Absolute value functions

Absolute value measures distance even when positions point in opposite directions, as they do along a street or when astronomers describe directions in space. You use it to write a machinist's tolerance, a poll's margin of error, and a thermostat's allowed temperature change. You build and read the V graph, then recover its equation from the corner and one other point. You solve equations by undoing outside operations and considering both directions of a possible distance. Finally, you solve inequalities, decide which endpoints count, and use boundaries to find where a function's output is positive or negative.

Lessons

Vocabulary

absolute value AB-suh-loot VAL-yoo: The absolute value of a number is its distance from zero. It keeps the size without the direction, so it is never negative. Like: Counting blocks from your house without saying left or right.
distance DIS-tuhns: Distance tells how far apart two places or numbers are. On a number line, the distance from A to B is |A − B|. Like: The number of blocks between two houses.
absolute value function AB-suh-loot VAL-yoo FUNK-shun: An absolute value function uses distance in its rule. The basic rule is f(x) = |x|; shifts and nonzero scaling turn its V graph into another V. Like: A machine that reports how far an address is from home.
toolkit function TOOL-kit FUNK-shun: A toolkit function is a basic, familiar rule you use as a starting point for building or recognizing other functions. Like: A basic tool you reuse for many jobs.
piecewise function PEES-wyze FUNK-shun: A piecewise function uses different rules for different input regions. Each input follows the rule assigned to its region. Like: A parking price that changes after the first hour.
number line NUM-ber lyne: A number line places real numbers in order on a straight line. Larger numbers lie to the right; smaller numbers lie to the left. Like: A straight street with numbered addresses.
origin OR-uh-jin: The origin is zero on a number line or (0, 0) where the two coordinate axes meet. It is the shared starting point for coordinates. Like: Home on your map.
input IN-put: An input is a value supplied to a function. On its graph, the input is the horizontal coordinate x. Like: The number you type into a machine.
output OUT-put: An output is the value a function returns for an input. On its graph, the output is the vertical coordinate y = f(x). Like: The result a machine hands back.
graph graf: A graph is a picture of input-output pairs. Each point (x, y) shows the output y that belongs to the input x. Like: A map of what a machine does to every input.
corner point KOR-ner poynt: The corner point is where the two straight arms of an absolute value graph meet and change direction. It is also called the vertex. Like: The hinge where two straight pieces meet.
vertex VER-teks: For an absolute value V, the vertex is its corner point. In a|x − h| + k with a ≠ 0, it is (h, k). Like: The tip of a folded sheet.
horizontal shift hor-uh-ZON-tuhl shift: A horizontal shift moves a graph left or right without changing its shape. Replacing x by x − h shifts right by h when h is positive. Like: Sliding a drawing sideways across a desk.
vertical shift VER-tih-kuhl shift: A vertical shift moves every graph point up or down by the same amount. Adding k outside the rule moves outputs up by k when k is positive. Like: Lifting a drawing without tilting it.
vertical stretch VER-tih-kuhl strech: A vertical stretch multiplies output distances from the horizontal axis by a factor greater than 1. An absolute value V becomes steeper before any vertical shift. Like: Pulling a drawing taller.
vertical compression VER-tih-kuhl kum-PRESH-un: A vertical compression multiplies output distances from the horizontal axis by a positive factor below 1. An absolute value V becomes flatter before any vertical shift. Like: Pressing a drawing shorter.
horizontal stretch hor-uh-ZON-tuhl strech: A horizontal stretch increases horizontal distances from the vertical axis. In |bx| with 0 < |b| < 1, corresponding points lie farther sideways from zero. Like: Pulling a drawing wider.
horizontal compression hor-uh-ZON-tuhl kum-PRESH-un: A horizontal compression decreases horizontal distances from the vertical axis. In |bx| with |b| > 1, corresponding points lie closer sideways to zero. Like: Squeezing a drawing narrower.
reflection rih-FLEK-shun: A reflection makes a mirror image across a line. Multiplying all outputs by −1 reflects a graph across the x-axis. Like: Flipping a drawing across a mirror line.
slope slohp: Slope is change in vertical coordinate divided by change in horizontal coordinate. It measures signed steepness for a line or one straight arm of a V. Like: How much a ramp rises for each step forward.
domain doh-MAYN: The domain is the set of inputs a function allows. Every real input is allowed in a full absolute value V. Like: All the numbers a machine accepts.
range raynj: The range is the set of outputs a function actually produces. A V opening upward cannot produce outputs below its vertex. Like: All the results a machine can hand back.
absolute value equation AB-suh-loot VAL-yoo ih-KWAY-zhun: An absolute value equation sets an expression containing absolute value bars equal to another expression. A solution makes that equality true. Like: Asking which houses are exactly three blocks from home.
