Week 4 · Lecture outline

Week 4 — Lecture Outline · Linear Functions & Their Graphs

College Algebra · MATH 120 Fall 2026 · Prof. Calloway Fictional sample

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objectives covered: Objective 4 — Analyze and graph linear functions using slope, intercepts, and the slope-intercept, point-slope, and standard forms; identify parallel and perpendicular lines.
SLOs touched: A (apply procedures accurately) · B (connect symbolic/graphical representations and interpret in context)
Meeting pattern: 2 sessions × 75 min = 150 min. Segment minutes below total ~150; scale to your own pattern.

Week at a Glance


The week's big question	"What does the steepness of a line tell you — and how do you build a line from as little as one point and a slope?"
By the end of the week, students can…	(1) compute slope from two points using m = (y₂ − y₁)/(x₂ − x₁) and read slope and intercept directly from y = mx + b; (2) write the equation of a line in slope-intercept or point-slope form given a point and slope or two points; (3) find x- and y-intercepts and recognize horizontal (slope 0) and vertical (undefined slope) lines; (4) identify parallel lines (equal slopes) and perpendicular lines (slopes that are negative reciprocals, m₁·m₂ = −1).
Key vocabulary	slope, rise, run, rate of change, slope-intercept form, point-slope form, standard form, x-intercept, y-intercept, horizontal line, vertical line, parallel lines, perpendicular lines, negative reciprocal
Materials	slides (Deck 4), the week's readings + video links, Desmos (desmos.com/calculator) for graphing and checking, one approved chatbot (Gemini / Claude / ChatGPT) for the AI-critique moment and the tutorial
Timing note	8 segments, ~150 min total. Session 1 = Segments 1–4 (~73 min). Session 2 = Segments 5–8 (~77 min).

Segment 1 — Hook & the Promise (8 min) · Session 1 opens

Hook. "Here's a scenario: a rideshare app charges a \$2.00 base fee plus \$0.50 per mile. If I told you I spent \$12 last night, how far did I go?" Wait for a few attempts (most students can figure this out). "Now: could you write the rule that connects miles driven to cost — a rule that works for any trip? That rule is a linear function, and its graph is a straight line. This week you learn to build that line, read it, and use it."

The promise (write it on the board): "By the end of this week you can look at any two points and immediately know the equation of the line through them — and you can tell whether two lines are parallel or perpendicular just by comparing their slopes."

Why it matters: "Linear functions are the first model we reach for in every science, business, and engineering field: revenue grows, temperature drops, distance increases — all at a steady rate. Knowing how to build and read a line is the foundation for every model the rest of the course builds."

Segment 2 — Slope: the Number That Names Steepness (22 min)

Plain language first. Slope is a single number that tells you how fast a line rises or falls as you move left to right. More precisely, slope m = rise/run = (change in y)/(change in x). Positive slope → the line goes up (left to right). Negative slope → the line goes down. Slope 0 → perfectly flat (horizontal). Undefined slope → perfectly vertical.

The formula. Given two points (x₁, y₁) and (x₂, y₂):

m = (y₂ − y₁) / (x₂ − x₁)

Memory hook: "Rise over run — y's on top because the y-axis goes up."

One fully worked example (every step):

Find the slope through (1, 2) and (4, 11).
1. Label: (x₁, y₁) = (1, 2) and (x₂, y₂) = (4, 11).
2. Numerator (rise): y₂ − y₁ = 11 − 2 = 9.
3. Denominator (run): x₂ − x₁ = 4 − 1 = 3.
4. m = 9/3 = 3.
Verify the direction: from left to right the line goes up (positive slope), which matches m = 3 > 0. ✓

Horizontal and vertical lines (do both immediately):

The line y = 5: every point has the same y-value, so y₂ − y₁ = 0 for any two points. m = 0/run = 0. Slope is 0 — the line is flat.

The line x = 5: every point has the same x-value, so x₂ − x₁ = 0. m = rise/0 — division by zero is undefined. Slope is undefined — the line is vertical.

Name the misconceptions out loud:
- ❌ "I flipped the rise and run — I computed (x₂ − x₁)/(y₂ − y₁) instead."
✅ Cure: y is always on top. Rise = Δy (how much you go up or down). Run = Δx (how much you go left or right). If you confuse them, your slope is the reciprocal of the real slope.
- ❌ "The slope of a vertical line is 0 — it's just a number, like 5."
✅ Cure: Zero slope is horizontal (flat). Vertical is undefined — you'd have to divide by zero to compute it. They're completely different.
- ❌ "It matters which point I call (x₁, y₁)."
✅ Cure: The slope formula is symmetric — you get the same answer either way, as long as you're consistent (both y's subtracted in the same order).

