Algebra 1 — Linear Functions Worksheet

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SLOPE rise ÷ run = (y₂−y₁)/(x₂−x₁) · "RISE over RUN"

y=mx+b m = slope · b = y-intercept · "Slope-Intercept Form"

PARALLEL Same slope (m₁ = m₂) · different y-intercept

PERP⊥ Negative reciprocal slopes → m₁ × m₂ = −1

VERTICAL x = a → undefined slope · NOT a function

HORIZONTAL y = b → slope = 0 · IS a function

STD FORM Ax + By = C · find intercepts: set x=0, set y=0

POINT-SLOPE y − y₁ = m(x − x₁) · use when given point + slope

✦ PROBLEMS ✦

📈 SLOPE FROM TWO POINTS

⚠️ COMMON TRAP: Subtraction order!

What is the slope of the line passing through (2, 5) and (6, 13)?

Keep the same order top and bottom: (y₂−y₁) / (x₂−x₁). Don't flip!

🔍 Solution

Use the slope formula: $ m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{13 - 5}{6 - 2} = \dfrac{8}{4} = 2 $

⚠️ Trap: If you accidentally do $\frac{6-2}{13-5}$ you get $\frac{1}{2}$ — that's choice A, the most common wrong answer!
RISE over RUN → numerator is always Δy (vertical change)

📝 SLOPE-INTERCEPT FORM

A line has slope $ m = -3 $ and y-intercept $ b = 7 $.
Which equation represents this line?

Plug directly into y = mx + b. Watch the sign of the slope!

🔍 Solution

$ y = mx + b $ → plug in $ m = -3 $ and $ b = 7 $:
$ y = -3x + 7 $ ✓

⚠️ Trap D: Students sometimes mix up m and b. Remember: m goes with x, b is alone!
y = mx + b → m = slope (with x!), b = where line crosses y-axis

📐 VERTICAL vs. HORIZONTAL LINES

⚠️ COMMON TRAP: Which is undefined?

What is the slope of the line $ x = -4 $?

Is this line going side-to-side or up-and-down? Picture it in your head!

🔍 Solution

$ x = -4 $ is a vertical line — it goes straight up and down.
Slope = $ \dfrac{\Delta y}{\Delta x} = \dfrac{\text{anything}}{0} $ → division by zero → UNDEFINED

⚠️ Trap A: Slope = 0 belongs to horizontal lines like $ y = 3 $!
VERTICAL = undefined HORIZONTAL = zero

🔄 POINT-SLOPE FORM → SLOPE-INTERCEPT

A line passes through $(1, -2)$ with slope $ m = 4 $.
Write it in slope-intercept form $ y = mx + b $.

Start with y − y₁ = m(x − x₁), then solve for y by distributing and moving the number to the right side.

🔍 Solution

Point-slope: $ y - (-2) = 4(x - 1) $
$ y + 2 = 4x - 4 $
$ y = 4x - 4 - 2 = 4x - 6 $ ✓

⚠️ Trap C: Students forget to distribute the 4 → they write $-4 - 2$ but stop at just $-2$. Always distribute THEN move!
y − y₁ = m(x − x₁) → distribute m first!

⚡ PARALLEL LINES

⚠️ COMMON TRAP: Same everything ≠ parallel!

Which line is parallel to $ y = \dfrac{2}{3}x - 5 $?

Parallel lines share the same slope but must have different y-intercepts. Same line ≠ parallel!

🔍 Solution

Original slope: $ m = \dfrac{2}{3} $
B has $ m = \dfrac{2}{3} $ with different b (+4 ≠ −5) → Parallel ✓

⚠️ Trap C: Same slope AND same intercept = the exact same line → NOT parallel (they overlap!)
⚠️ Trap A: That's the perpendicular slope (negative reciprocal)!
PARALLEL = same m, different b

⊥ PERPENDICULAR LINES

⚠️ COMMON TRAP: Flip AND change sign!

A line has slope $ m = \dfrac{3}{4} $. What is the slope of a line perpendicular to it?

Perpendicular = negative reciprocal. FLIP the fraction, then FLIP the sign. Two steps!

🔍 Solution

Negative reciprocal of $\dfrac{3}{4}$:
Step 1 — Flip: $\dfrac{4}{3}$
Step 2 — Change sign: $-\dfrac{4}{3}$ ✓

Check: $\dfrac{3}{4} \times \left(-\dfrac{4}{3}\right) = -1$ ✓ (Product of perpendicular slopes is always −1!)
⚠️ Trap B: Only flipped — forgot to change sign!
⊥ slope = FLIP then NEGATE

📊 X-INTERCEPT from STANDARD FORM

Find the x-intercept of: $ 3x - 2y = 12 $

At the x-intercept, y = 0. Substitute y = 0 and solve for x.

🔍 Solution

Set $ y = 0 $: $ 3x - 2(0) = 12 $
$ 3x = 12 \implies x = 4 $
x-intercept = $(4, 0)$ ✓

⚠️ Trap A: $(0, -6)$ is the y-intercept — found by setting x = 0!
Rule: x-intercept → set y = 0 | y-intercept → set x = 0
x-intercept: y=0 | y-intercept: x=0

🌍 REAL-WORLD LINEAR FUNCTION

A taxi charges a $3 flat fee plus $2.50 per mile.
Which equation gives total cost $C$ for $m$ miles? What does the slope represent?

The flat fee doesn't change → it's the y-intercept (b). The per-mile rate changes with distance → it's the slope (m).

🔍 Solution

Slope = rate of change = $2.50/mile → $m = 2.5$
Starting value (flat fee) = y-intercept → $b = 3$
$ C = 2.5m + 3 $ ✓ and slope = cost per additional mile

⚠️ Trap B: Switched the values — the flat fee is NOT the slope!
slope = rate of change y-intercept = starting value

📉 READING SLOPE FROM A GRAPH

⚠️ COMMON TRAP: Negative slope direction!

The graph below shows a line passing through $(0, 6)$ and $(3, 0)$. What is the slope?

Going from left to right, is the line going up or down? Down = negative slope!

🔍 Solution

From $(0, 6)$ to $(3, 0)$:
$ m = \dfrac{0 - 6}{3 - 0} = \dfrac{-6}{3} = -2 $ ✓

The line goes downward left to right → slope must be negative!
⚠️ Trap A: Students forget the negative sign when reading downward lines from graphs.
down-right = negative slope ↘

🏆 EQUATION OF A LINE — TWO POINTS

⚠️ HARDEST: Two-step process!

Find the equation of the line passing through $(-2,\ 1)$ and $(4,\ 10)$ in slope-intercept form.

Step 1: Find slope using the two points. Step 2: Plug slope + one point into point-slope form, then simplify.

🔍 Solution

Step 1 — Find slope:
$ m = \dfrac{10 - 1}{4 - (-2)} = \dfrac{9}{6} = \dfrac{3}{2} $

Step 2 — Use point-slope with $(4, 10)$:
$ y - 10 = \dfrac{3}{2}(x - 4) $
$ y - 10 = \dfrac{3}{2}x - 6 $
$ y = \dfrac{3}{2}x - 6 + 10 = \dfrac{3}{2}x + 4 $ ✓

⚠️ Note: Choice D shows the un-simplified slope $\frac{9}{6}$ — always simplify fractions!
Step 1: find m → Step 2: find b

✦ GREAT WORK! Review any incorrect answers carefully ✦