How do I know if I'm shifting left or right?

In y = f(x - h) + k, if h is positive, shift RIGHT; if h is negative, shift LEFT. This is counterintuitive because of the minus sign in the formula, but remember: (x - 2) shifts right, and (x - (-3)) = (x + 3) shifts left.

What's the difference between y = f(x - h) and y = f(x) - h?

These are completely different transformations. y = f(x - h) changes the input, creating a horizontal shift; y = f(x) - h changes the output, creating a vertical shift. Always apply horizontal shifts inside the function and vertical shifts outside.

Can I translate any function, or just parabolas and circles?

You can translate any function using y = f(x - h) + k: parabolas, circles, absolute value, square roots, exponential functions, trigonometric functions, and more. The formula is completely general and works for any function f.

When I translate a complex function like f(x) = 2x^2 - 3x + 1, do I have to expand everything, or can I leave it as y = 2(x - h)^2 - 3(x - h) + 1 + k?

You can leave it in either form, depending on what the problem asks for. If the problem requests a simplified equation, expand and combine like terms. If it only asks for the translated equation, the form y = 2(x - h)^2 - 3(x - h) + 1 + k is acceptable and often clearer for showing the transformation.

Translating a Graph

Translate graphs using y = f(x - h) + k to shift any function horizontally by h and vertically by k units, needed when repositioning known curves.

y = f(x - h) + k

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What each symbol means

What Translating a Graph takes

Symbol	Meaning
$f$	The original function being translated; $f$ can represent any function such as $x^2$ , $\|x\|$ , $\sqrt{x}$ , or a complex expression. If you forget which function $f$ represents, you will substitute the wrong rule into the formula.
$h$	The horizontal shift amount in units; positive $h$ shifts the graph RIGHT, and negative $h$ shifts it LEFT. The sign is crucial—confusing $h$ reverses the direction of your translation.
$k$	The vertical shift amount in units; positive $k$ shifts the graph UP, and negative $k$ shifts it DOWN. Note that the sign of $k$ is applied directly in the formula (positive $k$ adds, negative $k$ subtracts).
$x$	The input variable (independent variable) of the original function; when translating, every instance of $x$ must be replaced with $(x - h)$ . Partial substitution in multi-term functions produces incorrect equations.
$y$	The output variable (dependent variable) representing the result of the function; in $y = f(x - h) + k$ , the $y$ value reflects the result after both the horizontal and vertical transformations have been applied.

When to use it

Use this when a problem asks you to shift or move a graph you already know, or when a function must be adjusted based on changes to its coordinates.

Level

Usually taught in: Algebra II

Worked examples

1. Translate a parabola right and up

Problem

Translate the graph of

y = x^2

right 2 units and up 3 units. Write the equation of the translated graph.

$y = f(x - h) + k$
Start with the translation formula, where $f(x) = x^2$ , $h = 2$ (for the rightward shift), and $k = 3$ (for the upward shift).
$y = (x - 2)^{2} + 3$
Substitute $h = 2$ and $k = 3$ into the formula, replacing $x$ with $(x - 2)$ and adding 3.
$y = x^{2} - 4x + 4 + 3$
Expand $(x - 2)^2$ using the formula $(a - b)^2 = a^2 - 2ab + b^2$ , which gives $x^2 - 4x + 4$ .
$y = x^{2} - 4x + 7$
Combine the constants: $4 + 3 = 7$ .

Answer: $y = x^{2} - 4x + 7$

This example applies the basic translation formula to a quadratic function. Shifting right by 2 units means using $(x - 2)$ , and shifting up by 3 units means adding 3 to the result. The expanded form shows the final parabola equation in standard form.

2. Translate an absolute value function left and down

Problem

Write the equation of the graph of

y = |x|

translated 3 units left and 2 units down.

$h = -3, \quad k = -2$
Left movement means a negative shift, so $h = -3$ ; downward movement means a negative vertical shift, so $k = -2$ .
$y = f(x - h) + k = f(x - (-3)) + (-2)$
Substitute $h = -3$ and $k = -2$ into the translation formula; subtracting a negative is the same as adding.
$y = f(x + 3) - 2$
Simplify $x - (-3) = x + 3$ and write $+(-2)$ as $-2$ .
$y = |x + 3| - 2$
Since the original function is $f(x) = |x|$ , replace $x$ with $(x + 3)$ inside the absolute value.

Answer: $y = |x + 3| - 2$

This example uses negative values for $h$ and $k$ . The key insight is that left shifts result in adding inside the function (because $x - (-3) = x + 3$ ), while down shifts subtract from the result. The absolute value function transforms exactly like any other function.

3. Translate a quadratic function in a game-scoring context

Problem

In a video game, a player's score on level

n

is given by

s(n) = n^2 + 2n + 10

. After an update, the level counter starts at 5 (instead of 0), and all scores increase by 50 points. Write the new score function.