Segment 3 — Slope-Intercept Form y = mx + b (20 min)

Plain language first. Once you know slope and y-intercept, you can write the complete equation of the line immediately. The y-intercept is where the line crosses the y-axis (x = 0). The form y = mx + b names both directly: m is the slope, b is the y-intercept.

One fully worked example — reading from the equation:

Identify slope and y-intercept of y = −2x + 5.
- m = −2 (slope: the line falls 2 units for each 1 unit to the right).
- b = 5 (y-intercept: the line crosses the y-axis at (0, 5)).
Graphing from these two facts: plot (0, 5), then move right 1 and down 2 to the next point (1, 3), draw the line.

One fully worked example — writing the equation from a graph or context:

A line through (0, 4) and (2, 10): slope = (10 − 4)/(2 − 0) = 6/2 = 3; y-intercept = 4 (the point (0, 4) is on the y-axis). Equation: y = 3x + 4.
Check: plug in (2, 10): 3(2) + 4 = 10. ✓

The classic swap trap:

❌ "In y = −2x + 5, the slope is 5 and the y-intercept is −2." (swapping m and b)
✅ Cure: m is always the coefficient of x. b is always the constant term. Write the form and point to each: m goes with x, b stands alone.

Segment 4 — Point-Slope Form y − y₁ = m(x − x₁) (23 min) · Session 1 closes (~73 min)

Plain language first. What if you know the slope and a point that isn't the y-intercept? You don't have to find b first — just plug directly into point-slope form. Think of it as "the slope hasn't changed since the given point."

The form: y − y₁ = m(x − x₁), where (x₁, y₁) is a known point and m is the slope.

One fully worked example (every step):

Write the equation of the line with slope 3 through (2, 1).
1. Point-slope: y − 1 = 3(x − 2).
2. Distribute: y − 1 = 3x − 6.
3. Solve for y: y = 3x − 6 + 1 = y = 3x − 5.
Check: plug in (2, 1): 3(2) − 5 = 6 − 5 = 1 ✓.

One fully worked example — two points, no slope given:

Write the equation through (−1, 4) and (3, −4).
1. Slope: m = (−4 − 4)/(3 − (−1)) = −8/4 = −2.
2. Use point (−1, 4): y − 4 = −2(x − (−1)) → y − 4 = −2(x + 1).
3. Distribute: y − 4 = −2x − 2.
4. y = −2x + 2.
Check both points: (−1, 4): −2(−1) + 2 = 2 + 2 = 4 ✓; (3, −4): −2(3) + 2 = −6 + 2 = −4 ✓.

Misconceptions:
- ❌ "I subtracted the x₁ and y₁ with the wrong sign — I wrote y + y₁ = m(x + x₁)."
✅ Cure: The form is y minus y₁ and x minus x₁. If the given point is (2, 1), you write y − 1 = m(x − 2). Plugging in x = 2 gives y − 1 = 0, so y = 1 — it checks.

Interaction — Think-Pair-Share (~6 min): Show 4 quick problems on slides: a point (3, −2) and m = 4 → set up point-slope only (don't simplify). Check that students write y + 2 = 4(x − 3) (not y − 2). Then reveal and discuss.

Segment 5 — x-Intercept, Graphing Lines & Standard Form (18 min) · Session 2 opens

Hook back. "Last session: how to write the equation. Today: how to draw it and find where it crosses each axis."

Finding intercepts (plain language):
- y-intercept: set x = 0 and solve for y. The line crosses the y-axis at (0, b) in slope-intercept form.
- x-intercept: set y = 0 and solve for x. The line crosses the x-axis at some point (a, 0).

One fully worked example:

Find both intercepts of 2x + 3y = 12.
- y-intercept (x = 0): 3y = 12 → y = 4. Point: (0, 4).
- x-intercept (y = 0): 2x = 12 → x = 6. Point: (6, 0).
Graph: plot (0, 4) and (6, 0), draw the line through them.

Standard form Ax + By = C: same line, different look. Useful for finding intercepts (just set one variable to 0). To convert to slope-intercept, isolate y.

Horizontal and vertical lines revisited:
- y = 3: slope 0; y-intercept (0, 3); no x-intercept (unless y = 0, meaning the line is the x-axis itself).
- x = 5: undefined slope; x-intercept (5, 0); no y-intercept.

Segment 6 — Parallel & Perpendicular Lines (22 min)

Plain language first. Two lines going in exactly the same direction (never crossing) are parallel — they have the same slope. Two lines that cross at a right angle are perpendicular — their slopes multiply to −1. That means each slope is the negative reciprocal of the other: flip the fraction and change the sign.

Memory hooks:
- Parallel = same slope. If you see the same m, the lines never meet.
- Perpendicular = negative reciprocal. Flip it, then change its sign. Two steps, always.

Perpendicular-slope formula: if a line has slope m, the perpendicular line has slope −1/m.

