Graphs of linear functions

You will draw straight-line graphs three ways, then work backward from a graph to its equation. You will find axis crossings and distinguish horizontal and vertical lines. Next you will compare line directions and write parallel or perpendicular lines through a required point. Finally you will find shared points of two lines and use that idea to compare expenses with sales at break-even.

Lessons

Vocabulary

linear function LIN-ee-er FUNK-shun: A rule of the form f(x) = mx + b, with fixed numbers m and b. Its graph is a straight nonvertical line, and equal input changes give equal output changes. Like: A taxi fare with a starting fee and the same charge for every mile.
graph graf: A picture of number pairs. Each plotted point places its first number horizontally and its second number vertically. A function's graph shows its inputs and corresponding outputs. Like: A map where each address uses an across distance and an up or down distance.
input IN-put: The number you put into a function's rule. It appears inside the function's parentheses and is usually the horizontal coordinate x on its graph. Like: The button you choose on a vending machine.
output OUT-put: The number a function gives after using its input. It is written f(x), and it is the vertical coordinate y when the function is graphed. Like: The item delivered after you press a vending machine button.
coordinate pair koh-OR-duh-nit pair: An address written (x, y). The first number tells your horizontal position and the second tells your vertical position. On a function's graph, they are input and output. Like: Directions that say how far across, then how far up or down.
x-axis eks AK-sis: The horizontal number line on a coordinate graph. Every point on it has y = 0. Positive x goes right and negative x goes left. Like: The east and west road through the center of a map.
y-axis why AK-sis: The vertical number line on a coordinate graph. Every point on it has x = 0. Positive y goes up and negative y goes down. Like: The north and south road through the center of a map.
origin OR-uh-jin: The point (0, 0), where the x-axis and y-axis meet. It is the starting address for measuring horizontal and vertical positions. Like: The doorway you start from before following directions.
y-intercept why IN-ter-sept: Where a graph meets the y-axis. Its input is x = 0. For y = mx + b, the intercept value is b and the intercept point is (0, b). Like: The starting height before you begin walking horizontally.
initial value ih-NISH-ul VAL-yoo: The output when the input is 0. In a linear function f(x) = mx + b, it is b. It describes the amount present before input-dependent change begins. Like: A taxi's starting fee before you travel any miles.
slope slohp: A line's signed vertical change divided by its signed horizontal change. It tells the output change per input unit. A vertical line has no defined slope because its horizontal change is 0. Like: A ramp's height gained for each step across.
rate of change rayt uv chaynj: How much an output changes for each unit of input change. For a linear function it is constant and equals the slope. Its units are output units per input unit. Like: The same extra charge for every additional mile.
rise ryz: The signed vertical change between two points, $y sub 2$ − $y sub 1$ . An upward move has positive rise, and a downward move has negative rise. Like: The change in your height as you walk along a ramp.
run run: The signed horizontal change between two points, $x sub 2$ − $x sub 1$ . A rightward move has positive run, and a leftward move has negative run. Like: How far across the ground you walk along a ramp.
slope-intercept form slohp IN-ter-sept form: The writing y = mx + b, or f(x) = mx + b. It displays the slope m and the initial value b, giving the y-intercept point (0, b). Like: A route description giving your starting height and the rise per step across.
identity function eye-DEN-tuh-tee FUNK-shun: The function f(x) = x, whose output equals its input. Its graph passes through the origin and has slope 1. Like: A machine that returns exactly what you put in.
transformation trans-for-MAY-shun: A change to a graph made by changing its rule. Shifts move it, reflections flip it, and stretches or compressions change how far points sit from an axis. Like: Moving, flipping or resizing a drawing on a sheet of clear plastic.
vertical stretch VER-tih-kul strech: Multiplying every output by a positive factor greater than 1. Points move farther from the x-axis, while their inputs stay fixed. For y = x, the resulting line is steeper. Like: Stretching a drawing taller without making it wider.
vertical compression VER-tih-kul kum-PRESH-un: Multiplying every output by a positive factor between 0 and 1. Points move closer to the x-axis, while their inputs stay fixed. For y = x, the resulting line is flatter. Like: Pressing a drawing shorter without making it narrower.
vertical reflection VER-tih-kul ree-FLEK-shun: Multiplying every output by −1. Each point keeps its input but reverses its height, so the graph flips across the x-axis. Like: Seeing a drawing reflected across a horizontal mirror.
vertical shift VER-tih-kul shift: Adding the same number to every output. A positive addition moves a graph up; a negative addition moves it down. For a line, its slope stays the same. Like: Sliding a drawing upward or downward without tilting it.
absolute value AB-suh-loot VAL-yoo: A number's distance from 0, so it is never negative. For a slope, its absolute value measures steepness without recording whether the line rises or falls. Like: How far your address is from home, ignoring which side of home it lies on.
