AP®MathsCollege BoardPrecalculusRevision NotesPolynomial & Rational FunctionsTransformations of FunctionsVertical & Horizontal Translations

Vertical & Horizontal Translations (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Vertical translations

What are graph translations?

When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations
With a translation the shape, size, and orientation of the graph remain unchanged
- the graph is merely shifted (up or down, left or right) in the xy plane

Graph of sine waves intersecting at origin on x and y axes, with arrows indicating vertical and horizontal shifts of the central black wave. — **Translations of a graph**

What is a vertical translation?

If you start with a function $f (x)$ and add a constant $k$ to the output
- the new function $g (x) = f (x) + k$ is called an additive transformation of $f$
The effect on the graph is a vertical translation (shift)
- every point on the graph moves up or down by $k$ units
  - If $k > 0$ , the graph shifts up by $k$ units
  - If $k < 0$ , the graph shifts down by $| k |$ units
The shape of the graph does not change
- it is simply moved vertically
E.g. for a function $f (x)$
- $g (x) = f (x) + 1$ shifts the entire graph of $f$ up by 1 unit
- $h (x) = f (x) - 4$ shifts it down by 4 units

Graph showing a vertical translation of a parabola. Original curve y = f(x) , point (2, -3); translated curve y = f(x) + 1 , point (2, -2). — **Example of a vertical translation**

How does a vertical translation affect key features?

Every output value changes by $k$ , so
- $y$ -intercept
  - changes by $k$
- Local maxima and minima
  - their $y$ -values shift by $k$
  - but the $x$ -values where they occur stay the same
- Horizontal asymptotes
  - shift by $k$
    - e.g. if the original asymptote is $y = 0$ , the new one is $y = k$
- Zeros ( $x$ -intercepts)
  - generally change, because the outputs for each $x$ value are different
Features that depend only on $x$ -values are not affected:
- Vertical asymptotes stay in the same place
- Intervals of increase/decrease stay the same
- Concavity intervals (and the concavity on them) stay the same

How do I find the analytical form of a vertical translation?

$f (x) + k$ means 'add $k$ to the entire function'
- So if you have an expression for $f (x)$ , simply add $k$ to it to get the analytical form of the transformed function
- Simplify if necessary
E.g. if $f (x) = x^{2} - 3 x + 4$
- Then to find $g (x) = f (x) + 2$
  - $g (x) = x^{2} - 3 x + 4 + 2 = x^{2} - 3 x + 6$

Horizontal translations

What is a horizontal translation?

If you start with a function $f (x)$ and add a constant $h$ to the input before the function is applied
- the new function $g (x) = f (x + h)$ is also called an additive transformation of $f$
  - i.e., vertical and horizontal translations are both types of additive transformation
The effect on the graph is a horizontal translation (shift)
- every point on the graph moves left or right
  - If $h > 0$ , the graph shifts to the left by $h$ units
  - If $h < 0$ , the graph shifts to the right by $| h |$ units
- Note the counterintuitive direction
  - adding a positive number inside the function moves the graph to the left, not the right

The shape of the graph does not change
- it is simply moved horizontally
E.g. for a function $f (x)$
- $g (x) = f (x + 3)$ shifts the entire graph of $f$ left by 3 units
- $h (x) = f (x - 2)$ shifts it right by 2 units

Graph showing horizontal translation of a parabola y=f(x) right 2 units to y=f(x−2), moving vertex from (2,−3) to (4,−3), x-coordinates shift. — **Example of a horizontal translation**

Examiner Tips and Tricks

The most common error with horizontal translations is getting the direction wrong. Remember: $f (x + h)$ shifts the graph by $- h$ units (i.e. in the opposite direction to the sign of $h$ ). So $f (x + 3)$ shifts left by 3, and $f (x - 2)$ shifts right by 2.

How does a horizontal translation affect key features?

Every input value is shifted, so features that depend on $x$ -values move:
- Zeros ( $x$ -intercepts)
  - shift left or right by the same amount as the graph
- Local maxima and minima
  - their $x$ -values shift, but the $y$ -values stay the same
- Vertical asymptotes
  - shift horizontally the same way as the rest of the graph
  - e.g. if the original asymptote is $x = 0$ , the new one is $x = - h$
- Intervals of increase/decrease
  - shift horizontally (same pattern, different $x$ -values)
- Concavity intervals
  - shift horizontally (same pattern, different $x$ -values)
  - though the concavity on those intervals stays the same
Features that depend only on the overall shape or on the $y$ -values are not affected:
- Horizontal asymptotes stay at the same $y$ -value

Graph showing translations of y=f(x), with shifts down by 2 units and left by 3 units. Asymptote and point changes depicted with annotated arrows. — **Effects of vertical and horizontal translations**

How do I find the analytical form of a horizontal translation?

