Translations of Function Graphs

Definition of Translation: A translation is a transformation that 'slides' a graph horizontally, vertically, or both, without rotating or resizing it. Every point $(x, y)$ on the original curve is moved to a new position $(x', y')$ based on a fixed displacement.

Rigid Motion: Because the internal geometry of the function remains identical, translations are classified as rigid motions. This means that features like the curvature, the distance between stationary points, and the gradient at corresponding points remain unchanged.

Vector Representation: Translations are mathematically described using a translation vector $\begin{pmatrix} p \\ q \end{pmatrix}$. The top value $p$ indicates the horizontal shift (right if positive, left if negative), and the bottom value $q$ indicates the vertical shift (up if positive, down if negative).

Vertical Shift Logic: The transformation $y = f(x) + k$ adds a constant value to the output of the function. This directly changes the $y$-coordinate of every point on the graph, resulting in a vertical displacement of $k$ units.

Horizontal Shift Logic: The transformation $y = f(x + h)$ modifies the input before the function is applied. To achieve the same output value as the original function, the new input $x$ must be adjusted by $-h$, which causes the graph to shift horizontally in the opposite direction of the sign of $h$.

Mapping Notation: A general translation can be expressed as a mapping $(x, y) \to (x + p, y + q)$. This mapping shows that the horizontal component $p$ is added to the $x$-coordinates and the vertical component $q$ is added to the $y$-coordinates.

Applying Vertical Translations: To shift a graph vertically by $k$ units, simply add $k$ to the entire function expression, resulting in $g(x) = f(x) + k$. If $k$ is positive, the graph moves up; if $k$ is negative, the graph moves down.

Applying Horizontal Translations: To shift a graph horizontally by $h$ units, replace every instance of $x$ in the function with $(x - h)$. Note that $y = f(x - 3)$ shifts the graph 3 units to the right, while $y = f(x + 3)$ shifts it 3 units to the left.

Combining Translations: When both horizontal and vertical shifts are required, the function takes the form $y = f(x - p) + q$. This corresponds to a single translation by the vector $\begin{pmatrix} p \\ q \end{pmatrix}$.

Inside vs. Outside: Changes made 'outside' the function brackets (like $+k$) affect the vertical axis, while changes made 'inside' the function argument (like $x+h$) affect the horizontal axis.

Direct vs. Inverse: Vertical translations are direct (adding 5 moves it up 5), whereas horizontal translations are inverse (adding 5 moves it left 5, which is the negative $x$ direction).

Verify with Key Points: Always check the translation by applying it to a known point, such as the $y$-intercept or a turning point. If the original turning point is $(2, 5)$, a translation by $\begin{pmatrix} 3 \\ -1 \end{pmatrix}$ must result in a new turning point at $(5, 4)$.

Asymptote Awareness: Remember that asymptotes are lines, not just points, and they must be translated along with the curve. A horizontal asymptote $y = 2$ shifted up by 3 becomes $y = 5$, while a vertical asymptote $x = 1$ shifted left by 4 becomes $x = -3$.

Order of Operations: While the order of multiple translations does not matter (they are commutative), be extremely careful when combining translations with stretches or reflections. It is often safest to perform horizontal translations first if they are inside the function argument.