Week 4 · Chapter

Chapter 4 — Linear Functions & Their Graphs

College Algebra Free sample edition

The module's primary reading · ~35–45 minutes · Objective 4 (analyze linear functions (slope, intercepts, and the forms of a line, including parallel and perpendicular lines) and solve systems of linear equations and inequalities in two variables) · Read it first this week; the lecture, the tutorial, the quiz, and the Model/Build Studio all build on it.

In plain language

Some things change by the same amount every time, like a price that goes up 3 dollars for each topping you add. This week you learn to read that kind of steady change as a straight line on a graph. You will measure a line's slope (how steep it is, or how much it rises for each step to the right). You will find its intercepts (the places where the line crosses the two axes). Last, you will learn to tell when two lines run side by side forever and when they cross at a perfect square corner.

Why you already care about this

Picture a bowl shop near campus. A plain bowl costs 4 dollars, and every topping adds 3 dollars. One topping costs 7 dollars, two cost 10, three cost 13. You could keep a price list on your phone. Or you could notice that every step up the list is the same 3 dollars, and that one fact plus the 4-dollar starting price tells you every price on the menu.

A rideshare in your city works the same way. It charges a pickup fee plus a rate per mile. A seedling on a windowsill grows about the same amount each week. A car on the highway at a steady speed covers the same distance each hour. Each one has a starting value and a steady rate, and each one draws a straight line.

This chapter gives you the tools to read those lines and to build them yourself. You will see that a line's equation is a short sentence with two facts in it, and you will learn to check every equation you write by plugging a point back in.

By the end of this chapter you will be able to:

find the slope of a line from two points and say what it means in units;
tell a rising, falling, horizontal, or vertical line apart from its slope;
read the slope and the y-intercept straight from an equation such as y = 3x + 4;
write the equation of a line from two points and check it by substitution;
find a line's x- and y-intercepts and graph it from them;
decide whether two lines are parallel, perpendicular, or neither by checking their slopes.

Carry this sentence with you: rise over run — the y's go on top, not the x's.

Where we are

Already covered: Week 1 set the sign rules; Week 2 solved linear equations; Week 3 named functions such as g(x) = 3x + 4.
This week: the straight line itself — slope, slope-intercept form, point-slope form, intercepts and standard form, and parallel and perpendicular lines.
How they connect: a line's equation is a function like the ones from Week 3, and every rearranging step this week is a Week 2 move.
Where we're going next: Week 5, Systems of Linear Equations & Inequalities, puts two lines on one graph and asks where they meet.

Each earlier week hands you a tool for this one. Week 1 gave you the sign rules, so you know that −2 times −1 is 2 and that subtracting a negative adds. Week 2 taught you to solve an equation such as 3y = 12 by dividing both sides by 3, and that move turns one form of a line into another. Week 3 called the bowl-shop price a function, a rule that gives exactly one output for each input, and wrote it g(x) = 3x + 4. This week you graph that rule and read its two numbers.

1. Spot a steady rate: equal steps make a line

Imagine you track a seedling on your windowsill. It is 5 cm tall when you start, and it grows 2 cm each week. After one week it is 7 cm, then 9 cm, then 11 cm. Every week adds the same 2 cm. If you plot week against height, the points line up so neatly that a ruler passes through all of them.

That is the whole idea of a line: the output changes by the same amount every time the input goes up by 1. We call that amount a first difference — the amount one output in a table increases or decreases from the output before it when the inputs go up in steps of 1.

A linear function is a function whose graph is a straight line, written f(x) = mx + b; its output changes by the same amount, m, every time the input increases by 1.

Worked example — the bowl-shop table. The bowl shop's price for x toppings is g(x) = 3x + 4 dollars.

Fill in the table. For 0, 1, 2, 3, 4, 5 toppings, the prices g(0), g(1), g(2), g(3), g(4), g(5) are 4, 7, 10, 13, 16, 19 dollars.
Subtract each price from the next. 7 − 4 = 3. 10 − 7 = 3. 13 − 10 = 3. 16 − 13 = 3. 19 − 16 = 3.
Read the pattern. Every first difference is 3.