solution suh-LOO-shun: A solution is an input value that makes the original equation or inequality true. Always check it in the original statement. Like: A key that actually fits the lock.
zero ZEER-oh: A zero of a function is an input where its output is 0. On the graph, that input locates an x-intercept. Like: An address where the graph reaches ground level.
x-intercept eks IN-ter-sept: An x-intercept is a graph point on the x-axis, where y = 0. Its x-coordinate is a zero of the function. Like: The place a drawn path touches the horizontal road.
y-intercept wy IN-ter-sept: A y-intercept is a graph point on the y-axis, where x = 0. Evaluate the function at input zero to find it. Like: The place a drawn path meets the vertical road.
absolute value inequality AB-suh-loot VAL-yoo in-ih-KWOL-uh-tee: An absolute value inequality compares an expression containing absolute value bars with a limit. Its solution is every allowed input that makes the comparison true. Like: Asking which houses lie within a distance limit.
compound inequality KOM-pownd in-ih-KWOL-uh-tee: A compound inequality joins two comparisons with and or or. And requires both; or allows either comparison to be true. Like: Meeting both admission conditions, or qualifying through either of two doors.
interval IN-ter-vuhl: An interval is one connected stretch of the number line. It contains every number between its ends, with endpoints included or excluded as stated. Like: One unbroken section of street.
interval notation IN-ter-vuhl noh-TAY-shun: Interval notation writes an allowed stretch using its ends. A bracket includes an endpoint; a parenthesis excludes it. Infinity always takes a parenthesis. Like: A compact address range with inclusion labels.
union YOON-yun: A union combines sets: an input belongs if it belongs to either set. The symbol ∪ often joins the two outside intervals in an absolute value answer. Like: Combining the guests from either invitation list.
boundary point BOWN-duh-ree poynt: A boundary point separates input regions that may behave differently. Solve the corresponding equality to find boundaries, then check which regions and boundaries satisfy the inequality. Like: A fencepost separating two stretches of land.
test point test poynt: A test point is one chosen input used to check whether a region satisfies an inequality. For these V graphs without jumps, test each region between equality boundaries. Like: Sampling one house on each side of a fence.
set-builder notation SET-bil-der noh-TAY-shun: Set-builder notation names a set by a condition. In {x : condition} or {x | condition}, the separator means such that; here x is real. Like: An invitation saying everyone who meets this condition.
tolerance TOL-er-uhns: Tolerance is the allowed size of a difference from a stated center. A percent tolerance is calculated from that center, then allowed above and below it. Like: A permitted amount of wiggle room around a center.
nominal value NOM-uh-nuhl VAL-yoo: The nominal value is the stated or labeled center for an item. Its actual measured value can differ within the allowed tolerance. Like: The quantity printed on a package label.
absolute value bars AB-suh-loot VAL-yoo barz: Absolute value bars are the pair of vertical marks around an expression. Compute the expression inside first, then take its distance from zero. Like: A wrapper instructing you to measure the contents' size.
x-axis eks AK-sis: The x-axis is the horizontal coordinate line. Its points have y = 0, and it records the inputs of a function graph. Like: The left-to-right road on a map.
y-axis wy AK-sis: The y-axis is the vertical coordinate line. Its points have x = 0, and it records the outputs of a function graph. Like: The up-and-down road on a map.
horizontal axis hor-uh-ZON-tuhl AK-sis: The horizontal axis is the left-to-right coordinate line, also called the x-axis. Inputs are read along it. Like: The sideways ruler on a map.
vertical axis VER-tih-kuhl AK-sis: The vertical axis is the down-to-up coordinate line, also called the y-axis. Outputs are read along it. Like: The upright ruler on a map.
horizontal intercept hor-uh-ZON-tuhl IN-ter-sept: A horizontal intercept is a point where a graph meets the horizontal axis. It is an x-intercept, so the output is 0 there. Like: A path crossing a horizontal road.
vertical intercept VER-tih-kuhl IN-ter-sept: A vertical intercept is a point where a graph meets the vertical axis. It is a y-intercept, so the input is 0 there. Like: A path crossing a vertical road.
endpoint END-poynt: An endpoint is a numbered end of an interval. A closed (filled) dot or bracket includes it; an open (hollow) dot or parenthesis excludes it. Like: The house at either end of an allowed street section.
inequality in-ih-KWOL-uh-tee: An inequality compares values. The signs < and > exclude equality; the signs ≤ and ≥ allow it. It states which value is smaller or larger. Like: Comparing two heights instead of saying they match.
isolate EYE-suh-layt: To isolate an expression means to get it alone on one side by undoing outside operations. Preserve the equation or inequality while doing so. Like: Unwrapping the outside layers to reach one object.
