Linear Functions & Graphs

8.C.1Slope from a graph01/04

Find slope from a graph.

Step01of 04

Slope measures how steep a line is. From a graph, count the "rise over run" between any two points on the line.

m = \frac{rise}{run} = \frac{vertical change}{horizontal change}

8.C.1 — Find slope from a graph.

Step 01 of 04

Slope measures how steep a line is. From a graph, count the "rise over run" between any two points on the line.

m = \frac{rise}{run} = \frac{vertical change}{horizontal change}

Step 02 of 04

Pick two clean lattice points (where the line crosses a grid intersection cleanly). Count the squares up (or down) and right (or left).

"Up 3, right 2" gives slope $3/2$ . "Down 4, right 1" gives slope $- 4$ .

Step 03 of 04

Sign tells direction.

Slope > 0	→rises from left to right
Slope < 0	→falls from left to right
Slope = 0	→horizontal line
Undefined	→vertical line (run is zero — can't divide)

Step 04 of 04

Magnitude tells steepness. Slope of 5 is steeper than slope of 2. Slope of $- 3$ is steeper than slope of $- 1$ (in the other direction).

Key insight

Slope = rise / run. Pick two points, count squares up/down (rise) over right/left (run). Sign = direction; magnitude = steepness.

8.C.2 — Slope from two points.

Step 01 of 04

Slope formula — when given coordinates instead of a graph.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

"Change in y over change in x." Pick the points in any order — just stay CONSISTENT (the same point gives BOTH the $y_{1}$ and the $x_{1}$ ).

Step 02 of 04

Worked example. Find slope through $(2, 3)$ and $(7, 18)$ .

m = \frac{18 - 3}{7 - 2} = \frac{15}{5} = 3

Slope is 3. Check: starting at $(2, 3)$ and going to $(7, 18)$ is right 5 units, up 15 — rise/run = 15/5 = 3.

Step 03 of 04

Negative slope example. Through $(- 1, 8)$ and $(3, 0)$ :

m = \frac{0 - 8}{3 - ( - 1 )} = \frac{- 8}{4} = - 2

Negative slope, as expected — the y-value DECREASED as x increased.

Step 04 of 04

Zero and undefined slopes. Through $(2, 5)$ and $(8, 5)$ :

m = \frac{5 - 5}{8 - 2} = 0/6 = 0 (horizontal)

Through $(4, 1)$ and $(4, 9)$ :

m = \frac{9 - 1}{4 - 4} = 8/0 = undefined (vertical)

Key insight

Slope formula: difference of y's over difference of x's, in matching order. Same x in both points → division by zero → vertical (undefined). Same y in both → numerator zero → horizontal (slope 0).

8.C.3 — Slope-intercept form — write and graph.

The form

y = m x + b

Step 01 of 04

$m$ = slope, $b$ = y-intercept (where the line crosses the y-axis). The form makes both visible at a glance.

Step 02 of 04

Write the equation. Slope 2 through point $(0, - 3)$ :

y = 2 x + (- 3) = 2 x - 3

Through a non-y-intercept point. Slope $- 3$ through $(2, 5)$ :

Use point-slope $y - 5 = - 3 (x - 2)$ and convert:

y = - 3 x + 6 + 5 = - 3 x + 11

Step 03 of 04

Graph from the equation.

1.	→Plot the y-intercept $(0, b)$
2.	→From there, use the slope to step to the next point: rise of $m$ , run of 1
3.	→Connect with a straight line

Slope $- 1/3$ means down 1, right 3 (or up 1, left 3). Slope 4 means up 4, right 1 (or up 4 = up 4/1).

Step 04 of 04

Visualize y = -2x + 4.

Slope -2: down 2, right 1.

Key insight

Slope-intercept form makes the slope m and intercept b visible immediately. Plot the intercept first, use slope to step to a second point, draw a line.

8.C.4 — Compare two linear functions.

Step 01 of 04

Linear functions can be compared regardless of their representation (equation, table, graph, description). The key is reducing each to its slope and y-intercept.

Step 02 of 04

Worked example. Function A: $y = 3 x + 2$ . Function B: passes through $(0, 5)$ and $(2, 9)$ . Which has the steeper graph? Higher y-intercept?