In words: each topping adds 3 dollars, so the six prices sit on one straight line. If a friend orders five toppings, you can predict the 19-dollar bill before the register does.

Here is the table, with the first differences beside it.

Bowl-shop prices g(x) = 3x + 4 in dollars for 0 to 5 toppings, with the change from the previous row
Toppings (x)	Price in dollars (y)	Change from the previous row
0	4	(first row)
1	7	3
2	10	3
3	13	3
4	16	3
5	19	3

Misconception check. Students often think any table that goes up is a line. It is not. Prices of 4, 8, 16, 32 go up too, but the steps are 4, 8, and 16, not one repeated number. The cure: subtract neighbors. Equal steps mean a line; changing steps mean a curve.

2. Measure steepness: slope as rise over run

Think about a road trip. You drive at a steady 60 miles per hour. After 2 hours you have gone 120 miles, and after 5 hours you have gone 300 miles. In those 3 hours you covered 180 more miles, and 180 ⁄ 3 = 60 miles per hour. That 60 is how steep your distance line is. It is also your speed.

The steepness of a line has a name. The slope is the steepness of a line, written m: the change in y divided by the change in x between any two points on the line, m = (y₂ − y₁) ⁄ (x₂ − x₁). It is the same for every pair of points on one line.

You will also hear it called rise over run, a plain-language description of slope: the vertical change between two points (the rise) divided by the horizontal change between them (the run), with the vertical change on top. And a rate of change is how much an output changes for each one-unit increase in the input; for a line it equals the slope and carries units such as dollars per topping.

In the formula, (x₁, y₁) and (x₂, y₂) are two named points on the line. The small 1 and 2 are labels, not exponents. They tell you which point is which.

Worked example — the slope of the bowl-shop line. Take two points from the table: (0, 4) and (2, 10).

Rise. Subtract the y-values: 10 − 4 = 6.
Run. Subtract the x-values in the same order: 2 − 0 = 2.
Divide. m = (10 − 4) ⁄ (2 − 0) = 6 ⁄ 2 = 3.
Sanity check. The line rises from left to right, and 3 > 0 agrees.

In words: the slope is 3, measured in dollars per topping. That means each topping you add raises your price by 3 dollars.

Any two points on the line give the same slope. Through (1, 7) and (5, 19), m = (19 − 7) ⁄ (5 − 1) = 12 ⁄ 4 = 3. The order does not matter either, as long as you keep it the same on top and bottom. In the reverse order, (4 − 10) ⁄ (0 − 2) = (−6) ⁄ (−2) = 3. Two negatives divide to a positive, so you land on the same 3.

Misconception check. Students often put the x-values on top: (2 − 0) ⁄ (10 − 4) = 2 ⁄ 6 = 1⁄3. That is run over rise, and it is wrong. It is tempting because x comes first in an ordered pair. The cure: say "rise over run" out loud, and remember that the rise is the up-and-down change, which is y.

3. Read the sign: rising, falling, horizontal, and vertical lines

Picture yourself walking along a line from left to right. If you climb, the slope is positive. If you walk downhill, the slope is negative. Your phone plan gives a third case. Say an unlimited plan costs 40 dollars a month no matter how many gigabytes you use. Use 2 gigabytes or 20; the bill stays at 40. That graph is perfectly flat.

A horizontal line is a flat line on which every point has the same y-value; its equation has the form y = k for a number k, and its slope is 0 because its rise is 0.
A vertical line is an up-and-down line on which every point has the same x-value; its equation has the form x = k for a number k, and its slope is undefined because its run is 0.
An undefined slope is the slope of a vertical line, which has no numerical value because finding it would require dividing the rise by a run of 0.

Worked example — a horizontal line. Take the points (0, 4) and (5, 4).

Rise. 4 − 4 = 0.
Run. 5 − 0 = 5.
Divide. m = (4 − 4) ⁄ (5 − 0) = 0 ⁄ 5 = 0.

In words: the slope is 0, every point has y-value 4, and the line is y = 4.

Worked example — a vertical line. Take the points (6, 0) and (6, 4).

Rise. 4 − 0 = 4.
Run. 6 − 6 = 0.
Try to divide. (4 − 0) ⁄ (6 − 6) = 4 ⁄ 0, and no number times 0 gives 4. So the slope is undefined.