real number REE-uhl NUM-ber: A real number is a value that can be placed on the number line. Real numbers include negative numbers, zero, positive numbers, fractions, and decimals. Like: Any address on an unbroken number street.
no solution noh suh-LOO-shun: No solution means no allowed input makes the statement true. An absolute value alone cannot equal a negative number because distance is never negative. Like: Searching for a house at a negative distance from home.
graphical approach GRAF-ih-kuhl uh-PROHCH: A graphical approach uses a graph to read intersections or regions above and below a comparison line. Exact boundaries may require algebra when the picture is unclear. Like: Using a map to see which paths meet.
algebraic approach al-juh-BRAY-ik uh-PROHCH: An algebraic approach uses equivalent equations or inequalities to find exact answers. For absolute value, isolate the bars and translate the distance condition into cases or bounds. Like: Following precise written directions instead of estimating from a map.
actual value AK-choo-uhl VAL-yoo: The actual value is the measured amount for one item. It may differ from the nominal value because manufactured items are not perfectly identical. Like: The amount on your scale rather than the label.
resistance rih-ZIS-tuhns: Resistance measures how strongly a part limits the flow of electricity. Its unit is the ohm; its actual measured value can differ from its stated nominal value. Like: A narrow passage that makes a flow harder.
resistor rih-ZIS-ter: A resistor is a part that limits how much electricity flows. Its label gives a nominal resistance in ohms and may give an allowed tolerance. Like: A controlled narrowing in a flow path.
ohm ohm: An ohm is a unit for resistance, the amount by which a part limits the flow of electricity. Here it identifies what the measurement number describes. Like: A measurement label, as inch labels a length.
capacitor kuh-PAS-ih-ter: A capacitor is a part that stores electric charge, an electrical amount, for later. Capacitance measures this stored charge per unit of voltage; manufactured capacitance values may vary within a tolerance. Like: A small reservoir that stores an amount for later.
capacitance kuh-PAS-ih-tuhns: Capacitance measures stored electric charge per unit of voltage. Electric charge is the stored electrical amount; voltage measures the electrical push. Manufacturing tolerance allows actual capacitance to differ from its nominal value. Like: A reservoir's storage capacity per unit of pressure.
nonnegative non-NEG-uh-tiv: Nonnegative means zero or positive. It excludes every negative number. Like: A ruler's lengths start at 0.
opposite OP-uh-zit: The opposite of a number is equally far from zero on the other side. The opposite of 0 is 0. Like: Turning a signed walk around.
strict inequality strikt in-ih-KWOL-uh-tee: A strict inequality uses < or > and excludes equality. Like: You stop before the fencepost.
inclusive inequality in-KLOO-siv in-ih-KWOL-uh-tee: An inclusive inequality uses ≤ or ≥ and allows equality. Like: The fencepost itself belongs.
branch branch: A branch is one alternative equation or inequality obtained when a problem separates into cases. Like: One road from a fork.
coefficient koh-uh-FISH-unt: A coefficient is the number multiplying a letter or grouped calculation. Like: The number of identical copies.
parent function PAIR-unt FUNK-shun: A parent function is the starting rule before transformations. Here it is y = |x|. Like: The original drawing before you move it.
vertex form VER-teks form: Vertex form writes a V as a|x − h| + k, with a ≠ 0, so its corner (h, k) is visible in the formula. Like: A recipe showing the meeting point.
set notation set noh-TAY-shun: Set notation names a collection. Curly braces with commas list its individual members. Like: A guest list of exactly who belongs.
empty set EMP-tee set: The empty set has no members. Write ∅ or { } when no values satisfy the problem. Like: A guest list with no names.
plus or minus plus or MY-nus: The symbol ± names both signs. In a tolerance, c ± r allows values from c − r through c + r. Like: An allowance above and below a center.
margin of error MAR-jin uhv AIR-er: A margin of error is the allowed size of a difference from a stated measurement or estimate. Like: An allowed amount on either side of a report.
infinity in-FIN-uh-tee: Infinity, written ∞, describes continuing without a final bound. It is not a real endpoint you can include. Like: A road with no last house.
viewing window VYOO-ing WIN-doh: A viewing window is the input and output stretch shown on a graphing screen. It can show only part of a function's graph. Like: A camera frame covering part of a road.
expression ik-SPRESH-un: An expression is a calculation made of numbers, letters, and operations. It does not by itself claim an equality. Like: A recipe before you ask for a result.
electric current ih-LEK-trik KUR-ent: Electric current measures how much electric charge flows through a part over time. A resistor limits this flow. Like: The rate of water through a pipe.
electric charge ih-LEK-trik charj: Electric charge is an electrical amount that can be stored or moved. A capacitor can store charge for later. Like: An amount kept in a reservoir.
voltage VOHL-tij: Voltage measures the electrical push between two places. Capacitance measures stored charge for each unit of that push. Like: Pressure driving flow through a pipe.

