Linear functions

You will describe one steady change in words, with a formula, in a table and on a graph. You will learn what the slope and the starting value say about that change. Then you will build an equation from points, a graph, a table or a real situation. Each method finishes with a check that puts the answer back into the information you were given.

Lessons

Vocabulary

Function FUNK-shun: A rule that assigns exactly one output to each allowed input. Like: A machine that gives one result for each permitted setting.
Linear function LIN-ee-er FUNK-shun: A function with constant rate of change, written f(x) = mx + b. Its graph lies on a straight nonvertical line; this section includes constant functions. Like: A meter adding the same amount for each equal step.
Polynomial pol-ee-NOH-mee-ul: An expression made by adding terms that multiply numbers by nonnegative whole-number powers of a variable. Like: A sum built from whole-number layers of multiplication.
Degree duh-GREE: The largest variable exponent in a polynomial after combining terms, ignoring terms with zero coefficients. A nonzero constant has degree zero. Like: The highest multiplication layer that actually remains.
Input IN-put: The value you put into a function, often written x. Like: The setting you choose on a machine.
Output OUT-put: The value a function produces for a particular input, often written y or f(x). Like: The result that comes out of a machine.
Independent variable in-duh-PEN-dunt VAIR-ee-uh-bul: The variable used for the input. You choose its allowed value, and the function determines the corresponding output. Like: The knob you set before the machine responds.
Dependent variable duh-PEN-dunt VAIR-ee-uh-bul: The variable used for the output. Its value depends on the input through the function's rule. Like: The meter reading that responds to the knob setting.
Ordered pair OR-derd pair: Two numbers written in a fixed order, (x, y). The first gives horizontal position and the second vertical position. Like: An address that lists across before up or down.
Coordinate pair koh-OR-duh-nut pair: An ordered pair giving the horizontal and vertical coordinates of one point. Like: The two parts of one map address.
x-coordinate eks koh-OR-duh-nut: The first number of an ordered pair, giving horizontal position. For a function graph, it is the input. Like: The sideways part of an address.
y-coordinate why koh-OR-duh-nut: The second number of an ordered pair, giving vertical position. For a function graph, it is the output. Like: The height part of an address.
Word form werd form: A description of a function's input, output and rule using sentences. Like: Giving spoken directions rather than a map.
Function notation FUNK-shun noh-TAY-shun: Writing such as f(x) that names a function and its input. f(3) means the output of f at input 3. Like: A machine label followed by its setting.
Tabular form TAB-yuh-ler form: A table that aligns each selected input with its corresponding output. Like: A list pairing each machine setting with its result.
Graphical form GRAF-ih-kul form: A picture that places input horizontally and output vertically to show a function's relationship. Like: A map of how the output moves as the input changes.
Slope slohp: The output change divided by the corresponding nonzero input change. It says how much output changes per input unit. Like: The height gained for each step across a ramp.
Rate of change rayt uv chaynj: How much output changes for each unit of input change, measured by a ratio of changes. Like: The extra distance covered for each second on a timer.
Constant rate of change KON-stunt rayt uv chaynj: A rate that stays the same between every pair of distinct allowed inputs. Equal input changes give equal output changes. Like: A meter adding the same charge for every equal mile.
Rise ryz: The signed change in output between two points, $y sub 2$ − $y sub 1$ . It is negative when the output falls. Like: The upward or downward change in a ramp's height.
Run run: The signed change in input between two points, $x sub 2$ − $x sub 1$ . It is the denominator in the slope ratio. Like: The sideways change along a ramp.
Vertical displacement VER-tih-kul dis-PLAYS-munt: The signed change in vertical position between two points. For a function graph, it is the rise. Like: An elevator's signed change in height.
Horizontal displacement hor-ih-ZON-tul dis-PLAYS-munt: The signed change in horizontal position between two points. For a function graph, it is the run. Like: Your sideways change along a walking path.
Absolute value AB-suh-loot VAL-yoo: A number's distance from zero, ignoring direction. It is always nonnegative. Like: How far you walked from the start, regardless of direction.
Steepness STEEP-nus: How sharply a line tilts. With the same units and axis scales, a larger absolute slope means a steeper line. Like: How sharply a ramp climbs or descends.
y-intercept why IN-ter-sept: The point where a graph meets the y-axis, (0, b). The intercept value b is the output at input zero. Like: The meter reading before any distance is added.
Vertical intercept VER-tih-kul IN-ter-sept: Another name for the y-intercept, where a graph meets the vertical axis at input zero. Like: The starting mark on the upright axis.
Initial value ih-NISH-ul VAL-yoo: The output at input zero, often a starting amount in a model. It is b in f(x) = mx + b. Like: What the taxi meter shows before the ride adds miles.
Slope-intercept form slohp IN-ter-sept form: Writing a line as y = mx + b or f(x) = mx + b, with slope m and input-zero output b. Like: Writing the per-mile charge beside the starting taxi charge.
Point-slope form point slohp form: Writing a line as y − $y sub 1$ = m(x − $x sub 1$ ), using slope m and one known point ( $x sub 1$ , $y sub 1$ ). Like: Anchoring a ramp at one known place and setting its tilt.
Increasing linear function in-KREE-sing LIN-ee-er FUNK-shun: A linear function with positive slope. Its output becomes larger as its input becomes larger. Like: A ramp rising as you walk to the right.
