What is the primary difference between the effect of $f(x) + k$ and $f(x + k)$ on a graph?

$f(x) + k$ is a vertical translation that changes the $y$-coordinates, moving the graph up or down. $f(x + k)$ is a horizontal translation that changes the $x$-coordinates, moving the graph left or right.

How do horizontal and vertical translations affect the asymptotes of a function?

Vertical translations shift horizontal asymptotes ($y=c$) up or down. Horizontal translations shift vertical asymptotes ($x=c$) left or right.

In the transformation $f(x) \to f(x - h) + k$, which variable represents the horizontal shift and which represents the vertical shift?

$h$ represents the horizontal shift (where a positive $h$ moves the graph right) and $k$ represents the vertical shift (where a positive $k$ moves the graph up).

Why is $f(x + 2)$ a shift to the left rather than the right?

To achieve the same output value as the original $f(x)$, the new input $x$ must be 2 units smaller to compensate for the '+2' inside the function. Thus, every point moves 2 units in the negative x-direction.

What common mistake occurs when applying a horizontal translation to a function like $y = x^2 + 5x$?

Students often only replace the first $x$ or add/subtract the constant at the end. To correctly translate horizontally, every instance of $x$ must be replaced with $(x - h)$.

If a graph is translated by the vector $\binom{3}{-4}$, what are the new coordinates of the point $(1, 2)$?

The new coordinates are $(1+3, 2-4)$, which simplifies to $(4, -2)$. The vector indicates a shift of 3 units right and 4 units down.

Define a 'rigid transformation' in the context of graph translations.

A rigid transformation is a change in the position of a graph that does not alter its size or shape. Translations are rigid because the distance between any two points on the graph remains constant.

What is the general equation for a function $f(x)$ translated $h$ units horizontally and $k$ units vertically?

The general equation is $y = f(x - h) + k$. This form accounts for both the horizontal displacement $h$ and the vertical displacement $k$.

What is mapping notation for a translation?

Mapping notation is written as $(x, y) \to (x + h, y + k)$. It describes the transformation of individual points from the original graph to the new graph.

Does a translation change the gradient (slope) of a linear function?

No, a translation preserves the gradient. Since the orientation and shape are unchanged, the slope of a line remains the same after being shifted.

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Translations of Graphs

Summary

A translation is a rigid transformation that shifts every point of a graph a constant distance in a specified direction. This process changes the position of the function on the Cartesian plane while preserving its shape, size, and orientation. Mathematically, translations are represented by adding or subtracting constants from the input variable or the entire function.

1. Definition & Core Concepts

A translation is a geometric transformation that 'slides' a graph without rotating, resizing, or reflecting it. Every point $(x, y)$ on the original graph is moved by the same distance and in the same direction to a new position $(x', y')$ .

Because the shape remains identical, translations are classified as isometries or rigid motions. The resulting graph is congruent to the original, meaning all corresponding angles and side lengths (if applicable) are preserved.

In vector notation, a translation can be described by a vector $\binom{h}{k}$ , where $h$ represents the horizontal displacement and $k$ represents the vertical displacement.

A diagram showing a red parabola f(x) being shifted diagonally to a new blue position f(x-h)+k, illustrated with a translation vector.

2. Vertical Translations

3. Horizontal Translations

4. Underlying Principles

5. Key Distinctions

6. Exam Strategy & Tips

Translations of Graphs

Summary

1. Definition & Core Concepts

In vector notation, a translation can be described by a vector $\binom{h}{k}$ , where $h$ represents the horizontal displacement and $k$ represents the vertical displacement.

A diagram showing a red parabola f(x) being shifted diagonally to a new blue position f(x-h)+k, illustrated with a translation vector.

2. Vertical Translations

Mathematical Form: A vertical translation is represented by the equation $y = f(x) + k$ , where $k$ is a constant.
Direction of Shift: If $k > 0$ , the graph shifts upward by $k$ units. If $k < 0$ , the graph shifts downward by $|k|$ units.
Coordinate Impact: This transformation affects only the $y$ -coordinates of the points. Every point $(x, y)$ on the original graph maps to $(x, y + k)$ .
Asymptotes: If the function has a horizontal asymptote (e.g., $y = L$ ), the new horizontal asymptote will be $y = L + k$ .

3. Horizontal Translations

Mathematical Form: A horizontal translation is represented by the equation $y = f(x - h)$ , where $h$ is a constant.
Direction of Shift: If $h > 0$ , the graph shifts to the right by $h$ units. If $h < 0$ , the graph shifts to the left by $|h|$ units.
Counter-Intuitive Nature: Note that $f(x - 3)$ moves the graph 3 units to the right (positive x-direction), while $f(x + 3)$ moves it 3 units to the left. This is because the 'input' must be adjusted to achieve the same output value as the original function.
Coordinate Impact: This transformation affects only the $x$ -coordinates. Every point $(x, y)$ on the original graph maps to $(x + h, y)$ .

