Absolute Value as Distance and a Graph, foundations for calculus lesson

Two hikers stand on a straight trail, one at the 3 kilometer marker and one at the 10 kilometer marker. The distance between them is 7 kilometers, and it stays 7 no matter which hiker you measure from. Absolute value is the tool that turns a signed difference like $3 - 10 = -7$ into that distance. Calculus uses it constantly to say how close one number is to another.

Key terms

Absolute value

Definition For a real number $x$ , the absolute value of $x$ is

\lvert x \rvert = \begin{cases} x, & x \ge 0, \\ -x, & x < 0. \end{cases}

In plain words A nonnegative number stays as it is. A negative number has its sign flipped, so the result is positive.

Example The definition gives $\lvert 6 \rvert = 6$ , $\lvert 0 \rvert = 0$ and $\lvert -7 \rvert = -(-7) = 7$ . For an expression, work out the inside first, so $\lvert 3 - 8 \rvert = \lvert -5 \rvert = 5$ .

Nonnegative values

Definition For every real $x$ ,

\lvert x \rvert \ge 0,

and $\lvert x \rvert = 0$ exactly when $x = 0$ . Also $\lvert -x \rvert = \lvert x \rvert$ and $\sqrt{x^2} = \lvert x \rvert$ for every real $x$ . A number is nonnegative when it is greater than or equal to $0$ .

In plain words An absolute value can be zero or positive, never negative. Removing a minus sign is the right picture for numbers, but not for letters, since $-x$ is positive whenever $x$ is negative.

Example Take $x = -5$ . The second line of the definition gives $\lvert x \rvert = -x = -(-5) = 5$ . Here the formula $-x$ produces a positive number. Also $\sqrt{(-5)^2} = \sqrt{25} = 5$ , not $-5$ . An equation such as $\lvert x \rvert = -3$ has no solution at all.

Distance on the number line

Definition The number line pairs every real number with a point on a line, with $0$ at a chosen origin, positive numbers to the right and equal steps for equal differences. For real numbers $a$ and $b$ , the distance between them on the number line is

\lvert a - b \rvert = \lvert b - a \rvert.

In particular $\lvert x \rvert$ is the distance from $x$ to $0$ , and $\lvert x - a \rvert$ is the distance from $x$ to $a$ .

In plain words Subtract the two numbers in either order and drop the sign. What remains is how far apart they sit on the line.

Example The distance between $-3$ and $4$ is

\lvert 4 - (-3) \rvert = \lvert 7 \rvert = 7,

and in the other order $\lvert -3 - 4 \rvert = \lvert -7 \rvert = 7$ as well. Count the steps from $-3$ to $4$ on a number line and you also get $7$ .

Solving a distance equation

Definition Let $a$ and $d$ be real numbers. The equation $\lvert x - a \rvert = d$ asks for every $x$ at distance $d$ from $a$ . If $d > 0$ it has exactly two solutions, $x = a - d$ and $x = a + d$ . If $d = 0$ it has exactly one solution, $x = a$ . If $d < 0$ it has no solution, since a distance is never negative.

In plain words Stand at $a$ and step $d$ units to the left and $d$ units to the right. Those two landing spots are the answers, unless $d$ is zero or negative.

Example Solve $\lvert x - 2 \rvert = 5$ . The numbers at distance $5$ from $2$ are $2 - 5 = -3$ and $2 + 5 = 7$ . Both check, since $\lvert -3 - 2 \rvert = 5$ and $\lvert 7 - 2 \rvert = 5$ . For an equation like $\lvert x + 1 \rvert = 4$ , first write $x + 1$ as $x - (-1)$ . The center is $-1$ , so the solutions are $-5$ and $3$ .

Graph of absolute value

Definition The graph of $y = \lvert x \rvert$ is the set of points $(x, \lvert x \rvert)$ for all real $x$ . It is made of two rays that start at the origin, the ray $y = x$ for $x \ge 0$ and the ray $y = -x$ for $x \le 0$ . Its domain is all real numbers and its range is $[0, \infty)$ . More generally, the graph of $y = \lvert x - a \rvert$ consists of the ray $y = x - a$ for $x \ge a$ and the ray $y = a - x$ for $x \le a$ . The range $[0, \infty)$ holds for the full real domain. On a smaller domain the range can be smaller.

In plain words The height of the graph above $x$ is the distance from $x$ to $0$ . The farther you go from $0$ in either direction, the higher the graph, and it rises one unit for each unit you move.

Example A table of distances gives the points to plot.

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$\lvert x \rvert$	$3$	$2$	$1$	$0$	$1$	$2$	$3$

The points on the left lie on the line $y = -x$ and those on the right lie on $y = x$ . Joined up they make a V. If you keep only the inputs in $[-1, 2]$ , the outputs fill $[0, 2]$ instead of all of $[0, \infty)$ .

Vertex

Definition The vertex of the graph of $y = \lvert x - a \rvert$ is the point $(a, 0)$ where its two rays meet. It is the lowest point of the graph. To its left the graph is a straight piece with slope $-1$ , and to its right a straight piece with slope $1$ . These slopes belong to the separate sides. The two rays meet at the vertex with different directions, forming a corner, so no single slope describes the graph at the vertex itself.

In plain words The vertex is the tip of the V, where the distance is zero. It is a sharp corner, not a smooth turn.

Example The graph of $y = \lvert x - 2 \rvert$ has its vertex at $(2, 0)$ . Going right from the vertex the points $(3, 1)$ and $(4, 2)$ rise with slope $1$ . Going left the points $(1, 1)$ and $(0, 2)$ rise as you move left, so that side has slope $-1$ . That corner is exactly the kind of point where calculus later finds that a graph has no tangent line.

Symmetry

Definition For every real $x$ , $\lvert -x \rvert = \lvert x \rvert$ , so the graph of $y = \lvert x \rvert$ is symmetric about the $y$ axis. More generally, the graph of $y = \lvert x - a \rvert$ is symmetric about the vertical line $x = a$ , because for every real $t$

\lvert (a + t) - a \rvert = \lvert (a - t) - a \rvert = \lvert t \rvert.

In plain words Fold the picture along the vertical line through the vertex and the two sides land on each other. Points the same distance to the left and right of $a$ have the same height.

Example For $y = \lvert x - 2 \rvert$ , the inputs $2 + 3 = 5$ and $2 - 3 = -1$ both give height $3$ . Once you have plotted the right side of the V, you get the left side for free by mirroring it across $x = 2$ .

Check yourself

Solve $\lvert x + 3 \rvert = 6$ .

Show the answer

Write $x + 3$ as $x - (-3)$ . The equation asks for the numbers at distance $6$ from $-3$ , which are $-3 - 6 = -9$ and $-3 + 6 = 3$ .

How many solutions does each equation have? First $\lvert x - 4 \rvert = 0$ , then $\lvert x - 4 \rvert = -2$ .

Show the answer

The first has exactly one solution, $x = 4$ , because only $4$ is at distance $0$ from $4$ . The second has none, because an absolute value is never negative.

Find the vertex of $y = \lvert x - 1 \rvert$ and its range when the domain is only $[-2, 3]$ .

Show the answer

The vertex is $(1, 0)$ , and $1$ lies in $[-2, 3]$ , so the smallest output is $0$ . The largest distance from $1$ within the interval is at $x = -2$ , where the height is $3$ , while $x = 3$ gives only $2$ . The range is $[0, 3]$ .

Absolute Value as Distance and a Graph

Key terms

Absolute value

Nonnegative values

Distance on the number line

Solving a distance equation

Graph of absolute value

Vertex

Symmetry

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus