Digital SAT flashcards

The rate at which a quantity decreases over time in an exponential decay model.

• Represented by 'r' in the formula A(t) = P(1 - r)^t.
• It is derived from the decay factor (1-r).
• Expressed as a decimal or percentage.

Memory trick: Decay models decrease, so look for a factor less than one!

Rearranging an equation to isolate a specific variable, often by using inverse operations.

• Apply inverse operations (addition/subtraction, multiplication/division, exponents/roots) to both sides.
• The order of operations is reversed when isolating a variable.
• Common in geometry, physics, and other applied math problems.

Memory trick: Unwrap the variable, inverse operations are your tools.

A logarithm is the inverse operation to exponentiation. The equation log_b(y) = x is equivalent to b^x = y. Converting between these forms is crucial for solving logarithmic and exponential equations.

• log_b(y) = x means 'b raised to the power of x equals y'.
• If no base 'b' is specified for 'log', it is assumed to be base 10 (common logarithm).
• If 'ln' is used, the base is 'e' (natural logarithm).

Memory trick: Log is the Exponent: 'Base to the power of log equals the argument'.

An exponential growth model describes a quantity that increases at a rate proportional to its current value. It is represented by the formula y = a * b^x, where 'a' is the initial amount, 'b' is the growth factor (b > 1), and 'x' is the number of growth periods.

• Growth factor 'b' is (1 + rate of increase).
• The variable representing time or periods is in the exponent.
• Commonly used for population growth, compound interest, and radioactive decay (decay when b < 1).

Memory trick: Exponent's power: small changes grow big, fast!

In a logistic growth model P(t) = C / (1 + Ae^(-kt)), the carrying capacity (C) is the maximum population size that the environment can sustain indefinitely.

• The carrying capacity is the upper limit of population growth.
• It is represented by the numerator (C) in the logistic function.
• As time (t) approaches infinity, the exponential term approaches zero, and P(t) approaches C.

Memory trick: Logistic Limit: The top number 'C' is the population ceiling!

For a quadratic function in the form ax^2 + bx + c, the maximum or minimum value occurs at the vertex. If a > 0, it's a minimum; if a < 0, it's a maximum.

• The x-coordinate of the vertex is -b/(2a).
• Substitute the x-coordinate back into the function to find the y-coordinate (maximum/minimum value).
• The sign of 'a' determines if it's a maximum (a<0) or minimum (a>0).

Memory trick: Vertex Peak: Find the 'x' at -b/2a, then plug it in to get 'y' for the max/min!

The highest or lowest point on the graph of a quadratic function, representing the maximum or minimum value of the function.

• For f(x) = ax² + bx + c, the x-coordinate of the vertex is -b/(2a).
• If a > 0, the parabola opens upward, and the vertex is a minimum.
• If a < 0, the parabola opens downward, and the vertex is a maximum.
• The y-coordinate is found by substituting the x-coordinate into the function.

Memory trick: Vertex is the top or bottom, -b/2a finds the spot.

The total distance around the outside of a rectangle, calculated by summing the lengths of all four sides.

• Formula: P = 2(length + width) or P = 2l + 2w.
• In word problems, express length and width in terms of a single variable.
• Used in geometry and real-world applications like fencing or framing.

Memory trick: Geometry words become algebra lines, then solve for the shapes' secrets.

A relationship between two quantities where their ratio is constant. If one quantity increases, the other increases proportionally.

• Can be written as y = kx, where k is the constant of proportionality.
• Used to solve problems involving scaling ingredients or comparing rates.
• The graph of a direct proportion is a straight line passing through the origin.

Memory trick: Same scale, same story, both grow or shrink proportionally.

A linear equation can model situations where a quantity changes at a constant rate relative to another quantity.

• Identify the initial value (y-intercept).
• Identify the constant rate of change (slope).
• Determine if the rate is an increase (positive slope) or decrease (negative slope).

Memory trick: Words to Math: Find the Start, Find the Change, Build the Line!

Exponential growth with a doubling period describes situations where a quantity increases by a factor of 2 over a fixed time interval.