In words: every point has x-value 6, and the line is x = 6. A vertical line is not the graph of a function, since the one input 6 has many outputs.

Misconception check. Students often swap the two special lines, or say a vertical line has slope 0. The cure is to ask where the zero lands. A horizontal line has zero rise, and 0 on top gives a slope of 0. A vertical line has zero run, and 0 on the bottom gives no slope at all. Zero on top is fine; zero on the bottom breaks the division.

4. Read a line's equation: slope-intercept form

Go back to the bowl shop. The 4-dollar plain bowl is where the price starts. The 3 dollars per topping is how fast it climbs. Those two facts are the whole equation, and one form of a line puts them right where you can see them.

Slope-intercept form is the equation of a line written y = mx + b, so that the slope m and the y-intercept b can be read off directly.
The y-intercept is the point where a graph crosses the vertical y-axis, found by setting x = 0; for a line y = mx + b it is the point (0, b).

The slope m is the number multiplying x. The y-intercept b is the number standing alone. When x is 0, the mx term is 0, so y = b. That is why the line crosses the y-axis at (0, b).

Worked example — reading y = 3x + 4. Read the two numbers, then test a point.

Find the number multiplying x. It is 3, so m = 3.
Find the number standing alone. It is 4, so b = 4, and the line crosses the y-axis at (0, 4).
Check a point by substitution. Try (2, 10): 3(2) + 4 = 6 + 4 = 10. It matches.

In words: the line starts at 4 dollars and climbs 3 dollars per topping. A customer who wants two toppings pays 10 dollars.

In practice (your tools). Open the free Desmos graphing calculator and type y = 3x + 4. Click the point where the line crosses the y-axis and Desmos labels it (0, 4). Then type y = mx + 4 and accept the slider Desmos offers for m. Drag the slider and watch the line tip up and down around the point (0, 4).

Misconception check. Students often swap m and b. In y = −2x + 5, the slope is −2 and the y-intercept is 5, at the point (0, 5). If you read the slope as 5 and the intercept as −2, you have swapped m and b. It is tempting because 5 is the bigger, friendlier number. The cure: the slope is always glued to x. Find x first, then read its coefficient.

5. Build a line from two points: point-slope form

Sometimes you do not know where a line crosses the y-axis. You only know two points on it. Maybe a delivery app shows your driver at two moments, and you want the whole path. You can still write the equation in two steps: find the slope, then anchor it to one point.

Point-slope form is the equation of a line written y − y₁ = m(x − x₁), built from one known point (x₁, y₁) on the line and the line's slope m.

Worked example — the line through (−1, 4) and (3, −4). Find the slope first, then anchor it to one point.

Slope. m = (−4 − 4) ⁄ (3 − (−1)). On top, −4 − 4 = −8. On the bottom, subtracting −1 adds 1, so 3 − (−1) = 3 + 1 = 4. So m = (−8) ⁄ 4 = −2.
Pick a point. Use (−1, 4), so x₁ = −1 and y₁ = 4.
Fill in point-slope form. y − 4 = −2(x − (−1)), which is y − 4 = −2(x + 1).
Distribute the −2. −2 times x is −2x, and −2 times 1 is −2. So y − 4 = −2x − 2.
Add 4 to both sides. y = −2x − 2 + 4, so y = −2x + 2.
Check both points. At x = 3, −2(3) + 2 = −6 + 2 = −4, which matches (3, −4). At x = −1, −2(−1) + 2 = 2 + 2 = 4, which matches (−1, 4).

In words: the line falls 2 units for every 1 unit you move right, and it crosses the y-axis at (0, 2). Both points check, so you can trust the equation.

Misconception check. Students often write y − 4 = −2(x − 1) when the point is (−1, 4). It is tempting to copy the 1 and forget its sign. The cure: write the point's x-value in parentheses, x − (−1), and only then simplify to x + 1. Your check catches this every time, because the wrong line misses (−1, 4).