Decreasing linear function duh-KREE-sing LIN-ee-er FUNK-shun: A linear function with negative slope. Its output becomes smaller as its input becomes larger. Like: A ramp falling as you walk to the right.
Constant function KON-stunt FUNK-shun: A function with the same output at every allowed input. Its graph is horizontal and its slope is zero. Like: A flat monthly fee regardless of days used.
Domain doh-MAYN: The set of all inputs allowed for a function or its stated real-world model. Like: The settings a machine permits you to choose.
Real number REEL NUM-ber: A number represented by a position on the number line, including integers, fractions, terminating or repeating decimals, and irrational numbers. Like: Any location along one continuous measuring path.
Nonnegative non-NEG-uh-tiv: Zero or greater. Zero is included. Like: A timer that starts at zero and moves forward.
Model MOD-ul: A mathematical description of a situation, used within stated assumptions and allowed inputs. Like: A simplified map that keeps the features needed for one trip.
Fixed cost fikst kawst: A cost paid even when no items are produced. It stays unchanged as production count changes within the model. Like: Rent that is due even when the workshop makes nothing.
Overhead OH-ver-hed: The operating expenses that keep a business running. In this cost model, monthly overhead is part of the fixed cost. Like: Paying to keep the workshop open before making an item.
Production cost pruh-DUK-shun kawst: The cost of making items. This section models it as a fixed amount per item multiplied by the number of items. Like: Buying materials each time you make another object.
Variable cost VAIR-ee-uh-bul kawst: The part of total cost that changes with production count. Here it equals the per-item cost times the number produced. Like: The supply bill that grows as more objects are made.
Marginal cost MAR-jih-nul kawst: The extra cost of producing one additional item. In a linear cost model, it is the slope, with units dollars per item. Like: The amount the bill rises when you add one more item.
Base salary bays SAL-uh-ree: The salary paid before any commission is added. It is the income at zero sales in this model. Like: A guaranteed starting amount before rewards for extra sales.
Commission kuh-MISH-un: Pay earned from sales. Here a fixed payment is added for each policy sold, so commission per policy is the income slope. Like: An extra reward added each time you make a sale.
PSI pee ess eye: Pounds per square inch, a unit of pressure. It describes force distributed over area. Like: How strongly something presses on each square-inch patch.
Linear equation LIN-ee-er ih-KWAY-zhun: An equation describing a straight line. Here nonvertical lines are written y = mx + b; a vertical line is written x = c. Like: A written address rule for every spot along one straight road.
Horizontal line hor-ih-ZON-tul lyn: A line whose height stays the same. Its equation is y = b and its slope is zero. Like: A level floor.
Vertical line VER-tih-kul lyn: A line whose horizontal coordinate stays fixed. It has undefined slope and is not a function of x. Like: An upright wall.
Slope formula slohp FOR-myuh-luh: The fraction that divides the change in output by the matching change in input, using the same point order. Like: Height change per forward step on a ramp.
Evaluate ih-VAL-yoo-ayt: Find an expression's value or a function's output after replacing its variable with a given number. Like: Choose a machine setting and read the result.
Coefficient koh-uh-FISH-unt: The number multiplying a variable or a power of a variable. Like: How many copies of a named amount you count.
x-axis eks AK-sis: The horizontal number line on a coordinate graph. Points on it have y-coordinate zero. Like: The sideways ruler on a map.
y-axis why AK-sis: The vertical number line on a coordinate graph. Points on it have x-coordinate zero. Like: The upright ruler on a map.
Origin OR-ih-jin: The point (0, 0) where the horizontal and vertical axes meet. Like: The shared starting mark of two rulers.
Coordinate koh-OR-duh-nut: One number specifying a point's position along an axis. Like: One part of a map address.
Integer IN-tuh-jer: A whole-number value without a fractional part, positive, negative or zero. Like: A numbered floor rather than a position between floors.
Rational number RASH-uh-nul NUM-ber: A number expressible as an integer divided by a nonzero integer. Its decimal ends or repeats. Like: A measurement that can be named exactly with equal fraction pieces.
Irrational number ih-RASH-uh-nul NUM-ber: A real number that cannot be written as an integer divided by a nonzero integer. Its decimal never ends or repeats. Like: A ruler location that no integer fraction can name exactly.
Ray ray: Part of a line that begins at one endpoint and continues without end in one direction. Like: A route with a starting point and no endpoint in its forward direction.
Segment SEG-munt: Part of a line between two endpoints. Here both endpoints are included. Like: A route from one chosen stop to another.
Dividend DIV-uh-dend: The amount being divided in a division. Like: The total you start sharing.
Divisor duh-VY-zer: The number you divide by. It must be nonzero. Like: The number of groups sharing a total.
Quotient KWOH-shunt: The result of a division. Like: The amount each equal group receives.
Remainder rih-MAYN-der: The amount left after taking complete equal groups during division. Like: The items left after filling equal boxes.
Numerator NOO-muh-ray-ter: The top of a fraction, counting the indicated equal pieces. Like: How many equal slices you have.
Denominator dih-NOM-uh-nay-ter: The nonzero bottom of a fraction, naming the number of equal parts in a whole. Like: How many equal slices divide the whole.
Exponent EK-spoh-nunt: The raised number in a power. A positive integer exponent counts repeated factors. Like: How many copies enter a multiplication.
Factor FAK-ter: A number or expression multiplied by another to make a product. Like: One ingredient of a multiplication.