A: slope 3, intercept 2. B: slope $(9 - 5) / (2 - 0) = 2$ , intercept 5.

Steeper	→A (slope 3 vs. 2)
Higher y-int	→B (5 vs. 2)
Where they cross	→set equal: $3 x + 2 = 2 x + 5 \Rightarrow x = 3, y = 11$

Step 03 of 04

From a table. Compute slope as constant difference per unit change in x. y-intercept = the y-value when $x = 0$ (or extrapolate).

Table for C: $(1, 7), (3, 13), (5, 19)$ . Differences in y per unit x: $(13 - 7) / (3 - 1) = 3$ . So slope is 3. y-intercept: extrapolate back to $x = 0$ → $y = 7 - 3 = 4$ . Equation: $y = 3 x + 4$ .

Step 04 of 04

Compare across representations. Get every function into $y = m x + b$ form, then compare slopes and intercepts directly. Two functions are the SAME line iff both slope AND intercept match.

Key insight

Reduce every linear function to slope + intercept, regardless of how it's presented (equation, graph, table, words). Once both are in slope-intercept form, comparison is direct.

8.C.5 — Construct a linear function from a description.

Step 01 of 04

Pattern. A real-world description often gives you a STARTING VALUE (y-intercept) and a RATE OF CHANGE (slope).

"Starts at..."	→y-intercept (b)
"Increases by ... per ..."	→slope (m), positive
"Decreases by ... per ..."	→slope (m), negative
"Earns / costs / charges per ..."	→slope per unit

Step 02 of 04

Worked example. A gym charges $50 to join, plus $20 per month. Total cost $C$ after $m$ months?

C = 20 m + 50

Slope 20 (cost per month), y-intercept 50 (cost at month 0).

Step 03 of 04

Decay example. Phone battery starts at 100% and drops 5% per hour. Battery $B$ after $h$ hours?

B = - 5 h + 100

Negative slope (decreasing). At hour 10: $B = - 50 + 100 = 50%$ . Battery dies (B = 0) when $h = 20$ .

Step 04 of 04

Two-point setup. Sometimes the description gives you two specific values. Compute slope from those, then find y-intercept by plugging back in.

"After 3 hrs, $42; after 7 hrs, $86." Slope $(86 - 42) / (7 - 3) = 11$ . Then $42 = 11 (3) + b \Rightarrow b = 9$ . Equation: $C = 11 t + 9$ .

Key insight

Identify the starting value (y-intercept) and the rate of change (slope). If you don't have a starting value, build one from a known point. Always check by plugging in a second known value.

8.C.6 — Interpret slope and intercept in context.

Step 01 of 04

In a real-world equation $y = m x + b$ :

$m$ (slope)	→rate of change in y per unit increase in x — usually has units like $/hour or miles/gallon
$b$ (y-intercept)	→starting value, when x = 0

Always include UNITS in the interpretation.

Step 02 of 04

Worked example. Plumber charges $C = 80 h + 50$ , where $h$ is hours.

Slope = 80	→plumber charges $80 per hour of work
Intercept = 50	→flat fee of $50 just for showing up (regardless of hours)
C(2)	→= $80 (2) + 50 = $210$ for a 2-hour job

Step 03 of 04

Whether the y-intercept makes sense. Sometimes $x = 0$ isn't physically meaningful, so the y-intercept is more of a mathematical artifact than a real value.

Example: $H = 10 a + 80$ for height (cm) at age $a$ (years), valid for ages 5–15. The intercept 80 would mean "height at age 0 is 80 cm" — but the model doesn't apply to newborns. Don't extrapolate outside the valid range.

Step 04 of 04

Sign of slope tells you the direction. Positive slope = quantity grows. Negative slope = quantity shrinks.

$B = - 5 h + 100$ for battery percentage: slope -5 means battery drops 5% per hour. Intercept 100 means starting at 100%. Both make sense physically.

Key insight

Slope = "per unit" rate (with units). Intercept = "starting" value (when input is zero). Always interpret in the problem's context, and be skeptical of intercepts when zero input doesn't physically apply.

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