REAL ACT FULL NOTES AND FLASHCARDS

1. Subject-Verb Agreement

The subject and verb must agree in number (singular/plural).

✅ Example:
❌ The group of students are going on a trip.
✔ The group of students is going on a trip.

2. Pronoun-Antecedent Agreement

A pronoun must agree with the noun it replaces in number and gender.

✅ Example:
❌ Each student must bring their book.
✔ Each student must bring his or her book.

("Each" is singular, so "his or her" is correct.)

3. Misplaced and Dangling Modifiers

Modifiers must be placed next to the word they describe.

✅ Example:
❌ Running down the street, the backpack fell off John’s shoulder.
✔ Running down the street, John's backpack fell off his shoulder.

(The backpack isn’t running! John is.)

4. Parallel Structure

When listing or comparing items, they must have the same grammatical form.

✅ Example:
❌ She enjoys reading, to swim, and biking.
✔ She enjoys reading, swimming, and biking.

5. Comma Rules

• Before a coordinating conjunction (FANBOYS) in a compound sentence
  ✅ Example: I wanted to go, but I was too tired.

• After an introductory phrase
  ✅ Example: After the movie, we went to dinner.

• In a list (Oxford comma preferred)
  ✅ Example: I bought apples, bananas, and oranges.

6. Semicolon vs. Colon vs. Dash

• Semicolon (;) joins two complete sentences without a conjunction.
  ✅ Example: I love coffee; it helps me stay awake.

• Colon (:) introduces lists or explanations.
  ✅ Example: There are three things I need: food, water, and sleep.

• Dash (—) adds emphasis or extra info.
  ✅ Example: I finally arrived—an hour late!

7. Apostrophes (Possession & Contractions)

• Singular Possession: John’s book (one person)

• Plural Possession: The students’ books (multiple students)

• It’s vs. Its:

  • It’s = it is

  • Its = possessive (The cat licked its paw.)

8. Wordiness & Redundancy

The ACT prefers concise writing.

✅ Example:
❌ The reason why he left is because he was tired.
✔ The reason he left is that he was tired.

("Reason why" and "because" are redundant.)

9. Commonly Confused Words

• Than vs. Then → Than = comparison, Then = time

• Effect vs. Affect → Effect = noun, Affect = verb

• Who vs. Whom → Who = subject, Whom = object

✅ Example:
Who/Whom do you love?
✔ You love whom → Whom is correct.

10. Sentence Structure & Fragments

A sentence must have a subject and verb.

✅ Example:
❌ Because I was tired. (Fragment)
✔ Because I was tired, I went to bed. (Complete sentence)

Equation of a Circle - Standard Form

(x−h)²+(y−k)²=r², where (h,k) is the center and r is the radius.

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Diameter

Double the radius (2r).

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Regular Polygon

A polygon with all sides and angles equal.

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Sum of Interior Angles of a Polygon

(n−2)×180°, where n is the number of sides.

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Sum of Exterior Angles of a Polygon

Always equals 360° for any polygon.

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Area of a Triangle

A=1/2×Base×Height.

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Surface Area of a Cube

SA=6a², where a is the length of a side.

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Volume of a Cube

V=a³, where a is the length of a side.

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Surface Area of a Rectangular Prism

SA=2lw + 2lh + 2wh, where l, w, and h are length, width, and height.

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Volume of a Rectangular Prism

V=l×w×h.

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Surface Area of a Sphere

SA=4πr², where r is the radius.

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Volume of a Sphere

V=(4/3)πr³.

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Surface Area of a Pyramid

SA=B+1/2Pl, where B is base area, P is perimeter, l is slant height.

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Volume of a Pyramid

V=(1/3)Bh, where B is base area and h is height.

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Surface Area of a Right Circular Cone

SA=πr² + πrl.

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Volume of a Right Circular Cone

V=(1/3)πr²h.

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Surface Area of a Cylinder

SA=2πr² + 2πrh.

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Volume of a Prism

V=Bh, where B is the area of the base and h is the height.

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Radius

Distance from the center to the edge of a circle or sphere.

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Height

Perpendicular distance between the bases or the top and bottom of a solid figure.

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Slant Height

Diagonal distance from the apex to the edge of a base in a cone or pyramid.

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Base Area

Area of the base of a solid.

In a triangle all 3 angles added together equal

180 degrees

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Mastered

An isosceles triangle always has 2

congruent angles(the third angle is different)

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Mastered

The mean of a string of numbers is referring to the

average of the numbers

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Mastered

To find mean all you have to do is

add all the numbers together and divide the sum of the numbers by the amount of numbers there are

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Mastered

An absolute value is always

a positive number

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Mastered

The formula to find slope using 2 coordinate points is (slope formula)

y2-y1 / x2-x1

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When multiplying fractions you

multiply straight across (numerator*numerator and denominator*denominator)

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When dividing fractions you

keep the 1st fraction the same, flip the 2nd fraction and multiply them straight across.