Quick checks

Find the slope through (−1, 4) and (2, −5).

−3: = = −3, using the same point order in both changes.

Write the line with slope −3 through (−1, 4).

y = −3x + 1: point-slope form is y − 4 = −3(x + 1). Distribute, then add 4 to both sides.

A linear function has f(0) = 6 and f(4) = 14. Find f(x).

f(x) = 2x + 6: slope is = 2 and the input-zero output is 6.

Is g(x) = −x + 5 increasing or decreasing?

Decreasing, because its slope − is negative.

What kind of line is h(x) = 7?

Constant and horizontal, with slope zero, because every input gives output 7.

Can the train's elapsed-time input be 2.5 seconds? Can it be −2 seconds in the stated model?

2.5 seconds is allowed.
−2 seconds is excluded.
The domain is nonnegative real elapsed time, rather than only whole seconds.

In C(x) = 1250 + 37.5x, give the meanings and units of both numbers.

1250: fixed cost of $1,250 for the month.
37.5: marginal cost of $37.50 per item.
The first is the output at zero items.
The second is the added cost per input unit.

Before you start

Explain it like I am five: a taxi meter
You get into a taxi. The meter already shows $4 before the car moves. That is the amount you pay for starting the ride.
Now imagine the meter adds $2 for each mile. After one mile, you pay $6. After two miles, you pay $8. The added amount stays the same for each extra mile.
You can keep the whole story in one sentence: start with $4, then add $2 for every mile. A formula, a table and a picture can all tell that same story. This section teaches you to move between those versions.
An ordered pair is an address
A point on a graph is like an address with two directions. An ordered pair, written (x, y), tells you the sideways position first and the up-or-down position second. A coordinate is one of these position numbers. The horizontal line is the x-axis. The vertical line is the y-axis. They meet at the origin, (0, 0). Positive x means right and negative x means left. Positive y means up and negative y means down. In a function problem, x usually records the input, the number you supply, and y records the output, the number the rule gives back. A letter such as x is a variable, a placeholder for a number.
Subtracting signed numbers and dividing with signs
Think of positive and negative numbers as positions on a walking path. Adding a positive amount moves you right. Adding a negative amount moves you left. Subtraction undoes the move named after it. Subtracting a negative amount undoes a leftward move, so you move right instead. Division also needs a sign check. In a ÷ b, a is the dividend, the amount being divided; b is the divisor, the number you divide by; and the result is the quotient. With a nonzero dividend and divisor, a quotient is positive when the two signs match and negative when they differ. Division by zero has no answer. These ideas matter when you compare coordinates that lie on opposite sides of zero. An integer is a number with no fractional part, such as −3, 0 or 4. Even integers are multiples of 2, meaning 2 times an integer, such as −4, 0 and 6. Odd integers are one more or one less than an even integer, such as −3, 1 and 5. The positive-or-negative division rule applies when the dividend is nonzero. Zero divided by a nonzero number is zero.
Fractions and matching piece sizes
A fraction is a division written in two levels. In $3 over 5$ , the bottom number says the whole is divided into five equal pieces, and the top says you have three pieces. You can rename a fraction without changing its size by multiplying or dividing both numbers by the same nonzero number. To add or subtract fractions, the pieces must have the same size. That means the bottom numbers, called denominators, must match. A whole number can also be renamed as a fraction so that it uses the same piece size. The top number is the numerator. The bottom number is the denominator and must be nonzero. A reduced fraction has no common integer factor greater than 1 left in its top and bottom. A factor is a number multiplied by another number to make a product.
Distribute a multiplier and combine matching terms
Imagine three bags, each containing x dollars with two dollars taken out. Three copies of the whole bag means three copies of x and three copies of the two-dollar deduction. This is distribution: a number outside parentheses multiplies every term inside. A term is one piece joined to another by addition or subtraction. After distributing, you can combine matching terms. Three copies of x plus one copy of x make four copies of x. A plain number and an x term do different jobs, so they stay separate.