$f (x + h)$ means 'replace $x$ everywhere in $f (x)$ with $x + h$ '
- So if you have an expression for $f (x)$ , simply put $x + h$ everywhere that $x$ is to get the analytical form of the transformed function
- Simplify if necessary
E.g. if $f (x) = x^{2} - 3 x + 4$
- Then to find $g (x) = f (x + 2)$
  - $\begin{array}{rcl} g (x) & = & {(x + 2)}^{2} - 3 (x + 2) + 4 = x^{2} + 4 x + 4 - 3 x - 6 + 4 = x^{2} + x + 2 \end{array}$

How can you tell vertical and horizontal translations apart?

Vertical translation $g (x) = f (x) + k$
- the constant $k$ is added outside the function
  - it changes the output
Horizontal translation $g (x) = f (x + h)$
- the constant $h$ is added inside the function (to the input)
  - it changes which input produces each output
A common mistake is to confuse the two
- If you see a constant being added to $f (x)$ , that's vertical
- If you see a constant being added to $x$ inside $f$ , that's horizontal

Worked Example

The function $f$ is given by $f (x) = - 3 x^{2} + 2 x - 5$ . The graph of which of the following functions is the image of the graph of $f$ after a horizontal translation of the graph of $f$ by 2 units?

(A) $p (x) = - 3 x^{2} + 2 x - 3$ , because this is an additive transformation of $f$ that results from adding 2 to $f (x)$ .

(B) $q (x) = - 3 x^{2} - 10 x - 13$ , because this is an additive transformation of $f$ that results from adding 2 to each input value $x$ .

(C) $r (x) = - 3 x^{2} + 2 x - 7$ , because this is an additive transformation of $f$ that results from subtracting 2 from $f (x)$ .

(D) $s (x) = - 3 x^{2} + 14 x - 21$ , because this is an additive transformation of $f$ that results from subtracting 2 from each input value $x$ .

Answer:

'A horizontal translation of the graph of $f$ by 2 units' means a translation in the positive $x$ direction; i.e., 'to the right'

Adding or subtracting a constant from $f (x)$ causes a vertical translation, not a horizontal one

So you can rule out (A), which is a vertical translation up by 2 units
and also (C), which is a vertical translation down by 2 units

Adding 2 to each input value gives

$\begin{array}{rcl} - 3 {(x + 2)}^{2} + 2 (x + 2) - 5 & = & - 3 (x^{2} + 4 x + 4) + 2 x + 4 - 5 \\ = & - 3 x^{2} - 12 x - 12 + 2 x + 4 - 5 \\ = & - 3 x^{2} - 10 x - 13 \end{array}$

That is function $q$ in option (B)
But adding 2 to each input value causes a horizontal translation by -2 units (i.e., to the left)

Subtracting 2 from each input value gives

$\begin{array}{rcl} - 3 {(x - 2)}^{2} + 2 (x - 2) - 5 & = & - 3 (x^{2} - 4 x + 4) + 2 x - 4 - 5 \\ = & - 3 x^{2} + 12 x - 12 + 2 x - 4 - 5 \\ = & - 3 x^{2} + 14 x - 21 \end{array}$

That is function $s$ in option (D)
And indeed subtracting 2 from each input value (i.e., adding -2) causes a horizontal translation by $- (- 2) = 2$ units

(D) $s (x) = - 3 x^{2} + 14 x - 21$ , because this is an additive transformation
of $f$ that results from subtracting 2 from each input value $x$

Examiner Tips and Tricks

On the exam, you may be given a function and asked which of several expressions represents a particular translation. Be careful to distinguish between adding a constant to the output (vertical shift) versus adding it to the input (horizontal shift).

The incorrect answer options are often designed to test exactly this confusion

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Vertical & Horizontal Translations (College Board AP® Precalculus): Revision Note

Vertical translations

What are graph translations?

What is a vertical translation?

How does a vertical translation affect key features?

How do I find the analytical form of a vertical translation?

Horizontal translations

What is a horizontal translation?

Examiner Tips and Tricks

How does a horizontal translation affect key features?

How do I find the analytical form of a horizontal translation?

How can you tell vertical and horizontal translations apart?

Worked Example

Examiner Tips and Tricks

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Author: Roger B

Reviewer: Mark Curtis