Algebra 1 & Geometry — Study Sheet

Algebra 1

Linear Functions · Equations · Inequalities

⚡ Quick Memory Points

SLOPE

rise ÷ run = (y₂−y₁)/(x₂−x₁)

Y-INTERCEPT

where line crosses y-axis (x=0)

FLIP SIGN

multiply/divide by negative → flip < or >

ISOLATE

get variable alone on one side

PARALLEL

same slope, different y-intercept

Algebra · Slope

📌 Key Formula: slope m = (y₂ − y₁) / (x₂ − x₁).
Think: rise over run — how much does y change per 1 unit of x?

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

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Algebra · Slope-Intercept

📌 Key Form: y = mx + b → m is slope, b is y-intercept.
⚠️ Common trap: people confuse which number is slope vs. intercept.

In the equation $y = -3x + 7$ what is the y-intercept?

Algebra · Solving Equations

📌 ISOLATE: Do the same operation to both sides.
Step 1 — move constants. Step 2 — divide coefficient.

Solve for x: $2x + 9 = 3$

Algebra · Inequalities ⚠️

📌 FLIP SIGN RULE: When you multiply or divide both sides by a negative number, flip the inequality sign!
This is the #1 most missed concept.

Solve the inequality: $-4x < 20$ What is the solution?

Algebra · Linear Function

📌 POINT-SLOPE: y − y₁ = m(x − x₁)
Use this when given slope + one point.

A line has slope m = 3 and passes through (1, 4). What is the equation of the line in slope-intercept form?

Algebra · Parallel Lines

📌 PARALLEL = SAME SLOPE
If two lines are parallel, they have identical slopes but different y-intercepts.

Which line is parallel to y = 2x − 5?

Algebra · Systems of Equations

📌 SUBSTITUTION: Solve one equation for one variable, then plug it into the other equation.

Solve the system: $y = x + 2 and y = 3x - 4$

Algebra · Compound Inequality

📌 AND vs OR: "AND" → both must be true (middle region). "OR" → at least one true (outside regions).

Which value of x satisfies the compound inequality: $-1 \leq x < 4$

Algebra · Function Notation

📌 f(x) means: plug x into the formula.
f(3) → replace every x with 3, then calculate.

If f(x) = 4x − 6, what is f(−2)?

Algebra · Slope from Graph ⚠️

📌 ZERO & UNDEFINED: Horizontal line → slope = 0. Vertical line → slope = undefined (can't divide by 0).

A line is perfectly horizontal on the coordinate plane. What is its slope?

Geometry

Triangles · Polygons · Circles

⚡ Quick Memory Points

TRIANGLE SUM

3 angles always add to 180°

POLYGON SUM

(n − 2) × 180° for n sides

CIRCUMFERENCE

C = 2πr or πd

AREA CIRCLE

A = πr²

PYTHAGOREAN

a² + b² = c² (right triangle)

Geometry · Triangle Angles

📌 TRIANGLE SUM = 180°
Always. No exceptions. Add all 3 angles → 180°.

A triangle has angles of 55° and 75°. What is the measure of the third angle?

Geometry · Pythagorean Theorem ⚠️

📌 a² + b² = c²
c = hypotenuse (longest side, opposite right angle). a and b are the two legs.

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

Geometry · Polygon Angle Sum

📌 FORMULA: Sum of interior angles = (n − 2) × 180°
Pentagon has 5 sides → (5−2) × 180° = 3 × 180° = 540°

What is the sum of the interior angles of a hexagon (6 sides)?

Geometry · Circle Area

📌 A = πr² — radius not diameter!
⚠️ Most common mistake: using diameter instead of radius.

What is the area of a circle with radius r = 5? (Use π ≈ 3.14)

Geometry · Circumference ⚠️

📌 C = 2πr = πd
Given diameter? Use πd directly. Given radius? Use 2πr.

A circle has a diameter of 10 cm. What is its circumference? (Use π ≈ 3.14)

Geometry · Triangle Types

📌 EQUILATERAL: All 3 sides equal → all 3 angles = 60°.
ISOSCELES: 2 sides equal → 2 base angles equal.
SCALENE: All sides different.

An equilateral triangle has a perimeter of 36 cm. What is the length of one side?

Geometry · Exterior Angle ⚠️

📌 EXTERIOR ANGLE THEOREM: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
This trips up many students!

In a triangle, two interior angles are 40° and 65°. What is the exterior angle at the third vertex?

Geometry · Regular Polygon Angle

📌 Each interior angle of a regular polygon:
= (n − 2) × 180° ÷ n

What is the measure of each interior angle of a regular octagon (8 sides)?

Geometry · Circle Sector ⚠️

📌 ARC / SECTOR = fraction of full circle:
Fraction = central angle ÷ 360°
Arc length = (θ/360°) × 2πr

A circle has radius 6 cm. A sector has a central angle of 90°. What is the area of the sector? (Use π ≈ 3.14)

Geometry · Triangle Inequality ⚠️

📌 TRIANGLE INEQUALITY: The sum of any two sides must be greater than the third side.
a + b > c, a + c > b, b + c > a

Which set of side lengths can form a valid triangle?

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