Solve an equation and put the answer back
An equation is like a balanced scale. The two sides have equal value. Solving means finding the number that a letter must stand for to keep the scale balanced. You may add, subtract, multiply or divide by the same nonzero number on both sides. Work backward through the operations attached to the letter. The purpose of each move is to leave the letter alone. Then replace the letter with your answer in the original equation. That check catches a sign error even when the algebra looks tidy.
Function notation and the order of operations
A function is like a machine with a named rule. Writing f(3) tells you to put 3 into the machine named f. The parentheses here name the input; they do not mean f multiplied by 3. To evaluate means to find the output for that input. Replace every x in the formula by the input, using parentheses around a negative number. Then follow the order of operations: work inside parentheses, do powers, multiply or divide from left to right, and add or subtract from left to right.
Coefficients, powers, polynomials and degree
Think of labeled boxes on a shelf. Three boxes holding x dollars each contain 3x dollars. The multiplying number 3 is the coefficient: it counts copies of the amount. A power counts repeated factors instead. In $x^{2}$ , the raised 2 is the exponent and means x × x. A polynomial adds terms made from numbers times nonnegative integer powers of the variable. Its degree is the largest exponent that remains after matching terms are combined. You can use those names to distinguish a line whose input term remains from a constant whose input term disappears.
Real numbers, rational numbers and irrational numbers
Imagine a ruler with no gaps between its marks. Every place on it represents a real number. Some places can be described by a fraction of integers with a nonzero denominator, such as one half. Those are rational numbers. Their decimals end or settle into a repeating pattern. Other places cannot be described by any such integer fraction. Those are irrational numbers. Their decimals keep going without a repeating pattern. Both kinds can measure time or distance. Nonnegative means zero or greater. A square root asks which nonnegative number multiplied by itself gives the number inside the root.
Allowed inputs, intervals and whole-number counts
A machine accepts only suitable inputs. A taxi can travel two and a half miles, but a salesperson cannot sell two and a half complete policies in this model. The domain is the set of allowed inputs. Nonnegative means zero or greater. An interval describes every real number between its ends. Square brackets include a finite endpoint; round parentheses leave one out. Infinity means the interval keeps going and is never an endpoint you can reach, so it always gets a parenthesis. A count needs separate whole numbers rather than every number in an interval.
Decimal multiplication, units and rounding
Decimal digits name smaller pieces of a whole. A number with three decimal places, such as 0.434, counts thousandths. To multiply decimals by hand, multiply the digits as whole numbers first, then restore the total number of decimal places. A unit tells you what a number measures. In a rate, the output unit is divided by the input unit, such as dollars per item. Rounding replaces an exact value with a nearby one. Keep the exact value during your calculation and round only when the requested answer needs it.
Know cold, rebuild and put on the cheat sheet
Use three jobs for your studying. Know cold means recall a short fact without notes. Understand, then rebuild it when needed means use an explained idea to recover a formula or answer. Put on the cheat sheet means keep a compact study reference rather than memorize a long worked solution. Your exam is closed book, so this sheet is for practice beforehand. Put it away and rebuild the methods from memory during practice. The four-line formula card is a compact printable reference; keep the meaning of every symbol beside the formula while you study. This is a study map. The lessons that follow teach the terms, derive the formulas and work the methods named in these lists. Read those explanations before using the map to practice from memory.