increasing function in-KREE-sing FUNK-shun: A function whose outputs grow as its inputs grow over the input values being discussed. A linear function with positive slope is increasing across its whole domain. Like: Climbing higher as you walk right along a ramp.
decreasing function dee-KREE-sing FUNK-shun: A function whose outputs shrink as its inputs grow over the input values being discussed. A linear function with negative slope is decreasing across its whole domain. Like: Walking downhill while moving right along a ramp.
x-intercept eks IN-ter-sept: Where a graph meets the x-axis. The output there is 0. The intercept value is the input x, and its full point is (x, 0). Like: The place a ramp reaches ground level.
horizontal line hor-ih-ZON-tul lyn: A line with the same vertical coordinate at every point. Its equation is y = c for a fixed number c, and its slope is 0. Like: A level shelf with no change in height as you move across.
vertical line VER-tih-kul lyn: A line with the same horizontal coordinate at every point. Its equation is x = c. Its slope is undefined, and it is not a function of x. Like: A straight elevator shaft with one across position and many heights.
constant function KON-stunt FUNK-shun: A function that gives the same output for every input in its domain. Its graph is horizontal, and its rate of change is 0. Like: A machine that always returns the same item whichever button you press.
undefined slope un-dee-FYND slohp: The situation on a vertical line: its run is 0, so rise divided by run requires division by 0. No numerical slope exists. Undefined slope is different from slope 0. Like: Trying to measure height change per step across when you take no steps across.
vertical line test VER-tih-kul lyn test: A graph represents y as a function of x exactly when every vertical line meets it at most once. Two meetings would give one input two different outputs. Like: Holding one vending machine button fixed and checking that it promises only one result.
parallel lines PAIR-uh-lel lynz: Distinct lines in the same plane that never meet. Nonvertical parallel lines have equal slopes and different y-intercepts. Two distinct vertical lines are parallel too. Like: Two straight tracks that keep the same direction and never cross.
coincident lines koh-IN-sih-dunt lynz: Two line equations whose graphs are the same line. For slope-intercept equations, both the slopes and y-intercepts match. Every point on one is also on the other. Like: Tracing one line exactly on top of another.
perpendicular lines per-pun-DIK-yuh-ler lynz: Lines that meet at a right angle. For finite nonzero slopes, one is the negative reciprocal of the other, and their product is −1. Horizontal and vertical lines are also perpendicular. Like: The floor and a straight wall meeting at a square corner.
right angle ryt ANG-gul: An angle measuring 90°, which is one quarter of a full turn. Perpendicular lines form right angles where they meet. Like: The square corner of a sheet of paper.
reciprocal rih-SIP-ruh-kul: The number that multiplies with a given nonzero number to make 1. For a fraction, exchange its numerator and denominator while keeping its sign. Zero has no reciprocal. Like: An undo button for multiplication.
negative reciprocal NEG-uh-tiv rih-SIP-ruh-kul: The opposite of a nonzero number's reciprocal. Flip the fraction, then reverse its sign. Its product with the original number is −1, which connects finite nonzero perpendicular slopes. Like: An undo multiplier followed by a reversal of direction.
point-slope form poynt slohp form: The equation y − $y sub 1$ = m(x − $x sub 1$ ). It describes a nonvertical line using slope m and a known point ( $x sub 1$ , $y sub 1$ ). Each change is measured from that point. Like: A ramp described by one starting address and its rise per step across.
system of linear equations SIS-tum uv LIN-ee-er ih-KWAY-zhunz: Two or more linear equations considered together. A solution must make every equation true at the same time. For two equations in x and y, shared graph points show the solutions. Like: Two address requirements that the same location must satisfy.
solution to a system suh-LOO-shun too uh SIS-tum: A set of variable values that makes every equation in the system true. In two variables, write a full coordinate pair and check it in both equations. Like: One address that passes every requirement on your list.
point of intersection poynt uv in-ter-SEK-shun: A point shared by two graphs. It has the same horizontal and vertical coordinates on both. For two line equations, such a point is a solution to their system. Like: The address where two roads cross.
contradiction kon-truh-DIK-shun: A statement that cannot be true, such as 0 = 4. If correct equation steps produce a contradiction, no values satisfy the original equations together. Like: Being required to stand at two different addresses at the same time.
infinitely many solutions IN-fuh-nit-lee MEN-ee suh-LOO-shunz: A system has infinitely many solutions when an unlimited collection of variable values satisfies every equation. Two coincident lines share every point on their line. Like: Two copies of the same road map allowing all the same addresses.
break-even point brayk EE-vun poynt: The quantity and money amount where total revenue equals total cost. It is the intersection of their graphs. At that quantity, profit is 0. Like: Earning back exactly what you spent.
