Vertical Translations of Linear Functions

Q: What's the quickest way to find $k$ from a graph?

Compare $y$-intercepts: $k$ equals the new intercept minus the original one. Any matching pair of points works, but the intercepts are the easiest to read.

A vertical translation slides a graph straight up or down without turning or stretching it. For a function $f$ , the translated function is $g(x) = f(x) + k$ : adding a positive number shifts the graph up, and subtracting shifts it down.

The reason is simple — $f(x) + k$ takes every output of $f$ and adds $k$ to it. Every point on the graph moves the same distance in the same direction, so the shape is untouched. For a line, that means the slope stays exactly the same.

What stays the same, what changes

The slope stays the same, so the original line and the translated line are parallel. What changes is the $y$ -intercept: it moves by exactly $k$ . Translate $f(x) = 2x + 1$ up $3$ units and you get $g(x) = 2x + 4$ — same slope $2$ , intercept raised from $1$ to $4$ .

In the graph below, the blue line is $f(x) = x$ and the red line is $f(x) + 3$ . Every point on the blue line moved up $3$ to land on the red one.

Writing the translated equation

To translate up $k$ units, add $k$ to the whole function and simplify. Down $k$ units means subtract $k$ . The $k$ attaches to the constant term only: $f(x) = x + 5$ shifted down $3$ becomes $g(x) = x + 5 - 3 = x + 2$ .

To go the other way — reading $k$ off a graph — compare the two $y$ -intercepts. If $f$ crosses the $y$ -axis at $3$ and $g$ crosses at $-1$ , then $k = -1 - 3 = -4$ , a shift of $4$ units down.

Worked examples

Example 1: translate up

Translate $f(x) = x$ up $4$ units to get $g(x)$ .

Add

4

to the function

g(x) = f(x) + 4

Substitute

f(x) = x

g(x) = x + 4

Answer: $g(x) = x + 4$

Example 2: translate down

Translate $f(x) = x + 5$ down $3$ units.

Subtract

3

from the function

g(x) = f(x) - 3

Substitute

f(x) = x + 5

g(x) = x + 5 - 3

Simplify

g(x) = x + 2

Answer: $g(x) = x + 2$

Example 3: find $k$ from two graphs

The line $f$ has $y$ -intercept $3$ . Its translation $g(x) = f(x) + k$ has $y$ -intercept $-1$ . Find $k$ .

The intercept moves by exactly

k

k = -1 - 3

Simplify

k = -4

g

f

shifted

4

units down

Answer: $k = -4$

Try one yourself

Common questions

Does a vertical translation change the slope?

No. Every point moves the same distance straight up or down, so the steepness is unchanged and the two lines are parallel.

How is

f(x) + k

different from

f(x + k)

$f(x) + k$ adds to the output, so the graph moves vertically. $f(x + k)$ changes the input, so the graph moves horizontally. The position of the $+\,k$ — outside or inside the parentheses — tells you which.

What if

k

is negative?

Then the graph shifts down. $g(x) = f(x) - 7$ is a translation $7$ units down — the same rule with $k = -7$ .

What's the quickest way to find

k

from a graph?

Compare $y$ -intercepts: $k$ equals the new intercept minus the original one. Any matching pair of points works, but the intercepts are the easiest to read.

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Vertical Translations of Linear Functions

What stays the same, what changes

Writing the translated equation

Worked examples

Example 1: translate up

Example 2: translate down

Example 3: find $k$ from two graphs

Try one yourself

Common questions

Keep learning

Horizontal Translations of Linear Functions

Dilations of Linear Functions

Graphing Linear Functions

Absolute Value Functions

Vertical Translations of Linear Functions

What stays the same, what changes

Writing the translated equation

Worked examples

Example 1: translate up

Example 2: translate down

Example 3: find kkk from two graphs

Try one yourself

Common questions

Keep learning

Horizontal Translations of Linear Functions

Dilations of Linear Functions

Graphing Linear Functions

Absolute Value Functions

Example 3: find $k$ from two graphs