Transformation of functions

Start with vertical shifts, then learn why horizontal shifts move in the opposite direction from the sign inside the function. Combine those moves and track what happens to the allowed inputs and possible heights. Next, reflect graphs and use those reflections to decide whether a function is even, odd, or neither. Finish with stretches, compressions, and a point rule that keeps several transformations in the correct order.

Lessons

Vocabulary

Transformation trans for MAY shun: A change to a Function that moves, flips, stretches, or compresses its Graph. Like: Redrawing a picture after moving or resizing its dots.
Function FUNGK shun: A rule that gives exactly one Output for every accepted Input. Like: A recipe with one specified result for each ingredient amount.
Input IN put: The value you give a Function before its rule runs, usually labeled x. Like: The number you enter into a calculator.
Output OUT put: The result returned by a Function, written f(x) or y. Like: The calculator's displayed result.
Graph graf: A picture of a Function's accepted Input and Output pairs, plotted as points (x, f(x)). Like: A map of sideways addresses and their heights.
Toolkit function TOOL kit FUNGK shun: A familiar starting Function whose basic Graph you use to recognize and build transformed graphs. Like: A standard shape stored in a drawing kit.
Vertical shift VER ti kul shift: Adding a Constant to every Output, moving the entire Graph up or down without changing its shape. Like: Raising a drawing to a higher shelf.
Horizontal shift hor i ZON tul shift: Replacing x by x − h moves the Graph right h; a negative h moves it left. Like: Sliding a drawing sideways on a desk.
Inside change in SIDE chaynj: A change to the Input expression before the Function runs; it changes horizontal coordinates. Like: Changing the ingredient amount before following a recipe.
Outside change out SIDE chaynj: A change applied after the Function returns its Output; it changes vertical coordinates. Like: Adjusting a finished recipe's result after cooking.
Domain doh MAYN: All Inputs a Function accepts without an invalid calculation. Like: The ingredients a recipe permits.
Range raynj: All Outputs a Function can actually produce from its accepted Inputs. Like: The results the recipe can make.
Vertex VER teks: The turning point of a quadratic U, or the corner where an absolute value V changes direction. Like: The lowest bend of a bowl or the tip of a V.
Endpoint END point: A boundary point where a graph or interval starts or stops. Like: The end of a road.
Asymptote AS im toht: A line a Graph approaches as Inputs approach a boundary or grow without bound. Reciprocal graphs approach their asymptotes without meeting them. Like: A guide line the reciprocal curve gets closer to.
Vertical reflection VER ti kul ree FLEK shun: Taking the opposite of every Output, so the Graph flips across the x-axis. Like: Seeing a reflection above and below a horizontal mirror.
Horizontal reflection hor i ZON tul ree FLEK shun: Replacing the Input by its opposite, so the Graph flips across the y-axis. Like: Seeing a reflection on opposite sides of a vertical mirror.
Reflection ree FLEK shun: A flip across a line that places each point the same distance on the opposite side. Like: A mirror image across the mirror's surface.
Even function EE vun FUNGK shun: A Function with opposite Inputs allowed and equal Outputs: f(−x) = f(x). Its Graph is symmetric about the y-axis. Like: A picture whose left and right halves mirror each other.
Odd function odd FUNGK shun: A Function with opposite Inputs allowed and opposite Outputs: f(−x) = −f(x). Its Graph is symmetric about the origin. Like: A picture that fits itself after a half turn.
Neither even nor odd NEE ther EE vun nor odd: A Function that fails both required symmetry tests, including the requirement that opposite Inputs belong to its Domain. Like: A drawing that fails both a mirror test and a half-turn test.
Symmetric about the y-axis si MET rik uh BOUT the why AK sis: Reflecting left to right across the y-axis leaves the Graph unchanged. Like: Two matching wings on opposite sides of a center line.
Symmetric about the origin si MET rik uh BOUT the OR i jin: Turning a Graph half a turn around (0, 0) leaves it unchanged. Like: A picture that matches itself after rotating 180°.
Origin OR i jin: The point (0, 0), where the x-axis and y-axis meet. Like: The starting address for measuring both directions.
Vertical stretch VER ti kul strech: Multiplying every height by a factor greater than 1 in magnitude, increasing distances from the x-axis. A negative factor also reflects. Like: Making a photograph taller while preserving its sideways positions.
Vertical compression VER ti kul kum PRESH un: Multiplying heights by a nonzero factor smaller than 1 in magnitude, reducing distances from the x-axis. A negative factor also reflects. Like: Making a photograph shorter while preserving its sideways positions.
Horizontal stretch hor i ZON tul strech: Multiplying sideways distances by a factor greater than 1. In f(bx), use 0 < |b| < 1; a negative b also reflects. Like: Making a photograph wider without changing its heights.
Horizontal compression hor i ZON tul kum PRESH un: Multiplying sideways distances by a factor between 0 and 1. In f(bx), use |b| > 1; a negative b also reflects. Like: Making a photograph narrower without changing its heights.
Scale factor skayl FAK ter: The positive multiplier describing how distances change. Vertical distance scales by |a|; horizontal distance scales by the Reciprocal of |b|. Like: The enlargement percentage on a copier.
Reciprocal ree SIP ruh kul: One divided by a nonzero number. Multiplying the number by its Reciprocal gives 1. Like: The multiplier that undoes a nonzero multiplication.
Sequence of transformations SEE kwens uv trans for MAY shunz: The ordered changes used to redraw a Graph. Two changes on the same coordinate can give different results when reversed. Like: Following recipe steps in the specified order.
Point rule point rool: The coordinate recipe for moving each graph point through a Transformation. Like: A forwarding address for every dot in a drawing.
Constant KON stunt: A fixed number that does not vary with the Input. The toolkit Constant function returns 1 for every Input. Like: A fixed admission fee, regardless of the time you arrive.