fixed cost fikst kost: A cost that stays the same as the number of items changes within a model. It is present even at quantity 0 and contributes the cost function's initial value. Like: Paying the same workshop rent before making any items.
variable cost VAIR-ee-uh-bul kost: The part of cost that changes with the number of items. In a linear model, a constant cost per item multiplied by quantity gives the total variable cost. Like: Buying more material whenever you make another item.
cost function kost FUNK-shun: A rule giving total cost for an input quantity. In a linear model, C(x) = vx + F adds total variable cost vx to fixed cost F. Like: A spending calculator that includes supplies and the starting bill.
revenue REV-uh-noo: The money received from sales before subtracting costs. If every item sells for the same price, total revenue is price multiplied by the number sold. Like: The money entering the cash drawer from customers.
revenue function REV-uh-noo FUNK-shun: A rule giving sales money for an input quantity. With a constant selling price p and quantity x sold, R(x) = px. Selling 0 items then produces revenue 0. Like: A sales calculator using the price and the number sold.
profit PROF-it: Revenue minus total cost. A positive result is money remaining after costs. A negative result is a loss, and 0 means break-even. Like: What remains in your cash drawer after paying every bill.
function FUNK-shun: A rule assigning exactly one output to each allowed input. Different inputs may share an output, but one input cannot receive two different outputs. Like: A button has one assigned result.
real number REE-ul NUM-ber: A number with a position on the number line, including whole numbers, negatives, fractions and decimals. Nonzero means different from 0. Finite means having a numerical size rather than infinity. Like: Any address along an unbroken straight road.
domain doh-MAYN: The collection of allowed inputs for a function. A formula may accept every real input while a practical situation permits only meaningful quantities. Like: The buttons a machine permits you to press.
coefficient koh-uh-FISH-unt: The number multiplying a variable. In a linear function's slope-intercept form, the coefficient of the input variable is its slope. Like: The price attached to each repeated item.
numerator NOO-mer-ay-ter: The top number of a fraction. It counts how many of the denominator's equal-sized parts you have, or names the amount being divided. Like: The number of slices you took.
denominator dih-NOM-uh-nay-ter: The bottom number of a fraction. It specifies the equal parts in a whole, or the divisor. A denominator cannot be zero in a defined fraction. Like: How many equal slices the whole pizza was cut into.
fraction FRAK-shun: One number divided by another nonzero number, written with a top and bottom. It can describe part of a whole, a quotient or a comparison. Like: Your share of equal pizza slices.
ratio RAY-shee-oh: A comparison made by division. Slope compares vertical change with horizontal change as the ratio rise divided by run. Like: Two cups of flour for each cup of water.
order of operations OR-der uv op-er-AY-shunz: The agreed calculation order: grouping, powers, multiplication and division, then addition and subtraction. Within equal-priority operations, work left to right. A power repeats multiplication. Like: A recipe whose preparation steps must happen in order.
horizontal shift hor-ih-ZON-tul shift: Moving a graph left or right without changing its shape. The rule g(x) = f(x − h) moves points right h when h is positive. Like: Sliding a drawing sideways on a desk.
equation ih-KWAY-zhun: A statement that two expressions have equal values. A solution makes the equality true. Adding the same amount to both sides preserves their equality. Like: Two sides of a balanced scale.
variable VAIR-ee-uh-bul: A letter representing a number that may change or is unknown. The same letter has the same value everywhere within one equation. Like: A labeled box whose contents you have not chosen yet.
vertical displacement VER-tih-kul dis-PLAYS-munt: The signed change in vertical position. Between graph points it is the rise, $y sub 2$ − $y sub 1$ . Going down produces a negative displacement. Like: How much your height changes on a ramp.
horizontal displacement hor-ih-ZON-tul dis-PLAYS-munt: The signed change in horizontal position. Between graph points it is the run, $x sub 2$ − $x sub 1$ . Going left produces a negative displacement. Like: How far your address changes across a map.
factor FAK-ter: One of the numbers multiplied to make a product. A whole-number factor of a positive whole number divides it with nothing left over. Like: The size of one repeated group.
product PROD-ukt: The answer to multiplication. Like: The total from equal groups.
term term: One piece being added or subtracted in an expression. Like: One item in a bill's sum.
divisor dih-VY-zer: The number you divide by. It must be nonzero in defined division. Like: The number of equal groups you share among.
quotient KWOH-shunt: The answer to division. Like: Each group's share after dividing equally.
constant KON-stunt: A fixed number. A constant term has no changing variable attached. Like: A setup charge paid once.
nonnegative non-NEG-uh-tiv: Zero or positive. Nonnegative numbers include 0 and exclude every negative number. Like: An item count can begin at zero but cannot fall below zero.
continuous model kun-TIN-yoo-us MOD-ul: A model allowing inputs between whole-number counts. A drawn line may cross at a fractional input even when actual sales require whole items. Like: A ruler lets you mark positions between its numbered marks.

