Absolute Value Functions

Q: Why does $|x - 3|$ move the graph right instead of left?

The vertex sits where the inside of the absolute value equals zero. For $|x - 3|$ that happens at $x = 3$, so the corner lands at $x = 3$ — to the right. Inside the bars, the shift always runs opposite to the sign you see.

Q: How do I find the vertex of $|x + 2| - 5$?

Rewrite the inside as $x - (-2)$, so $h = -2$ and $k = -5$: the vertex is $(-2, -5)$. Whenever you see a plus inside the bars, $h$ is negative.

Q: Is the vertex the highest or the lowest point?

It depends on the direction. If the V opens up ($a > 0$), the vertex is the lowest point. If it opens down ($a < 0$), the vertex is the highest point.

Q: Why is the graph a V and not a line?

Absolute value flips the sign of negative inputs, so the left half of the graph is the mirror image of what a line would do. Two straight arms meeting at a corner make the V.

The graph of an absolute value function is a V shape. The parent function $f(x) = |x|$ makes the simplest V: it sits with its corner at the origin, falling with slope $-1$ on the left and rising with slope $1$ on the right, because absolute value turns every input into a non-negative output.

Every absolute value function in this lesson can be written as $f(x) = a|x - h| + k$ . Three numbers, three jobs: $(h, k)$ is the corner of the V — called the vertex — and $a$ controls whether the V opens up or down and how narrow it is.

Reading the vertex from the equation

Match the equation to the form $a|x - h| + k$ and the vertex is $(h, k)$ . Watch the sign on $h$ : the form has a minus sign built in, so $|x - 3| + 2$ has $h = 3$ and the vertex $(3, 2)$ , while $|x + 5| - 4$ is really $|x - (-5)| - 4$ , putting the vertex at $(-5, -4)$ .

The value of $h$ shifts the V left or right, and $k$ shifts it up or down — the same translation rules that move linear functions. The graph below shows $f(x) = |x - 1| - 2$ , a V with its vertex at $(1, -2)$ .

What $a$ does

The sign of $a$ sets the direction: positive $a$ opens the V upward, negative $a$ opens it downward. So $f(x) = -2|x - 1| + 7$ is an upside-down V with its vertex at $(1, 7)$ — and since it opens down, that vertex is the highest point on the graph.

The size of $a$ sets the steepness of the two arms. With $|a| > 1$ the V is narrower than the parent function; with $|a| < 1$ it is wider. The arms are straight lines with slopes $a$ and $-a$ .

Sketching in three moves

Plot the vertex $(h, k)$ first. Decide the direction from the sign of $a$ . Then step out from the vertex using the slope: for $a = 1$ , move right $1$ and up $1$ for the right arm, and mirror it for the left arm. Three points — the vertex and one point on each arm — pin down the whole graph.

The table below lists points on that same $f(x) = |x - 1| - 2$ : the outputs are symmetric around the vertex at $x = 1$ , which is exactly what makes the graph a V.

f(x) = |x - 1| - 2

$x$	$f(x)$
$-1$	$0$
$0$	$-1$
$1$	$-2$
$2$	$-1$
$3$	$0$

Worked examples

Example 1: vertex from the equation

Find the vertex of $g(x) = |x - 3| + 2$ and tell which way it opens.

Match the form

a|x - h| + k

h = 3,\quad k = 2

Read the vertex

(3, 2)

a = 1

is positive, so the V opens up

Answer: Vertex $(3, 2)$ , opening up

Example 2: watch the sign of $h$

Find the vertex of $f(x) = |x + 5| - 4$ .

Rewrite to match the form

f(x) = |x - (-5)| - 4

Read off

h

and

k

h = -5,\quad k = -4

Write the vertex

(-5, -4)

Answer: Vertex $(-5, -4)$

Example 3: a downward V

Describe the graph of $f(x) = -2|x - 1| + 7$ .

Read the vertex

(1, 7)

a = -2

is negative, so the V opens down

|a| = 2 > 1

, so the V is narrower than

|x|

Answer: A narrow V opening down from the vertex $(1, 7)$ .

Try one yourself

Common questions

Why does

|x - 3|

move the graph right instead of left?

The vertex sits where the inside of the absolute value equals zero. For $|x - 3|$ that happens at $x = 3$ , so the corner lands at $x = 3$ — to the right. Inside the bars, the shift always runs opposite to the sign you see.

How do I find the vertex of

|x + 2| - 5

Rewrite the inside as $x - (-2)$ , so $h = -2$ and $k = -5$ : the vertex is $(-2, -5)$ . Whenever you see a plus inside the bars, $h$ is negative.

Is the vertex the highest or the lowest point?

It depends on the direction. If the V opens up ( $a > 0$ ), the vertex is the lowest point. If it opens down ( $a < 0$ ), the vertex is the highest point.

Why is the graph a V and not a line?

Absolute value flips the sign of negative inputs, so the left half of the graph is the mirror image of what a line would do. Two straight arms meeting at a corner make the V.

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Absolute Value Functions

Reading the vertex from the equation

What $a$ does

Sketching in three moves

Worked examples

Example 1: vertex from the equation

Example 2: watch the sign of $h$

Example 3: a downward V

Try one yourself

Common questions

Keep learning

Vertical Translations of Linear Functions

Horizontal Translations of Linear Functions

Piecewise and Step Functions

Graphing Linear Functions

Absolute Value Functions

Reading the vertex from the equation

What aaa does

Sketching in three moves

Worked examples

Example 1: vertex from the equation

Example 2: watch the sign of hhh

Example 3: a downward V

Try one yourself

Common questions

Keep learning

Vertical Translations of Linear Functions

Horizontal Translations of Linear Functions

Piecewise and Step Functions

Graphing Linear Functions

What $a$ does

Example 2: watch the sign of $h$