4. Underlying Principles

The principle of substitution governs how equations change. To translate a graph horizontally by $h$ , every instance of $x$ in the original equation is replaced with $(x - h)$ .

For vertical translations, the constant $k$ is added to the entire output of the function. This can be viewed as replacing $y$ with $(y - k)$ , which simplifies to $y = f(x) + k$ .

When both translations are applied simultaneously, the mapping is $(x, y) \to (x + h, y + k)$ , resulting in the general form $y = f(x - h) + k$ .

5. Key Distinctions

Feature	Vertical Translation	Horizontal Translation
Equation Change	$f(x) \to f(x) + k$	$f(x) \to f(x - h)$
Coordinate Change	$(x, y) \to (x, y + k)$	$(x, y) \to (x + h, y)$
Asymptote Affected	Horizontal ( $y = c$ )	Vertical ( $x = c$ )
Intuition	Sign matches direction (+ is up)	Sign is opposite (- is right)

It is vital to distinguish between translating the function and translating the axes. Translating a graph right by $h$ is mathematically equivalent to shifting the y-axis left by $h$ .

6. Exam Strategy & Tips

Check the Vertex/Intercepts: When sketching a translated graph, always identify a 'key point' like a vertex or y-intercept. Apply the translation to that point first to anchor your sketch.
Verify Asymptotes: For rational or exponential functions, check if the translation moves the asymptotes. A vertical shift $k$ always moves a horizontal asymptote, while a horizontal shift $h$ always moves a vertical asymptote.
Substitution Method: In complex equations like $y = x^2 + 2x$ , to shift right by 4, replace every $x$ with $(x - 4)$ . Do not just subtract 4 at the end, as that would be a vertical shift.
Sanity Check: If you shift a graph $f(x) = x^2$ to the right to get $f(x-2)$ , test a point. The original vertex was at $x=0$ . The new vertex should be at $x=2$ . Plugging $x=2$ into $(x-2)^2$ gives $0$ , confirming the shift is correct.

Mathematical Form: A vertical translation is represented by the equation $y = f(x) + k$ , where $k$ is a constant.
Direction of Shift: If $k > 0$ , the graph shifts upward by $k$ units. If $k < 0$ , the graph shifts downward by $|k|$ units.
Coordinate Impact: This transformation affects only the $y$ -coordinates of the points. Every point $(x, y)$ on the original graph maps to $(x, y + k)$ .
Asymptotes: If the function has a horizontal asymptote (e.g., $y = L$ ), the new horizontal asymptote will be $y = L + k$ .

Mathematical Form: A horizontal translation is represented by the equation $y = f(x - h)$ , where $h$ is a constant.
Direction of Shift: If $h > 0$ , the graph shifts to the right by $h$ units. If $h < 0$ , the graph shifts to the left by $|h|$ units.
Counter-Intuitive Nature: Note that $f(x - 3)$ moves the graph 3 units to the right (positive x-direction), while $f(x + 3)$ moves it 3 units to the left. This is because the 'input' must be adjusted to achieve the same output value as the original function.
Coordinate Impact: This transformation affects only the $x$ -coordinates. Every point $(x, y)$ on the original graph maps to $(x + h, y)$ .

The principle of substitution governs how equations change. To translate a graph horizontally by $h$ , every instance of $x$ in the original equation is replaced with $(x - h)$ .

For vertical translations, the constant $k$ is added to the entire output of the function. This can be viewed as replacing $y$ with $(y - k)$ , which simplifies to $y = f(x) + k$ .

When both translations are applied simultaneously, the mapping is $(x, y) \to (x + h, y + k)$ , resulting in the general form $y = f(x - h) + k$ .

Feature	Vertical Translation	Horizontal Translation
Equation Change	$f(x) \to f(x) + k$	$f(x) \to f(x - h)$
Coordinate Change	$(x, y) \to (x, y + k)$	$(x, y) \to (x + h, y)$
Asymptote Affected	Horizontal ( $y = c$ )	Vertical ( $x = c$ )
Intuition	Sign matches direction (+ is up)	Sign is opposite (- is right)

It is vital to distinguish between translating the function and translating the axes. Translating a graph right by $h$ is mathematically equivalent to shifting the y-axis left by $h$ .

Check the Vertex/Intercepts: When sketching a translated graph, always identify a 'key point' like a vertex or y-intercept. Apply the translation to that point first to anchor your sketch.
Verify Asymptotes: For rational or exponential functions, check if the translation moves the asymptotes. A vertical shift $k$ always moves a horizontal asymptote, while a horizontal shift $h$ always moves a vertical asymptote.
Substitution Method: In complex equations like $y = x^2 + 2x$ , to shift right by 4, replace every $x$ with $(x - 4)$ . Do not just subtract 4 at the end, as that would be a vertical shift.
Sanity Check: If you shift a graph $f(x) = x^2$ to the right to get $f(x-2)$ , test a point. The original vertex was at $x=0$ . The new vertex should be at $x=2$ . Plugging $x=2$ into $(x-2)^2$ gives $0$ , confirming the shift is correct.