• Formula: P(t) = P₀(2)^(t/d)
• P₀ is initial amount, t is time, d is doubling period.
• The exponent t/d ensures the doubling factor applies correctly over the period.

Memory trick: Doubling: Start with P₀, multiply by 2, raise to time-over-period!

An exponential equation is an equation where the variable appears in the exponent. To solve it, one typically isolates the exponential term and then uses logarithms to bring the exponent down, or by finding a common base.

• If bases are the same, equate the exponents.
• If bases are different, take the logarithm of both sides.
• Use log properties: log(b^x) = x * log(b).

Memory trick: Logarithms 'unlock' the exponent from its high perch.

The half-life of a substance undergoing exponential decay is the time required for half of the initial quantity to decay.

• Set the final amount to half of the initial amount.
• Solve the exponential equation using logarithms.
• Applies to radioactive decay, drug clearance, etc.

Memory trick: Half-life: Halve the start, then log the time!

A relationship between two quantities where their product is constant. As one quantity increases, the other decreases proportionally.

• Can be written as y = k/x or xy = k, where k is the constant of proportionality.
• The graph of an inverse variation is a hyperbola.
• Used to model relationships where an increase in one variable leads to a decrease in another.

Memory trick: Direct goes together, inverse goes apart, constant glue holds them all.

Mixture problems involve combining two or more quantities with different properties to create a desired mixture. They are typically solved using a system of two linear equations.

• One equation represents the total amount/volume of the mixture.
• The second equation represents the total amount of the key component (e.g., acid, solute) in the mixture.
• Convert percentages to decimals for calculations.

Memory trick: Mix it Up: Total amount equation, then total component equation!

Algebraic problems involving combining two or more quantities with different concentrations or values to achieve a desired overall concentration or value.

• Often solved using systems of linear equations or a single variable equation.
• Key is to track the total amount of a specific component (e.g., acid, solute).
• Equation typically balances the sum of component amounts with the desired total amount.

Memory trick: Mix the parts, balance the whole, percentage is key to the goal.

A linear equation is an algebraic equation in which each term has an exponent of 1, and the graph of the equation is a straight line. Solving it means finding the value(s) of the variable(s) that make the equation true.

• Involves isolating the variable using inverse operations.
• Operations must be applied to both sides of the equation to maintain equality.
• The goal is to get the variable by itself on one side of the equals sign.

Memory trick: Isolate the variable, step by step, to find its true value.

The point at which total cost and total revenue are equal, meaning there is no net loss or gain.

• Calculated by setting Cost Function C(x) equal to Revenue Function R(x).
• Represents the minimum number of units that must be sold to cover all expenses.
• Below this point, the business incurs a loss; above it, the business makes a profit.

Memory trick: Cost, Revenue, Profit: The business trio, where break-even balances the two.

To solve an equation containing a radical expression, isolate the radical, then raise both sides to the power corresponding to the index of the radical (e.g., square for square root).

• Isolate the radical term on one side of the equation.
• Raise both sides to the power of the radical's index.
• Solve the resulting linear or quadratic equation.
• Always check for extraneous solutions by substituting back into the original equation.

Memory trick: Radical Road: Isolate the root, square it, then solve, but always check!

To find when one exponential function exceeds another, set up an inequality and solve for the variable, often requiring the use of logarithms.

• Set the inequality: f(t) > g(t) or f(t) < g(t).
• Isolate the exponential terms on one side.
• Use logarithms to solve for the exponent variable.

Memory trick: Compare Exponentials: Set inequality, simplify bases, then log both sides!

The minimum value of an absolute value function of the form y = a|x - h| + k is k, and it occurs when x = h, making the absolute value term zero.

• The absolute value expression |X| is always ≥ 0.
• Its minimum value is 0.
• The minimum of y = |X| + k is k.

Memory trick: Absolute Min: The 'k' is the lowest, when the inside is zero!

A polynomial function is a function of the form P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where 'n' is a non-negative integer and the 'a' values are coefficients. Evaluating it means finding the output value for a given input value of 'x' (or 't').

• Substitute the given value for the variable into every term.
• Follow the order of operations (PEMDAS/BODMAS).
• The constant term (a_0) represents the y-intercept or initial value when x=0.

