 1$

The intuitive vertical stretch behavior that occurs when multiplying the output by a number greater than $1$.

Dilation Property for $0 < a < 1$

The intuitive vertical compression (shrink) behavior that occurs when multiplying the output by a number between $0$ and $1$.

Negative Sign in Vertical Reflection

The negation applied to the output of the function ($-f(x)$) to invert values vertically.

Negative Sign in Horizontal Reflection

The negation applied to the input of the function ($f(-x)$) to invert values horizontally.

Coordinate Transformation for $y = f(x) + k$

The rule mapping points $(x, y)$ on $y = f(x)$ to $(x, y + k)$ on $y = f(x) + k$.

Minimum Point Shift in Example

The change in position from $(-2, -4)$ to $(-2, 3)$ under the vertical shift $y = f(x) + 7$.

Shrink

A synonym given in the notes for vertical compression when $0 < a < 1$ in $g(x) = a \cdot f(x)$.

Scaling by Factor $\frac{1}{b}$

The horizontal scaling factor applied to a function when the input variable $x$ is multiplied by $b$.

Scaling by Factor $a$

The vertical scaling factor applied to a function when the entire function $f(x)$ is multiplied by $a$.

Transformed Function Notation $g(x)$

The notation used to denote the new function resulting from stretches, compressions, or reflections of $f(x)$.

Transformed Function Notation $y_1$

The notation used in the guide to denote the combined transformation function $y_1 = k \cdot f(x) + c$.