Quick checks

Find the input giving output 0 for f(x) = 4x − 8, and name its x-intercept point.

x = 2.
Intercept point: (2, 0).
Reason: 0 = 4x − 8 gives 4x = 8, and f(2) = 8 − 8 = 0.

What slope makes a right angle with y = x + 1?

−, because × (−) = −1.

Are y = 3x − 1 and 6x − 2y = 8 parallel?

Yes. Isolating y in the second gives y = 3x − 4: both slopes are 3 and their intercepts differ.

Find the shared point of y = 2x + 1 and y = −x + 7.

(2, 5), because equal outputs give 3x = 6, and both formulas return 5 at input 2.

What is the vertical line through (−4, 9)?

x = −4, because the across coordinate stays fixed while the height can vary.

Is y = −x + 7 increasing or decreasing?

Decreasing, because each move right 4 lowers the output by 3.

At the small-item break-even point, cost and revenue are both $25. What is the profit?

$0, because profit = revenue − cost = $25 − $25.

A two-line system simplifies to 0 = 0. Do all points in the plane solve it?

No. The coincident lines have infinitely many solutions on their common line
points away from that line fail the original equations.

Before you start

Signed numbers: walking, subtracting, multiplying and dividing
Think of a number line as a sidewalk with 0 at your doorway. Positive numbers are addresses to the right. Negative numbers are addresses to the left. Adding a positive number moves you right, and adding a negative number moves you left. Subtracting undoes an addition. So subtracting a negative number reverses a leftward move and sends you right. A minus sign may name a negative number, as in −9, or tell you to subtract, as in 4 − 9. Parentheses keep those two jobs visible. Multiplying or dividing numbers also uses their signs, so keep the direction separate from the size. The symbol = says the amounts or calculations on both sides have the same value. An equation is a statement requiring the two sides of = to have equal values. The symbols × and ÷ mean multiplication and division. Numbers you multiply are factors, and the answer is their product. A term is one piece separated from the next by addition or subtraction, such as the 3 and −3 in 3 + (−3). Nonzero means different from 0.
Fractions: equal pieces, common denominators and reducing
Imagine two identical sandwiches. Cutting one into 6 equal pieces and the other into 4 equal pieces makes pieces of different sizes. The fraction $5 over 6$ means 5 of the first sandwich's 6 pieces. Its top number is the numerator, and its bottom number is the denominator. To add or subtract fractions, first describe both amounts using the same size of piece. A common denominator is a shared bottom number that does this. Reducing runs the process backward: it groups smaller pieces into bigger ones without changing how much sandwich you have. For example, $2 over 6$ and $1 over 3$ are the same amount.
Multiplying, dividing and finding a negative reciprocal
Multiplying by a fraction takes that share of an amount. Half of one third of a sandwich is one sixth, so $1 over 2$ × $1 over 3$ = $1 over 6$ . Multiply the tops together and the bottoms together. Division asks how many copies of one amount fit into another. To divide by a nonzero fraction, multiply by its reciprocal: the fraction with its top and bottom exchanged. A reciprocal is the number that multiplies with the original to make 1. A negative reciprocal goes one step farther and changes the sign, so its product with the original is −1. You will use that last relationship for perpendicular slopes. The divisor is the amount you divide by. The quotient is the answer to division. In $2 over 3$ ÷ $4 over 5$ , $4 over 5$ is the divisor, and $5 over 6$ is the quotient.
Absolute value: distance without direction
Think of walking three blocks from home. You could end up three blocks to the left or three to the right, but your distance from home is three blocks in either case. Absolute value means that distance from 0 on a number line. The two bars around a number name its absolute value. Say |−3| as absolute value of negative three. It is 3 because the distance is three units. A distance cannot be negative. Zero is no distance from home, so its absolute value is 0. You will use absolute value to measure a slope's size separately from its direction.
Substitution and function notation: an input goes into a rule