One fully worked example — parallel:

A line parallel to y = 4x − 1 through (2, 5) has slope 4 (same slope). Use point-slope: y − 5 = 4(x − 2) → y = 4x − 3.
Check: slopes equal (both 4) → parallel ✓.

One fully worked example — perpendicular:

Find the slope perpendicular to y = (2/3)x + 1.
Original slope: m = 2/3. Negative reciprocal: flip → 3/2, then change sign → −3/2.
Verify: (2/3)(−3/2) = −6/6 = −1 ✓.

The signature trap — name it and cure it explicitly:

❌ "The slope perpendicular to y = (2/3)x + 1 is −2/3." (only changed the sign, didn't flip)
✅ Cure: Two steps for perpendicular slope: (1) flip the fraction → 3/2; (2) change the sign → −3/2. Doing only one step gives the wrong answer. Memory check: always verify by multiplying: (2/3)(−3/2) = −1. If you don't get −1, something's wrong.

Another trap:

❌ "These two lines are perpendicular because they look like it on the graph."
✅ Cure: Always check algebraically: multiply the slopes. Only if the product is −1 are the lines perpendicular.

Segment 7 — Technology Workflow + AI-Critique Moment (15 min)

Technology workflow — Desmos for graphing and checking lines:
1. Open desmos.com/calculator.
2. Graph any equation: type y = 3x - 5 and press Enter. The line appears immediately.
3. Check parallel: type y = 3x + 2 on the next line — both lines appear; confirm they never intersect (same slope).
4. Check perpendicular: type y = (2/3)x + 1 and y = (-3/2)x + 4 — the lines should form a visible right angle. (Zoom to equal aspect ratio with the settings icon for best results.)
5. Check your intercepts: click a point where the line crosses an axis — Desmos labels the coordinates.

AI-critique moment (students verify, not consume):

Paste this to an approved chatbot: "What is the slope of a line perpendicular to y = (2/3)x + 1? Show your steps."
Then check its answer. Chatbots frequently give −2/3 (only changed the sign) or 3/2 (only flipped the fraction) instead of the correct −3/2 (flip AND change sign). If the chatbot's product isn't −1, it got it wrong. Your job all semester: the tool drafts, you judge.

Segment 8 — Callback, Tease & Hand-off (12 min) · Session 2 closes (~77 min)

Callback:
- Connects to Week 3: "A linear function is a function — it passes the vertical line test, and its domain is all real numbers. The slope is the constant rate of change we'll revisit when we study nonlinear functions and ask 'what if the rate isn't constant?'"
- The theme of the whole week: "Slope is a ratio. Every time you see 'per' in a real-world setting — cost per mile, pounds per week, degrees per hour — you're looking at a slope."

Tease next week — Week 5: "Next week we take two linear functions and ask when they're equal: that's a system of linear equations. You already know how to build each line. The new skill is finding where they intersect."

Hand-off (the week's graded work):
- Lecture Tutorial 4 (AI tutor, share-link submission) — slope, line equations, intercepts, parallel/perpendicular.
- Quiz 4 (no AI) and Discussion 4 ("Lines in the Real World" — real-world modeling with your own linear relationship).
- Assignment 4 ("Lines in Every Direction") — AI-coached, self-scored.

Instructor FAQ — Common Stumbles

Student says / does	Quick cure
Writes m = (x₂ − x₁)/(y₂ − y₁) (flipped).	Rise is always Δy (top), run is always Δx (bottom). Y up, x across.
Gives slope 0 for a vertical line.	0 is horizontal (flat). Vertical lines have undefined slope (division by zero).
Says "perpendicular slope is −2/3" when original is 2/3.	Two steps: flip the fraction (→ 3/2) then change the sign (→ −3/2). Verify: (2/3)(−3/2) = −1.
Writes y + y₁ = m(x + x₁) instead of y − y₁ = m(x − x₁).	The form has minus on both sides. Sub in the given point to check: it must satisfy the equation.
Swaps m and b in y = mx + b ("slope is 5, intercept is −2" when y = −2x + 5).	m is the coefficient of x; b is the constant. Point to each term explicitly.
Can't find x-intercept (confuses it with y-intercept).	x-intercept → set y = 0 and solve. y-intercept → set x = 0 and solve. Which variable are you setting to zero?
Says two lines are perpendicular "because they look that way."	Always verify algebraically: multiply the slopes; if the product is −1, they're perpendicular.
Can't use point-slope when given slope and a non-y-intercept point.	Point-slope form is the tool for this: y − y₁ = m(x − x₁). You do not need b first.

Scope flag

This outline stays within Objective 4. Standard form (Ax + By = C) is introduced for intercept-finding practice and as a conversion exercise, but slope-intercept and point-slope are the primary forms throughout. Complex-number slopes, parametric lines, and vector forms are out of scope.

~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com