Identity function eye DEN ti tee FUNGK shun: The Function f(x) = x, which returns its Input unchanged. Like: A return desk that hands back exactly what you brought.
Absolute value function AB suh loot VAL yoo FUNGK shun: The Function f(x) = |x|, returning the distance from x to 0. Its Graph is a V. Like: Counting steps from 0 regardless of which direction you walked.
Quadratic function kwah DRAT ik FUNGK shun: A Function built from a nonzero $x^{2}$ term, an optional x term, and a Constant. The toolkit example is f(x) = $x^{2}$ . Like: The curved U profile of a bowl, possibly moved or flipped.
Cubic function KYOO bik FUNGK shun: A Function built from a nonzero $x^{3}$ term and optional lower powers. The toolkit example is f(x) = $x^{3}$ . Like: An S-shaped path passing through the center of a map.
Square root function skwair root FUNGK shun: The toolkit Function f(x) = $\sqrt{x}$ , returning the nonnegative number whose square is x. Like: Finding a square tile's side from its area.
Cube root function kyoob root FUNGK shun: The toolkit Function f(x) = $\sqrt[3]{x}$ , returning the number whose cube is x. Negative Inputs are allowed. Like: Undoing the cube calculation, including its sign.
Reciprocal function ree SIP ruh kul FUNGK shun: The toolkit Function f(x) = $1 over x$ . It accepts every real Input except 0 and never produces Output 0. Like: Dividing one unit among x equal shares.
Axis AK sis: One of the two reference number lines on a graph. The x-axis is horizontal; the y-axis is vertical. Together they are called axes. Like: The two measuring directions on a map.
Reference point REF er ens point: A recognizable point you track to identify or sketch a Transformation. Like: A landmark used to see how a map has moved.
Factoring FAK ter ing: Rewriting an expression as a product without changing its value. For transformations, this exposes the actual inside shift. Like: Repacking the same objects into equal groups.
Order of operations OR der uv op er AY shunz: The agreed sequence for reading arithmetic: parentheses, powers, multiplication and division, then addition and subtraction; tied operations go left to right. Like: Following the agreed order of steps in a recipe.
Translation trans LAY shun: A slide that moves every graph point the same distance and direction. A horizontal or vertical shift is a Translation. Like: Sliding a sheet of paper without turning or resizing it.
Coordinate koh OR di nut: One number in a point's address. In (x, y), x measures sideways position and y measures height from the Origin. Like: A street address with one number for each direction.
Interval notation IN ter vul noh TAY shun: A compact way to name stretches of real numbers. Brackets include endpoints, parentheses exclude them, and ∪ joins separate stretches. Like: Writing the start and end of an allowed road segment.
Square root skwair root: The nonnegative number whose square equals the number inside the square root sign. A negative inside has no real Square root. Like: Finding a square tile's side length from its area.
Cube root kyoob root: The number whose cube equals the given number. It keeps the given number's sign. Like: Undoing three repeated equal multiplications.
Reciprocal squared function ree SIP ruh kul skwaird FUNGK shun: The toolkit Function f(x) = $1 over x squared$ . Input 0 is excluded, and every Output is positive. Like: A reciprocal calculation after squaring makes the denominator positive.
Coefficient koh uh FISH unt: A fixed number multiplying a letter or expression. Like: The number of equal groups you have.
Identity eye DEN ti tee: An equation true for every allowed input, rather than a single numerical coincidence. Like: A description that fits every permitted case.
Radicand RAD ih kand: The whole expression inside a root sign. Like: Everything contained inside the root enclosure.
Set-builder notation set BIL der noh TAY shun: Names a collection by the condition its members must satisfy. The middle bar means such that. Like: A guest list described by an admission condition.
Polynomial pah lee NOH mee ul: A finite sum of whole-number powers of the input multiplied by fixed coefficients. Like: Combining several repeated-multiplication recipes into one answer.
Magnitude MAG nih tood: The size of a number without its sign, measured as distance from zero. Like: Distance from a starting point, regardless of direction.
Mastery MAS ter ee: The amount of a skill learned in a model. A percentage reports it out of one hundred. Like: The filled amount in a practice progress gauge.
Real number REE ul NUM ber: A number represented by a point on the number line, including negative numbers, zero, fractions, and square roots of nonnegative numbers. Like: An address anywhere along the number line.
Term term: One piece of a sum, with its attached sign. Like: One ingredient in a combined recipe.
Substitution sub stih TOO shun: Replacing every occurrence of a letter by a chosen value or expression. Like: Putting a selected ingredient into every place the recipe names it.
Numerator NOO muh ray ter: The top of a fraction, naming the amount being divided. Like: The number of equal pieces you have.
Denominator dih NOM ih nay ter: The bottom of a fraction, naming the divisor. It cannot be zero. Like: The number of equal parts in each whole.
Set set: A collection of values specified together. Like: An allowed guest list.
Percentage per SEN tij: A share expressed out of one hundred. Like: The filled amount on a progress gauge with one hundred divisions.
Nonnegative non NEG uh tiv: At least zero: zero and positive real numbers. Like: An account balance with no debt.
Nonpositive non PAH zih tiv: At most zero: zero and negative real numbers. Like: A temperature at or below zero.
Integer IN tuh jer: A whole number or its negative, including zero. Like: A marked whole-number address on a number line.
Ratio RAY shee oh: A comparison made by dividing one quantity by another nonzero quantity. Like: Comparing a resized photograph with its original size.
Distribution dis trih BYOO shun: Multiplying a sum by multiplying each term and adding the resulting products. Like: Making equal copies of every item in a bundle.
x-intercept eks IN ter sept: A graph point on the x-axis, where its output is zero. Like: Where a path meets the horizontal reference line.