$h = 5, \quad k = 50$
The level counter starting at 5 represents a horizontal shift of $h = 5$ to the right; the 50-point bonus is a vertical shift of $k = 50$ upward.
$s_\text{new}(n) = (n - 5)^{2} + 2(n - 5) + 10 + 50$
Apply the formula: replace each instance of $n$ with $(n - 5)$ in $s(n) = n^2 + 2n + 10$ , then add the vertical shift $k = 50$ .
$(n - 5)^{2} = n^{2} - 10n + 25$
Expand $(n - 5)^2$ using $(a - b)^2 = a^2 - 2ab + b^2$ ; here $a = n$ and $b = 5$ , so $n^2 - 2(5)n + 25 = n^2 - 10n + 25$ .
$2(n - 5) = 2n - 10$
Distribute the 2 across the binomial: $2 \cdot n - 2 \cdot 5 = 2n - 10$ .
$s_\text{new}(n) = n^{2} - 10n + 25 + 2n - 10 + 10 + 50$
Substitute the expanded expressions from the previous steps into the full formula.
$s_\text{new}(n) = n^{2} + (-10 + 2)n + (25 - 10 + 10 + 50)$
Collect like terms: combine all $n^2$ terms, all $n$ terms ( $-10n + 2n$ ), and all constants.
$s_\text{new}(n) = n^{2} - 8n + 75$
Simplify: $-10 + 2 = -8$ and $25 - 10 + 10 + 50 = 75$ .

Answer: $s_\text{new}(n) = n^{2} - 8n + 75$

This problem demonstrates that translation applies to any function, not just simple ones. Understanding that 'the level counter starts at 5' means replacing the variable with $(n - 5)$ , and carefully expanding all instances of $n$ in multi-term functions, are essential skills for word-problem applications of translation.

Common mistakes

Where Translating a Graph usually goes wrong

The mistake	Why it is wrong	The fix
To shift the graph right 2 units, I write $y = f(x + 2)$ because the shift amount is positive.	The formula $y = f(x - h) + k$ has a minus sign built in, so positive $h$ in $(x - h)$ produces a rightward shift; using $x + 2$ would shift LEFT, not right.	Use $y = f(x - 2)$ for a rightward shift of 2 units; in the formula, positive $h$ moves the graph to the right.
I translated the graph right 3 and down 5, so I wrote $y = f(x - 3)$ .	The vertical shift is completely missing. The formula $y = f(x - h) + k$ requires both the horizontal component $(x - h)$ and the vertical component $k$ .	Write both transformations: $y = f(x - 3) + (-5)$ , or equivalently $y = f(x - 3) - 5$ , to include the downward shift.
For $f(x) = x^2 + 3x$ , translating right 1 and up 2 gives $y = (x-1)^2 + 3x + 2$ .	The term $3x$ was not translated; every instance of $x$ in the original function must be replaced with $(x - 1)$ , not left unchanged.	Replace all instances of $x$ consistently: $y = (x-1)^2 + 3(x-1) + 2$ , then expand and simplify to get $y = x^2 + x + 1$ .

Tips and when to use something else

The signs in $y = f(x - h) + k$ can be counterintuitive: positive $h$ shifts the graph RIGHT, not left, and positive $k$ shifts it UP. The minus sign inside the function is part of the formula's convention.
Always substitute $(x - h)$ for every instance of $x$ in the original function, even if it appears in multiple terms or within exponents; partial substitution is a common source of errors.
In word problems, identify what the independent variable represents (level number, time, distance, etc.) before determining $h$ and $k$ , as the shift amounts may be properties of the real-world scenario.
To shift and flip a graph simultaneously, use Reflecting a Graph and this translation method separately—apply reflections first, then translate, unless the problem specifies a different order.

Related formulas

Equation of a ParabolaThe equation (x - h)^2 = 4p(y - k) describes a parabola with vertex at (h, k) and tells you which direction it opens and how wide it is.Equation of a CircleDescribes a circle using its center and radius; essential for finding circle equations, checking if points lie on circles, or solving circle geometry problems.Reflecting a GraphReflecting a graph flips it across an axis—use it when you need to reverse a function's behavior, either top-to-bottom or left-to-right.Equation of an EllipseEquation of an Ellipse describes an oval-shaped curve given its center, horizontal stretch, and vertical stretch; use it to model or graph any elliptical shape.

Frequently asked questions

How do I know if I'm shifting left or right?: In $y = f(x - h) + k$ , if $h$ is positive, shift RIGHT; if $h$ is negative, shift LEFT. This is counterintuitive because of the minus sign in the formula, but remember: $(x - 2)$ shifts right, and $(x - (-3)) = (x + 3)$ shifts left.
What's the difference between $y = f(x - h)$ and $y = f(x) - h$ ?: These are completely different transformations. $y = f(x - h)$ changes the input, creating a horizontal shift; $y = f(x) - h$ changes the output, creating a vertical shift. Always apply horizontal shifts inside the function and vertical shifts outside.
Can I translate any function, or just parabolas and circles?: You can translate any function using $y = f(x - h) + k$ : parabolas, circles, absolute value, square roots, exponential functions, trigonometric functions, and more. The formula is completely general and works for any function $f$ .
When I translate a complex function like $f(x) = 2x^2 - 3x + 1$ , do I have to expand everything, or can I leave it as $y = 2(x - h)^2 - 3(x - h) + 1 + k$ ?: You can leave it in either form, depending on what the problem asks for. If the problem requests a simplified equation, expand and combine like terms. If it only asks for the translated equation, the form $y = 2(x - h)^2 - 3(x - h) + 1 + k$ is acceptable and often clearer for showing the transformation.

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Reviewed 2026-09-18