Think of a function as a labeled calculator button. You choose a number, press the button, and its rule produces one answer. A letter such as x stands for the number you will put in. In f(x) = −3x + 8, f names the button, x is the input, and f(x) is its output. The writing −3x means −3 × x. To evaluate f(−2), replace every x in the rule with −2 and do the arithmetic. The parentheses in f(−2) identify the input. They do not mean f times −2. Keep parentheses around a negative replacement so its sign travels with it. An expression is a written calculation, such as −3x + 8. Evaluate means calculate its value at a given input. Substitution means replacing a letter with that input everywhere it occurs.
Distribute to every term, then get y alone
Imagine packing the same contents into several bags. Three bags holding an apple and an orange give three apples and three oranges. Multiplication outside parentheses reaches every term inside. That is distribution. An equation is a balance: its left and right sides have equal value. Adding, subtracting, multiplying or dividing both sides by the same allowed amount keeps that balance. Isolating y means rewriting the balance so one side contains y alone. This lets you read the output directly from an input x. The coefficient is the number multiplying a letter. To undo a coefficient, divide every term on both sides by it. A constant is a fixed number without a letter attached; in 7x − 3y + 6, the term 6 is a constant.
Solve an equation with the letter on both sides
Think of two balanced trays that both contain identical sealed boxes. If you remove two boxes from each tray, they still balance, and you can compare what remains. A variable is a letter standing for an unknown number, like the contents of one box. When the same letter appears on both sides of an equation, gather its terms on one side by performing the same operation on both sides. Then gather the ordinary numbers on the other side. Finally undo the number multiplying the letter. Solving means finding the number that makes the original equality true. A check puts that number back into both original sides.
Decimal coefficients are fractions you can clear
Money shows what a decimal means. A dollar contains 100 cents, so $0.08 is 8 hundredths of a dollar: $8 over 100$ . A decimal coefficient is still an ordinary multiplier. You can solve an equation containing decimals with the same balance moves as before. Multiplying the entire equation by 10, 100 or 1,000 can turn its decimals into whole numbers. Choose the multiplier from the number of decimal places. One place uses 10; two places use 100; three places use 1,000. Multiply every term, including any whole-number fee. The equation gets a different appearance, but it keeps the same solution.
Coordinates are addresses, and a function gives one output
A graph is a map for number pairs. Its horizontal road is the x-axis and its vertical road is the y-axis. They meet at the origin, the address (0, 0). A coordinate pair (x, y) tells you to move horizontally first, then vertically. For (−2, 4), move 2 left and 4 up. When the graph shows a function, x is the input and y is its output. Think of a vending machine: a particular button must have one assigned result. Different buttons can give the same result, but one button cannot promise two different results at once. A function follows that same one-input, one-output requirement. The domain is the collection of inputs the rule allows. A real number is a value with an address on the number line, including whole numbers, fractions and decimals. The real number line has no last address in either direction. Infinity describes that endless continuation, not an extra real-number address. Every real number has a finite numerical value, meaning a size rather than infinity. The coordinate plane is the whole flat map containing the two axes and every coordinate pair. Nonnegative means 0 or positive. Whole-number counts are 0, 1, 2 and so on; they exclude fractions of an item.
Slope and point-slope form: measure a walk from a known point
Picture a ramp on a map. To move between two points on it, you travel horizontally and change height. The run is the signed horizontal change, and the rise is the signed vertical change. Slope divides rise by run, telling you the height change for one horizontal unit. A straight ramp keeps this same ratio everywhere. Once you know one point on a line and its slope, you can describe every other point by its walk from the known point. That description is point-slope form. It says the vertical change equals the slope times the horizontal change. Subscripts such as $x sub 1$ label the first point; they do not multiply x by 1. A ratio is a comparison by division, so slope is a ratio of the two changes.