Quick checks

Write shifted down 4.

− 4, because subtracting 4 outside lowers every Output by 4.

Write shifted left 5.

(x + 5, because the new Input −5 must feed old Input 0 into the square.

Write reflected across the y-axis.

, because replacing the Input with its opposite reflects horizontal coordinates.

(3, −2) is on f. Where is it on g(x) = f(x − 6) + 3?

(9, 1), because the Point rule gives new x = 3 + 6 = 9 and new y = −2 + 3 = 1.

Is x⁵ − x even, odd, or neither?

Odd, because f(−x) = −x⁵ + x = −(x⁵ − x) = −f(x) on its symmetric Domain of all real numbers.

Is |x| + 2 even, odd, or neither?

Even, because f(−x) = |−x| + 2 = |x| + 2 = f(x) on its symmetric Domain of all real numbers.

Is + x even, odd, or neither?

Neither, because f(−x) = − x equals neither f(x) = + x nor −f(x) = − − x as a whole formula.

Before you start

Explain it like I am five
Imagine drawing a V on a sheet of graph paper. Each dot has an address: how far sideways it is and how high it is. A graph records those addresses.
You can redraw the same V farther right or higher up. You can also turn its heights upside down or spread its dots farther apart. Those changes are called a Transformation. A slide alone is called a Translation.
A function formula tells you how to redraw each dot. Changes outside the function act on heights. Changes inside act on the sideways addresses, so you must work backward to find the new address.
Reading signs, letters, domains, and graph addresses
Read the notation as labels on a map. A real number is a point on the number line, including negatives, zero, fractions, and decimals. A letter can stand for one of those numbers. In a formula, × and · mean multiply; a number beside a letter or parentheses also means multiply. A coefficient is that multiplying number. A constant is a number that stays fixed while the input changes.
Signed arithmetic, powers, and replacing every x by −x
A minus sign can mean moving left, changing direction, or taking the opposite of a whole expression. Parentheses show what it acts on. Think of multiplying by −1 as turning around on a number line: two turns face the original direction. An even number of negative factors pairs off; an odd number leaves one negative factor.
Function notation and coordinates: f versus f(x)
A Function is a named rule, like a recipe. The name f refers to the recipe; f(3) refers to the result made with Input 3. The parentheses do not mean f times 3. A Graph plots each Input and Output as a pair. The Coordinate x tells you how far right or left to go; y tells you how far up or down.
Fractions and dividing by a fraction
A fraction is division written as a stack: $3 over 4$ means 3 ÷ 4. Think of three slices from a pizza cut into four equal pieces. The bottom counts equal pieces in a whole; the top counts how many you have. Equal-sized pieces are needed for addition. Division asks how many of one amount fit into another.
Toolkit 1: Constant
A Constant stays the same, like a fixed admission fee. The toolkit Constant is f(x) = 1. Every Input gives Output 1, so its Graph is a flat line one unit above the horizontal Axis. A constant function could use another fixed number, but the toolkit starts with 1.
Toolkit 2: Identity function
The Identity function is a return desk that hands you back the same number you brought. Its formula is f(x) = x. Input −2 returns −2, and Input 3 returns 3. Equal sideways and vertical changes make a straight diagonal line through the Origin.
Toolkit 3: Absolute value function
The Absolute value function reports distance from 0 on a number line. Distance cannot be negative: both −3 and 3 are three steps from 0. The formula f(x) = |x| turns negative Inputs positive and leaves nonnegative Inputs alone. Its Graph is a V with a Vertex, or corner, at the Origin.
Toolkit 4: Quadratic function