6. Find the intercepts: standard form and graphing from two points

Suppose your study group has 12 dollars for snacks and drinks. Snacks cost 2 dollars each and drinks cost 3 dollars each. If x is the number of snacks and y is the number of drinks, spending the whole budget means 2x + 3y = 12. That is a line too, written a different way.

Standard form is the equation of a line written Ax + By = C, with the x-term and the y-term on the left side and a constant on the right, where A and B are not both 0.
The x-intercept is the point where a graph crosses the horizontal x-axis, found by setting y = 0 and solving for x.

Two points are all you need to draw a line, and the two intercepts are the easiest points to find.

Worked example — the intercepts of 2x + 3y = 12. Set one variable to 0 at a time.

Find the y-intercept. Set x = 0: 2(0) + 3y = 12, so 3y = 12. Divide both sides by 3: y = 4. The y-intercept is (0, 4).
Find the x-intercept. Set y = 0: 2x + 3(0) = 12, so 2x = 12. Divide both sides by 2: x = 6. The x-intercept is (6, 0).
Graph. Plot (0, 4) and (6, 0), and draw the line through them.
Read the budget. The 12 dollars buys 4 drinks and no snacks, or 6 snacks and no drinks.

In words: the group can spend everything on drinks or everything on snacks, and every point on the line between is another way to spend exactly 12 dollars.

Worked example — from standard form to slope-intercept form. Solve 2x + 3y = 12 for y, using the Week 2 moves.

Subtract 2x from both sides. 3y = −2x + 12.
Divide every term by 3. y = −(2⁄3)x + 4.
Read it. The slope is −2⁄3, and the y-intercept is 4.
Check with the intercepts. From (0, 4) to (6, 0), the slope is (0 − 4) ⁄ (6 − 0) = (−4) ⁄ 6 = −2⁄3. It matches.

In words: every extra snack costs two-thirds of a drink. The line falls, which makes sense, because more snacks leave room for fewer drinks.

In practice (your tools). In Desmos, type 2x + 3y = 12 exactly as written; Desmos graphs standard form directly. Click the line where it meets each axis and Desmos marks (0, 4) and (6, 0). Then type y = -(2/3)x + 4 on the next line and watch it land on top of the first line.

Misconception check. Students often read the slope of 2x + 3y = 12 as 2, because 2 is the number next to x. That shortcut only works in slope-intercept form. The cure: solve for y first. Here the slope is −2⁄3, not 2.

7. Compare two lines: parallel and perpendicular

Picture two rail tracks. They run side by side and never meet, because they climb at exactly the same angle. Now picture the corner of a sheet of paper, where two edges meet at a perfect square corner. Those are the two relationships between lines this section names.

Parallel lines are two different lines in the same plane that never meet; non-vertical parallel lines have equal slopes and different y-intercepts.
Perpendicular lines are two lines that cross at a right angle; when neither line is vertical, their slopes multiply to −1.
The negative reciprocal of a nonzero fraction is the number found by flipping it upside down and changing its sign, so the negative reciprocal of 2⁄3 is −3⁄2; a number times its negative reciprocal is −1.

Worked example — a parallel line through (−1, 4). Find the line through (−1, 4) that is parallel to y = 3x + 4.

Copy the slope. Parallel means equal slopes, so m = 3.
Point-slope form. y − 4 = 3(x − (−1)), which is y − 4 = 3(x + 1).
Distribute and solve. y − 4 = 3x + 3, so y = 3x + 7.
Check the point. 3(−1) + 7 = −3 + 7 = 4. It matches.

In words: y = 3x + 4 and y = 3x + 7 climb at the same rate, but one starts at 4 and the other at 7. They stay 3 units apart forever and never meet.

Worked example — perpendicular slopes. Are slopes 2⁄3 and −3⁄2 perpendicular?

Multiply. (2⁄3) × (−3⁄2) = −6⁄6.
Simplify. −6⁄6 = −1.

In words: the product is −1, so lines with these slopes are perpendicular. For example, y = (2⁄3)x + 4 and y = −(3⁄2)x + 4 meet at (0, 4) in a square corner.