Memory trick: Plug and play: input goes in, output comes out.

For a quadratic function in factored form y = a(x - r1)(x - r2), the x-coordinate of the vertex occurs halfway between the roots r1 and r2. The y-coordinate at this point is the maximum or minimum.

• Roots are the x-intercepts where y=0.
• X-coordinate of vertex: (r1 + r2) / 2.
• Substitute x-vertex into function to find max/min y-value.

Memory trick: Factored Max: Roots are easy, middle is 'x', plug it in to get 'y' high!

An effective introductory sentence or paragraph sets the tone, introduces the topic, and often presents the main argument or scope of the subsequent text.

• Should clearly state the paragraph's purpose.
• Must align with the overall essay's thesis.
• Can acknowledge counterarguments before presenting the main point.

Memory trick: An intro is like a door: it opens to the room's purpose.

Words or phrases that signal to the reader that new information is being introduced that supports, expands upon, or is similar to the preceding idea.

• Used to add more points or examples.
• Indicate continuity of thought.
• Examples: 'furthermore,' 'moreover,' 'also,' 'besides.'

Memory trick: Transitions are road signs for your reader, guiding them through your ideas.

Selecting the most accurate and specific word to convey the intended meaning, enhancing clarity and impact.

• Avoids vague or general terms.
• Considers connotations and denotations.
• Contributes to the overall tone and purpose.

Memory trick: Precision in words is like a laser: sharp, focused, and hits the exact target.

Phrases like 'along with,' 'as well as,' 'in addition to,' 'accompanied by,' or 'together with' do not make a singular subject plural. The verb still agrees with the initial singular subject.

• These phrases introduce additional information, not compound subjects.
• The verb agrees only with the noun that comes *before* these phrases.
• Mentally remove these phrases to simplify subject-verb identification.

Memory trick: 'Along with' is a friend, not a partner; the first subject rules the charter!

In a clause introduced by 'to which,' the verb must agree with the noun that functions as the subject within that clause, not necessarily the noun preceding 'to which'.

• Identify the noun acting as the subject *within* the 'to which' clause.
• This noun is often distinct from the noun that 'which' refers to.
• The verb must agree in number with this specific subject.

Memory trick: 'To which' guides you to a new clause, find *its* subject, then pause!

Indefinite pronouns like 'each,' 'every one,' 'anyone,' 'no one,' 'somebody,' etc., are always singular and require singular verbs.

• Pronouns like 'each,' 'everyone,' 'nobody' are singular.
• These pronouns take singular verbs.
• Phrases like 'of them' or 'of which' after these pronouns do not change their singular nature.

Memory trick: Each, every, any one, body, thing: always singular, always sing!

An appositive is a noun or noun phrase that renames another noun right beside it. It does not affect the number of the subject; the verb must agree with the actual subject, not the appositive.

• Appositives are usually set off by commas.
• They provide extra information about the subject.
• The verb must agree with the noun *before* the appositive.

Memory trick: Appositives are just a side note, the main subject's number is the boat!

When a clause begins with 'of which,' the verb must agree with the noun that immediately precedes 'of which' or is the subject of that relative clause.

• Identify the noun that 'of which' refers to.
• This noun is often the subject of the 'of which' clause.
• The verb must agree with this identified noun in number.

Memory trick: 'Of which' points back, so look to the noun track!

Indefinite pronouns like 'each,' 'every,' 'either,' 'neither,' 'one,' 'anybody,' 'everyone,' 'nothing,' etc., are always singular and require singular verbs.

• Treat 'each' as singular.
• Treat 'every' as singular.
• Treat 'one' as singular.
• These pronouns always take singular verbs, regardless of intervening phrases.

Memory trick: Pronouns like 'each' and 'one' are always singular, never plural, so use a singular verb, that's the rule!

When a long and complex phrase separates the subject from its verb, ignore the intervening phrase and ensure the verb agrees with the true subject.

• Identify the main subject first.
• Mentally remove all prepositional phrases and clauses between the subject and verb.
• The verb must agree only with the main subject's number.

Memory trick: Cut through the noise, find the true subject's voice!