Quick checks

Solve |x| = 7.

x = −7 or x = 7
set {−7, 7}, because both positions are seven units from zero.

Solve |x + 5| = −2.

No solution
set ∅, because an absolute value cannot be negative.

Solve |x − 1| < 3.

−3 < x − 1 < 3 gives −2 < x < 4.
Interval: (−2, 4).
Reason: strictly closer than three means between with both ends excluded.

Solve |x| ≥ 5.

(−∞, −5] ∪ [5, ∞), because distance at least five allows either outside direction and includes the five-unit boundaries.

For f(x) = −|x + 2| + 6, give the corner and opening direction.

Corner: (−2, 6).
Opening: down.
The inside is zero at −2, leaving height 6, and the negative multiplier lowers height as distance grows.

Why does multiplying −|A| < −3 by −2 give |A| > 6?

A stands for the whole inside calculation. Multiplying by −2 cancels the outside factor − and sends −3 to 6. A negative multiplier reverses number order, so the sign < changes to > and the result is |A| > 6.

Find the y-intercept of f(x) = 2|x − 3| − 2.

(0, 4), because input zero gives 2|−3| − 2 = 6 − 2 = 4.

Solve |2x − 1| > −4.

(−∞, ∞), because every absolute value is at least zero and zero is greater than −4.