Misconception check. Students often think the perpendicular slope to 2⁄3 is −2⁄3, or 3⁄2. Check each one. (2⁄3) × (−2⁄3) = −4⁄9, and (2⁄3) × (3⁄2) = 6⁄6 = 1. Neither product is −1. Change only the sign, and you miss. Flip only the fraction, and you miss again. The cure: flip it and change the sign, then multiply to confirm −1.

Here is a quick reference for the three cases.

How the slopes of two non-vertical lines decide whether the lines are parallel, perpendicular, or neither
Relationship	What the slopes do	Example slopes
Parallel	equal, with different y-intercepts	3 and 3
Perpendicular	multiply to −1	2⁄3 and −3⁄2
Neither	not equal, and the product is not −1	2⁄3 and −4⁄3

8. Check perpendicular with arithmetic, not your eyes

Graphs can fool you. On a phone screen or a stretched window, a graphing app may draw one unit across wider than one unit up. Then angles look different from what they are. Two lines can look like they meet in a square corner when they do not, and two perpendicular lines can look slanted.

Honesty note: a graph is a picture of the algebra, not a replacement for it. So the rule for this course is to decide with the slopes, then use the picture as a second opinion.

Worked example — lines that look perpendicular. Take slopes 2⁄3 and −4⁄3.

Multiply. (2⁄3) × (−4⁄3) = −8⁄9.
Compare with −1. −8⁄9 is close to −1, but it is not equal to −1, since −9⁄9 would be −1.

In words: those lines are not perpendicular, however they look on your screen. A designer laying out a floor plan, or a surveyor marking a property corner, would get a slightly crooked corner from them.

In practice (your tools). In Desmos, type y = (2/3)x + 4 and y = -(4/3)x + 4. Open the graph settings (the wrench icon) and change the x-axis range without changing the y-axis range. Watch the corner between the lines seem to change shape. The slopes never change, and neither does their product.

Misconception check. Students often say "they look perpendicular, so they are." It is tempting because a picture feels like proof. Here is a caveat worth keeping: a picture depends on the window and the screen, but the product of two slopes does not. The cure: multiply the slopes every time, and call the lines perpendicular only when the product is exactly −1.

A reflection to choose from. Pick one option, (a) or (b), and write two or three sentences.

(a) Think of a price or a fee in your own life that changes by the same amount each time, such as a pay rate. Name its starting value and its rate, and say what the slope means in its units.
(b) Think of two straight things that meet or run side by side, such as streets on a map or lanes on a track. Say whether they look parallel or perpendicular, and say what you would measure to be sure.

Key terms

First difference — The amount one output in a table increases or decreases from the output before it when the inputs go up in steps of 1; for a line, every first difference equals the slope. (first used: 1. Spot a steady rate: equal steps make a line)
Linear function — A function whose graph is a straight line, written f(x) = mx + b; its output changes by the same amount, m, every time the input increases by 1. (first used: 1. Spot a steady rate: equal steps make a line)
Slope — The steepness of a line, written m: the change in y divided by the change in x between any two points on the line. (first used: 2. Measure steepness: slope as rise over run)
Rise over run — A plain-language description of slope: the vertical change between two points divided by the horizontal change between them, with the vertical change on top. (first used: 2. Measure steepness: slope as rise over run)
Rate of change — How much an output changes for each one-unit increase in the input; for a line it equals the slope and carries units such as dollars per topping. (first used: 2. Measure steepness: slope as rise over run)
Horizontal line — A flat line on which every point has the same y-value; its equation has the form y = k, and its slope is 0 because its rise is 0. (first used: 3. Read the sign: rising, falling, horizontal, and vertical lines)
Vertical line — An up-and-down line on which every point has the same x-value; its equation has the form x = k, and its slope is undefined because its run is 0. (first used: 3. Read the sign: rising, falling, horizontal, and vertical lines)
Undefined slope — The slope of a vertical line, which has no numerical value because finding it would require dividing the rise by a run of 0. (first used: 3. Read the sign: rising, falling, horizontal, and vertical lines)
Slope-intercept form — The equation of a line written y = mx + b, so that the slope m and the y-intercept b can be read off directly. (first used: 4. Read a line's equation: slope-intercept form)
Y-intercept — The point where a graph crosses the vertical y-axis, found by setting x = 0; for a line y = mx + b it is the point (0, b). (first used: 4. Read a line's equation: slope-intercept form)
Point-slope form — The equation of a line written y − y₁ = m(x − x₁), built from one known point (x₁, y₁) on the line and the line's slope m. (first used: 5. Build a line from two points: point-slope form)
Standard form — The equation of a line written Ax + By = C, with the x-term and the y-term on the left side and a constant on the right, where A and B are not both 0. (first used: 6. Find the intercepts: standard form and graphing from two points)
X-intercept — The point where a graph crosses the horizontal x-axis, found by setting y = 0 and solving for x. (first used: 6. Find the intercepts: standard form and graphing from two points)
Parallel lines — Two different lines in the same plane that never meet; non-vertical parallel lines have equal slopes and different y-intercepts. (first used: 7. Compare two lines: parallel and perpendicular)
Perpendicular lines — Two lines that cross at a right angle; when neither line is vertical, their slopes multiply to negative 1. (first used: 7. Compare two lines: parallel and perpendicular)
Negative reciprocal — The number found by flipping a nonzero fraction upside down and changing its sign; a number times its negative reciprocal is negative 1. (first used: 7. Compare two lines: parallel and perpendicular)