When a singular subject is followed by a long or complex prepositional phrase, the verb must still agree with the singular subject, ignoring any nouns within the phrase.

• Prepositional phrases never contain the grammatical subject.
• Mentally cross out prepositional phrases to reveal the true subject.
• Be vigilant for plural nouns within these phrases that act as distractors.

Memory trick: Prepositional phrases are just a disguise, hide the real subject right before your eyes!

Phrases that come between the subject and the verb (e.g., prepositional phrases, appositives, clauses beginning with 'along with,' 'as well as,' 'in addition to') do not affect the number of the subject.

• The verb must agree with the true subject, not a noun in an intervening phrase.
• Mentally remove intervening phrases to identify the core subject.
• Phrases like 'especially,' 'particularly,' 'including' are often distractors.

Memory trick: Don't let the words in the middle fool you, find the real subject, it's true!

When subjects are joined by 'neither...nor,' the verb agrees in number with the subject that is closer to the verb.

• Focus on the subject immediately preceding the verb.
• If the closest subject is singular, use a singular verb.
• If the closest subject is plural, use a plural verb.

Memory trick: Neither-Nor: The verb's a friend to the subject near its end!

When an abstract noun (like 'practice,' 'strategy,' 'concept') acts as the subject of a clause, the verb must agree with it in number.

• Identify the abstract noun as the subject.
• Abstract nouns are typically singular unless explicitly plural.
• The verb must match this singular or plural form.

Memory trick: Abstract ideas, concrete verbs: match their count, don't be absurd!

The principle that the verb in a sentence must agree in number (singular or plural) with its subject.

• Singular subjects take singular verbs.
• Plural subjects take plural verbs.
• Compound subjects joined by 'and' are usually plural.

Memory trick: Subjects and verbs must always agree, like two friends on a spree!

Some plural nouns, like 'dynamics,' 'politics,' 'ethics,' 'mathematics,' end in 's' but are inherently plural in meaning and require a plural verb, especially when referring to qualities or processes.

• Nouns like 'dynamics,' 'statistics,' 'politics' can be singular or plural.
• When referring to a general field of study or a single concept, they are singular.
• When referring to specific qualities, activities, or processes, they are plural.
• In this context, 'dynamics' refers to ongoing processes, making it plural.

Memory trick: Nouns ending in 's' might trick your brain, check their meaning, then explain!

Choosing words and phrases that highlight the importance, intensity, or extent of an idea, ensuring the reader grasps the intended significance.

• Uses strong verbs and precise adjectives.
• Avoids vague or hedging language.
• Conveys the desired tone and impact.

Memory trick: To emphasize, turn up the volume on your words and shine a spotlight on the idea.

A word or phrase used to introduce information that goes against or challenges what was previously stated, often highlighting a different perspective or unexpected outcome.

• Connects opposing ideas.
• Signals a shift in argument or focus.
• Examples: however, nevertheless, on the other hand, in contrast.

Memory trick: Contrastive transitions are like a seesaw: one idea goes down, the other goes up to balance it.

These phrases signal to the reader that the upcoming information presents a contrasting idea, a concession, or an exception to what was previously stated.

• Indicate a shift in perspective or argument.
• Connect ideas that are in opposition or present a qualification.
• Examples: however, nevertheless, although, despite, in contrast.

Memory trick: Transitions move thoughts, sometimes straight, sometimes opposite.

Problems involving combining two or more quantities with different properties (e.g., cost, concentration) to achieve a desired overall property.

• Typically involve two equations: one for total quantity and one for total value/concentration.
• Can be solved using substitution or elimination methods.
• Define variables clearly for each unknown quantity.

Memory trick: Quantity + Value equations, then Substitute to Solve.

Words or phrases that connect ideas by showing that one idea is being added to another, often to provide more information, examples, or further support.

• Used to introduce supplementary information.
• Indicates a continuation or expansion of a point.
• Common examples: 'furthermore,' 'moreover,' 'in addition,' 'also.'

Memory trick: Connect ideas, don't just stand there, add something!

For a quadratic function in the form f(x) = ax^2 + bx + c, if a > 0, the parabola opens upwards, and its vertex represents the minimum value of the function.