Before you start

Explain it like I am five
Picture your home at 0 on a straight street. The number line is a map of that street. Positive addresses are to your right. Negative addresses are to your left. A house at −3 is three blocks left of home. A house at 3 is three blocks right.
Both houses are three blocks from home. Distance counts how far you travel, without keeping the direction. The absolute value bars in |−3| ask for that distance. So |−3| = 3 and |3| = 3.
You can use the same picture to ask which houses are a certain distance away, which are nearby, and which are farther away. When you draw address across the page and distance upward, the two sides of the street make a V. This section teaches you to read that picture and write it with numbers.
Signed arithmetic and substitution from zero
Use the street picture for signed arithmetic. Adding a positive number moves right. Adding a negative number moves left. Subtracting undoes the move, so subtracting a negative moves right. Multiplication groups equal changes. Its sign tells whether the direction stays or reverses: matching signs give a positive product, different signs give a negative product. Division undoes multiplication and uses the same sign pattern. A letter such as x holds a number. Substitution means putting a known number in every place that letter appears. Parentheses keep a negative number together while you calculate. An expression is a calculation such as 2x − 6. A coefficient is the number multiplying a letter or grouped calculation: 2 in 2x. Letters can also be nicknames for a whole calculation. Later, A names the whole inside of the bars and B names the number on the other side; in |2x − 6| = 8, A means 2x − 6 and B means 8.
Fractions, including division by a negative half
A fraction names equal pieces of a whole. In $3 over 4$ , the bottom 4 says the whole is cut into four equal pieces, and the top 3 says you have three of them. It also means 3 ÷ 4. Equal fractions describe the same amount with different-sized pieces: $6 over 8$ = $3 over 4$ . Add or subtract fractions only after their pieces have the same size. Multiply tops together and bottoms together. Dividing asks how many copies of one amount make another. To divide by a nonzero fraction, multiply by its reciprocal, which is the fraction turned upside down.
Solve a linear equation by undoing operations
Think of an equation as a balanced scale. The left and right sides name the same amount. To isolate a letter means to get it alone, so you can see its value. You keep the scale balanced by making the same change to both sides. In 3x − 5 = 7, a number x was multiplied by 3 and then had 5 removed. Solving walks those operations backward: add 5, then divide by 3. Here a linear equation has a number multiplying the unknown plus or minus an added number, as in 3x − 5 = 7. The same balancing idea works when the unknown letter is a instead of x.
Three-part inequalities and reversing the sign
An inequality compares sizes instead of saying two values are equal. The small opening of < points to the smaller amount: 2 < 5. The symbol ≤ also allows equality. A three-part inequality is a fence with a lower bound, an expression in the middle, and an upper bound. You change all three parts together. Adding or subtracting moves the whole fence. Multiplying or dividing by a positive number keeps its order. A negative number turns the number line around, so every inequality sign reverses. The bounds may then need rewriting from smaller to larger. A strict inequality uses < or > and leaves equality out. An inclusive inequality uses ≤ or ≥ and allows equality.
Coordinates, input, output, and slope
A point is an address on a flat map. The origin (0, 0) is home. The x-axis, also called the horizontal axis, runs left and right. The y-axis, also called the vertical axis, runs down and up. In (x, y), read the sideways move first and the vertical move second. For a function graph, x is the input and y is the output, written y = f(x). Slope measures how much the output changes for each unit of input. Picture a ramp: slope is its signed rise divided by its signed run. Always compare the two coordinates in the same point order. A function is a rule giving one output for each allowed input. In f(3), the 3 is the input, so evaluate means find its output. Domain names the allowed inputs, like addresses on the road; range names the outputs the rule reaches, like heights along the road. On a graph, imagine its shadow on the x-axis for domain and its shadow on the y-axis for range.
Intervals, endpoints, and union
A set is a collection of numbers. A real number is any position on the unbroken number line, including negatives, zero, fractions, and decimals. Curly braces enclose a set's description. An interval is a connected stretch of the number line, like all houses along one section of street. Interval notation records the left end, the right end, and whether each endpoint belongs. A parenthesis leaves an end out. A bracket includes it. Infinity describes a direction with no final house, so it always takes a parenthesis. Sometimes an answer has two separate stretches. Union, written ∪, puts those allowed stretches together. Set-builder notation gives the same information as a sentence inside braces. You can read all three forms without changing which numbers belong.
Find a percent of a stated center
Percent means out of 100. Five percent is five equal pieces out of a hundred, so 5% = $5 over 100$ = 0.05. Finding a percent of an amount means multiplying the amount by that fraction. A product label may give a center and a tolerance. The nominal value is the stated center. The actual value is the amount measured on one item. Tolerance is the allowed difference from that center. A resistor is a part that limits how much electricity flows, and ohms measure its resistance to that flow. A 680-ohm resistor with 5% tolerance allows a difference equal to 5% of 680, in either direction.
Factor a number out, then multiply back to check
Think of packing equal objects into bags. Six objects can be described as two bags of three. Factoring rewrites a sum or difference as a product without changing its value. Here you need to take a common number out of every term inside absolute value bars. A term is one part joined by addition or subtraction. In 2x − 6, both terms have a factor of 2: 2x = 2 × x and 6 = 2 × 3. Putting 2 outside parentheses gives 2(x − 3). Multiplying back, called distributing, checks every term.
Answer sets and the empty set
A set is a collection, like a guest list. Curly braces list exactly which numbers belong. The set {−7, 3} lists two numbers; {4} lists one. You do not shade the street between listed numbers: each number is its own entry. An empty guest list contains no entries. Its mathematical name is the empty set, written ∅ or { }. It means no solution when no number can satisfy a problem. Set-builder notation, from the interval refresher, instead writes a condition for belonging. These are different ways to name collections, chosen so readers can tell a short list from a whole interval.