Slope-intercept form, point-slope form, parallel and perpendicular lines, linear inequalities, and real-world applications — the complete precalculus foundation.
y = mx + b
m is slope, b is y-intercept. Best for graphing and identifying slope immediately.
y − y₁ = m(x − x₁)
Use when you know the slope and one point. Fastest route to the equation of a line.
Ax + By = C
A, B, C are integers with A ≥ 0. Preferred in some textbooks and for finding intercepts quickly.
Slope-Intercept → Standard
Standard → Slope-Intercept
Point-Slope → Slope-Intercept
Slope measures the steepness and direction of a line. Given two points (x₁, y₁) and (x₂, y₂):
Line rises from left to right. The larger the value of m, the steeper the rise. Example: m = 3 is steeper than m = 1.
Line falls from left to right. The more negative m, the steeper the fall. Example: m = −4 is steeper than m = −1.
Horizontal line — no rise. m = 0. Equation is y = k. Example: y = 5 is a horizontal line at height 5.
Vertical line — the run is 0, so division by zero makes slope undefined. Equation is x = h. Example: x = −2 is a vertical line.
Parallel lines never intersect. They have equal slopes and different y-intercepts.
Example: y = 4x + 7 ∥ y = 4x − 3 (both slope 4, different intercepts)
Perpendicular lines intersect at a 90° angle. Their slopes are negative reciprocals.
Example: slope 3/4 is perpendicular to slope −4/3. Product: (3/4)(−4/3) = −1 ✓
| Given Slope | Parallel Slope | Perpendicular Slope |
|---|---|---|
| m = 2 | m = 2 | m = −1/2 |
| m = −3/4 | m = −3/4 | m = 4/3 |
| m = 5 | m = 5 | m = −1/5 |
| m = 0 (horizontal) | m = 0 | undefined (vertical) |
Use point-slope form directly — it was built for this scenario.
Calculate slope first, then apply point-slope form.
Read key features directly off the graph.
Every point has the same y-coordinate. Slope = 0. Passes through (0, k) and is parallel to the x-axis.
y = 3 → horizontal at height 3
y = −5 → horizontal at height −5
y = 0 → the x-axis itself
Every point has the same x-coordinate. Slope = undefined. Not a function — fails the vertical line test on its own axis.
x = 4 → vertical at x = 4
x = −2 → vertical at x = −2
x = 0 → the y-axis itself
A linear inequality describes a half-plane — all points on one side of a boundary line.
Graph the boundary line
Replace the inequality symbol with = and graph the line. Use a dashed line for strict inequalities (< or >) and a solid line for ≤ or ≥.
Choose a test point
Pick any point not on the boundary line. The origin (0, 0) is easiest when the line doesn't pass through it.
Substitute and evaluate
Plug the test point into the original inequality. If the inequality is true, shade the region containing the test point. If false, shade the other side.
Shade the correct region
For y > mx + b: shade above the dashed line. For y < mx + b: shade below. For y ≥ mx + b: shade above including the solid boundary.
Dashed vs. Solid Boundary
Dashed: points on the line are NOT solutions (< or >). Solid: points on the line ARE solutions (≤ or ≥).
Flipping the Inequality
When multiplying or dividing both sides by a negative number, reverse the inequality symbol: −2x > 6 → x < −3.
Slope equals the rate of change. If temperature rises 3°F per hour, the linear model is T = 3t + T₀ where T₀ is the starting temperature. The slope of 3 is the rate of change.
A business model: Cost C = mx + b where b is the fixed cost and m is cost per unit. Revenue R = px where p is price. Profit = R − C. Break-even occurs when C = R.
d = rt (distance = rate × time) is linear when rate is constant. If a car travels at 60 mph starting 20 miles out, its position is d = 60t + 20. Slope is speed; y-intercept is starting position.
y = −(2/3)x + 4 → m = −2/3, b = 4
Step 1: Plot y-intercept at (0, 4)
Step 2: From (0, 4), apply slope −2/3: go down 2, right 3 → point (3, 2)
Step 3: Apply again: go down 2, right 3 → point (6, 0)
Step 4: Draw line through (0, 4), (3, 2), (6, 0)
Line falls from left to right (negative slope). x-intercept at (6, 0).