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Factoring

Process of breaking down an expression into a product of simpler expressions, called factors. EX: 4x+8 is factored as 2(2x+4)

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Quadratic Functions in Vertex form

Flashcard:

Term: Quadratic Functions in Vertex Form

Definition: A quadratic function expressed as ______ is written in the form f(x) = a(x - h)² + k , where ( (h, k) ) represents the vertex of the parabola, and ( a ) determines the direction and width of the graph.

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Quadratic Standard Form

Flashcard:

Term: Quadratic Standard Form

Definition: The ________ is a way of expressing a quadratic equation in the format (ax_² + bx + c = 0 ), where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ). This form is useful for identifying the coefficients and analyzing the properties of the quadratic function.

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Quadratic Vertex to Standard Form Conversion Formula

Flashcard:

Term: Quadratic Vertex to Standard Form Conversion Formula

Definition: The formula used to convert a quadratic equation from vertex form, ( y = a(x - h)^2 + k ), to standard form, ( y = ax^2 + bx + c ), involves expanding the vertex form and simplifying. The vertex form highlights the vertex at the point ( (h, k) ), while the standard form provides coefficients ( a ), ( b ), and ( c ) for further analysis.

Fill in the blank: The conversion from vertex form to standard form is achieved by expanding ( y = a(x - h)^2 + k ) to obtain ( y = ax^2 + _____ + c ).

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slope intercept form

y=mx+b

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Midpoint Formula

(x1+x2 / 2 , y1+y2 / 2) This is used for finding the midpoint between two coordinate points.

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FOIL

Method used to take factored equations and simplify them into normal equations.

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Factoring

Breaking apart an equation into all it’s parts. For example, the expression 4x + 8 can be factored as 2(2x + 4).

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Elimination

A method to eliminate one x or y part of an equation to solve for just one or the other by multiplying a whole equation to make it so you can add the first and the second equation together and solve for just y or just x. And then plugging in just the y or the x into one of the equations to solve for the other missing variable giving you the x and y to the coordinate point. (If you don’t understand this refer back to example in ACT prep notebook)

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Substitution

If you have an equation like x=9+1 where the equation specifically equals a variable then substitute the variable that the equation that is like this is equal to into the second equation that has this variable, in it to solve for x or y. (If you don’t understand this refer back to examples from notebook)

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Zero Product Property

if the product of two factors equals zero, then at least one of the factors must also equal zero. For example, if (x - 3)(x + 5) = 0, either x - 3 = 0 or x + 5 = 0, leading to x = 3 or x = -5.

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Distance Formula

√((x2 - x1)² + (y2 - y1)²)

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Cross Multiplication

A method used to simplify equations involving fractions by multiplying the numerator of one fraction by the denominator of the other and setting the products equal to each other. If a/b = c/d, then ad = bc.

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Commas to Seperate itmes in a series of three or more things.

I bought “coffee, tea, and milk.”

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Every declarative sentece must end with a period

She loves chocolate”.”

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Capitalize the first letter of a sentence.

“I” love milk.

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Capitalize proper nouns.

Name, days of the week, names of months.

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Do not use an apostrophe to form a plural word.

Incorrect: I love cat’s. Correct: I love cats.

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Apostrophes are used for contractions or to show posessions.

Don’t, Jack’s, Dog’s

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Every question or interrogative sentence must end with a question mark.

When does you shcool start?

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Use an exclamation mark at the end of a sentence to express excitement, strong emotion, or a sense of urgency.

You look beautiful! / Watch out!

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Use a comma to separate independent clauses (complete thoughts) when they’re joined by the following conjuctions.

Incorrect: I want to go out tonight but I need to study. Correct: I want to go out tonight, but I need to study.

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Use a semicolon between closely related independent clauses.

I love coffee; you love tea.

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Use a colon to introduce a list of items.

They serve many tupes of food: Chinese, Mexican, and American.

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Use a hyphen to join two or more words that serve as single adjectives before a noun.

Kind-hearted woman / Brand-new television

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Use a hypehn with compound numbers.

Ninety-nine, twenty-one

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Run-ons & Fragments

A complete sentence contains a subject, a predicate verb, and a complete thought. If any of the three is lacking, the sentence is called a fragment.

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Verbs: Subject-Verb Agreement & Verb Tenses

The ACT English section often includes long sentences in which the main subject and the verb are separated by many words or clauses. If you identify the subject of each sentence and make sure the verb matches it, you can ace this grammar rule.

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Punctuation

Commas, apostrophes, colons, semicolons, dashes, periods, question marks, and exclamation points.

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Wordiness

As long as there are no new grammar errors introduced, the shortest answer choice is often correct. Redundancy is a type of wordiness where the same thing is said twice such as “happy and joyful.” Keep it simple and to the point.

TOA, SOH, CAH:

Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent

Law of Sines:

A/Sin(A) = B/Sin(B) = C/Sin(C)

Law of Cosines:

c² = a² + b² - 2ab · cos(C)