Summary

A line changes by the same amount for each step of 1 in the input.
Slope is rise over run: the change in y on top, the change in x below.
A positive slope rises, a negative slope falls, a horizontal line has slope 0, and a vertical line has no slope.
In slope-intercept form, the slope multiplies x and the y-intercept stands alone.
Point-slope form builds a line from one point and the slope; check both points afterward.
Find intercepts by setting the other variable to 0, then graph through the two points.
Parallel lines share a slope; perpendicular slopes multiply to negative 1.
Trust the product of the slopes, not the look of the picture.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

Find the slope of the line through (2, 1) and (6, 9), write its equation in slope-intercept form, and check one point.
A climbing gym charges a 25-dollar sign-up fee plus 10 dollars per month. Write the total cost y after x months as a line, say what the slope and the y-intercept mean, and find the cost of 6 months.
Find both intercepts of 3x + 4y = 24, then rewrite the equation in slope-intercept form.
For each pair, decide parallel, perpendicular, or neither: (a) y = (5⁄2)x − 1 and y = −(2⁄5)x + 3; (b) y = 4x + 1 and y = 4x − 7; (c) y = 2x + 3 and y = −2x + 3.

Answers (worked reasoning — read after attempting)

The rise is 9 − 1 = 8, and the run is 6 − 2 = 4, so m = 8 ⁄ 4 = 2. Point-slope form with (2, 1) gives y − 1 = 2(x − 2). Distribute: y − 1 = 2x − 4. Add 1 to both sides: y = 2x − 3. Check (6, 9): 2(6) − 3 = 12 − 3 = 9. It matches. Run over rise would have given 4 ⁄ 8 = 1⁄2, the trap from Section 2.
The cost starts at 25 dollars and climbs 10 dollars a month, so y = 10x + 25. The slope 10 means 10 dollars per month. The y-intercept 25 is the cost at month 0, the sign-up fee alone. For 6 months, y = 10(6) + 25 = 60 + 25 = 85 dollars. Swapping m and b would give 25x + 10, a 160-dollar bill for 6 months, which is far too high.
Set x = 0: 4y = 24, so y = 6, and the y-intercept is (0, 6). Set y = 0: 3x = 24, so x = 8, and the x-intercept is (8, 0). To solve for y, subtract 3x: 4y = −3x + 24. Divide every term by 4: y = −(3⁄4)x + 6. Check with the intercepts: (0 − 6) ⁄ (8 − 0) = (−6) ⁄ 8 = −3⁄4. It matches.
In pair (a), (5⁄2) × (−2⁄5) = −10⁄10 = −1, so the lines are perpendicular. In pair (b), both slopes are 4 and the intercepts differ (1 and −7), so the lines are parallel. In pair (c), the slopes 2 and −2 are not equal, and 2 × (−2) = −4, not −1, so the lines are neither. Changing only the sign does not make a perpendicular slope.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, the quiz, and the discussion — and next week a new method, solving a system of equations, finds where two of these lines meet.

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