Points: (−1, 5) and (3, −3)
Step 1: Slope = (−3 − 5) / (3 − (−1)) = −8 / 4 = −2
Step 2: Point-slope using (−1, 5): y − 5 = −2(x − (−1))
Step 3: Expand: y − 5 = −2x − 2
Step 4: Solve for y: y = −2x + 3
Equation: y = −2x + 3
Verify: f(3) = −2(3) + 3 = −3 ✓ f(−1) = −2(−1) + 3 = 5 ✓
Reference line: y = 3x − 7 → slope = 3
Parallel line (same slope, through (2, 1)):
y − 1 = 3(x − 2) → y − 1 = 3x − 6 → y = 3x − 5
Perpendicular line (slope = −1/3, through (2, 1)):
y − 1 = −(1/3)(x − 2) → y = −(1/3)x + 2/3 + 1 → y = −(1/3)x + 5/3
Parallel: y = 3x − 5 | Perpendicular: y = −(1/3)x + 5/3
2x − y ≤ 4
Step 1: Rewrite boundary: 2x − y = 4 → y = 2x − 4. Solid line (≤).
Step 2: Test (0, 0): 2(0) − 0 ≤ 4 → 0 ≤ 4. TRUE.
Step 3: Shade the region containing (0, 0) — above the line y = 2x − 4.
Graph: solid line y = 2x − 4, shaded above (includes the line).
A company has fixed costs of $1,200/month and variable costs of $8 per unit.
Each unit sells for $20.
Cost and Revenue models:
C(x) = 8x + 1200 (cost)
R(x) = 20x (revenue)
Break-even point (C = R):
8x + 1200 = 20x
1200 = 12x
x = 100 units
Break-even at 100 units. Sell more than 100 to profit; fewer to incur a loss.
Profit model: P(x) = R(x) − C(x) = 12x − 1200. Slope 12 = profit per additional unit.
Slope-intercept form is y = mx + b, where m is the slope (rise over run) and b is the y-intercept (where the line crosses the y-axis). To graph a line in this form: plot the y-intercept at (0, b), then use the slope m = rise/run to find a second point — move up by the numerator and right by the denominator. Draw a line through both points. Example: y = 3x − 2 has slope 3 and y-intercept −2. Plot (0, −2), move up 3 and right 1 to get (1, 1), and draw the line.
Given two points (x₁, y₁) and (x₂, y₂): Step 1 — calculate the slope m = (y₂ − y₁) / (x₂ − x₁). Step 2 — substitute m and one point into point-slope form: y − y₁ = m(x − x₁). Step 3 — simplify to slope-intercept form y = mx + b by solving for y. Example: given (2, 5) and (4, 9), slope = (9 − 5)/(4 − 2) = 4/2 = 2. Point-slope: y − 5 = 2(x − 2) → y = 2x + 1.
Parallel lines have equal slopes and different y-intercepts — they never intersect. For example, y = 3x + 1 and y = 3x − 5 are parallel because both have slope 3. Perpendicular lines have slopes that are negative reciprocals of each other — their slopes multiply to −1. If one line has slope m, any perpendicular line has slope −1/m. Example: a line with slope 2/3 is perpendicular to a line with slope −3/2, because (2/3)(−3/2) = −1.
A horizontal line has slope 0 — it has no rise regardless of the run. Its equation is y = k for some constant k. For example, y = 4 is a horizontal line through every point with y-coordinate 4. A vertical line has undefined slope — you cannot divide by zero because the run is 0. Its equation is x = h for some constant h. For example, x = −3 is a vertical line through every point with x-coordinate −3. Vertical lines are not functions.
To graph a linear inequality like y > 2x − 1: Step 1 — graph the boundary line y = 2x − 1. Use a dashed line for strict inequalities (> or <) and a solid line for ≥ or ≤. Step 2 — pick a test point not on the line, such as (0, 0). Substitute: 0 > 2(0) − 1 → 0 > −1, which is true. Step 3 — shade the region containing the test point (above the line in this case). If the test point makes the inequality false, shade the other side.
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