• The x-coordinate of the vertex is given by x = -b / (2a).
• The minimum value is the y-coordinate of the vertex, f(-b / (2a)).
• Relevant for optimization problems like finding lowest point or minimum cost.

Memory trick: Vertex is the valley (or peak) of the parabola.

The 'r' value in the exponential growth formula P(t) = P_0 * (1 + r)^t, representing the percentage increase per time period.

• Expressed as a decimal in the formula.
• Multiply by 100 to convert to a percentage.
• Represents the rate of change relative to the current amount.

Memory trick: Growth Factor is One Plus Rate.

Words or phrases that signal a change in topic, perspective, location, or time within a text, guiding the reader to a new but related idea.

• Used to introduce a new aspect or angle of a discussion.
• Helps maintain coherence and clarity.
• Examples: 'turning to,' 'regarding,' 'as for,' 'in terms of,' 'meanwhile,' 'elsewhere.'

Memory trick: When you change your mind, you 'shift focus' to a new thought, use words to show it!

Choosing strong, vivid verbs and precise, descriptive adjectives to create a more impactful and emotionally resonant message, enhancing the reader's understanding and engagement.

• Replaces weak verbs with stronger, more specific ones.
• Uses adjectives that add precise detail and sensory information.
• Helps to convey tone, emotion, and emphasis effectively.

Memory trick: Paint with words, don't just sketch; make your meaning 'pop'!

Formal words or phrases used to introduce additional information or arguments that support or expand upon previous points, maintaining an academic or professional tone.

• Used in academic essays, reports, and formal writing.
• Signals an expansion of ideas, not a contrast.
• Examples: 'furthermore,' 'moreover,' 'in addition,' 'additionally,' 'consequently,' 'therefore.'

Memory trick: Formal writing means fancy words, not slang, to connect your ideas like a pro!

Words or phrases that introduce a statement that admits something, often something that seems to contradict or hinder the main point, but does not ultimately invalidate it.

• Signals that a point is being acknowledged despite a contrary main argument.
• Shows that something is true 'even so' or 'in spite of.'
• Common examples: 'nevertheless,' 'however,' 'even though,' 'although,' 'despite,' 'in spite of.'

Memory trick: Concession means 'giving in' a little, but still holding your ground!

Using two linear equations to represent the relationships between the dimensions (length and width) and the perimeter of a rectangle, then solving for the unknown dimensions.

• Perimeter formula for a rectangle: P = 2L + 2W.
• One equation often relates length and width (e.g., L = W + x).
• Solve using substitution or elimination methods.

Memory trick: Perimeter's puzzle: Length and Width unlock.

Pairs of conjunctions that connect grammatically equal elements in a sentence, often used to emphasize a dual or reciprocal relationship between ideas.

• Used to show balance or comparison.
• Common pairs: 'not only... but also,' 'either... or,' 'neither... nor,' 'both... and.'
• Helps to create parallelism and impact.

Memory trick: Make your point POP, don't just state it; give it a special 'zing'!

The y-intercept (b) in a linear equation y = mx + b represents the value of y when x is 0.

• It's the point where the line crosses the y-axis.
• Often represents an initial amount or starting value.
• In real-world scenarios, it's the value of the dependent variable when the independent variable is absent or at its starting point.

Memory trick: Slope is how it SLIDES, intercept is where it STARTS.

A relationship where one quantity varies inversely with the square of another quantity, expressed as y = k/x^2.

• The constant of proportionality (k) is found by multiplying y by x^2.
• As x increases, y decreases rapidly.
• Used in physics and chemistry for relationships like inverse square law.

Memory trick: Inverse means Divide, constant 'k' connects the sides.

Words or phrases that introduce potential negative outcomes, risks, or contrasting drawbacks following a discussion of benefits or positive aspects.

• Signals a shift from positive to negative implications.
• Often used to introduce counter-arguments or potential problems.
• Examples: 'however,' 'nevertheless,' 'on the other hand,' 'yet,' 'caution must be exercised.'

Memory trick: When you see a warning sign, you 'transition' to caution!