The toolkit Quadratic function squares its Input: f(x) = $x^{2}$ . Think of the area of a square tile: doubling its side makes its area four times as large. Negative Inputs also give nonnegative Outputs because two negative factors multiply to a positive number. The Graph is a smooth U with its Vertex at the Origin.
Toolkit 5: Cubic function
The toolkit Cubic function multiplies its Input by itself three times: f(x) = $x^{3}$ . A cube-shaped box with side 2 has volume 2·2·2 = 8. Algebra also accepts negative Inputs. Three negative factors leave one unpaired negative, so the Graph goes below the Origin on the left and above it on the right.
Toolkit 6: Square root function
The Square root function asks which nonnegative number squares to the Input. A square tile with area 9 has side $square root of 9$ = 3. The formula f(x) = $\sqrt{x}$ accepts 0 and positive Inputs. Its Graph starts at an Endpoint at the Origin and curves rightward while rising more slowly. The quantity under the root sign is called the radicand.
Toolkit 7: Cube root function
The Cube root function reverses cubing: f(x) = $\sqrt[3]{x}$ . A box with volume 8 and equal sides has side $cube root of 8$ = 2. Unlike a Square root, a Cube root can accept a negative number. The Input −8 returns −2 because (−2 $) cubed$ = −8. Its Graph is a sideways S through the Origin.
Toolkit 8: Reciprocal function
A Reciprocal tells you how much of a unit belongs to one equal share. The Reciprocal function is f(x) = $1 over x$ . Dividing 1 among two shares gives $1 over 2$ . Negative Inputs give negative Outputs. Its two curved branches approach the axes, but they never touch either one.
Toolkit 9: Reciprocal squared function
The Reciprocal squared function is f(x) = $1 over x squared$ . First square the Input, then divide 1 by that square. Picture equal distances on either side of 0: squaring makes their signs match, so both reciprocal branches sit above the horizontal Axis. The Graph has no point at Input 0.
Solving a linear equation and Factoring the inside
An equation is a balanced scale. Undo the operations around the unknown while doing the same thing to both sides. Factoring repacks an expression into a product, like putting equal groups into boxes. These two skills reveal which new Input feeds an old Input into a transformed Function.
Square roots, inequalities, Domain, and Range
The Domain lists which Inputs the Function accepts, like allowed ingredients. The Range lists the Outputs it can produce. A Square root accepts an inside value at least 0. Once you find that starting Input, its Graph tells you whether heights rise above an Endpoint or fall below it after a reflection.
Solve a reciprocal equation to prove a range
Suppose you know the finished height and need the address that would make it. Work backward as you would undo a recipe: remove the final addition, then undo division. A fraction is a division. Its bottom must remain nonzero while you clear it from an equation. Finding an allowed input for every proposed output proves that every one of those outputs belongs to the range.
Interval notation: endpoints, infinity, and separated pieces
Interval notation writes a stretch of number line using its two ends. A bracket includes an endpoint, like a gate you may stand on. A parenthesis excludes it. Infinity describes a direction that never ends, so it is never an included endpoint. The symbol ∪ joins separate accepted pieces.
What to know cold, rebuild, and keep on one page
Pack for this section the way you pack for a short trip. Keep a few essentials within reach, learn how to assemble the larger tools, and put the reference formulas together. Each starting rule on the toolkit cards above is a Toolkit function. Your exam is closed book, so use the one-page summary while studying, then practice reconstructing its formulas from a point.