ACT (Enhanced) flashcards

Evaluating an exponential decay function involves substituting a given value into the independent variable and calculating the resulting dependent variable, often using the natural base 'e'.

• Function takes the form y = ab^(kx) or y = ae^(kx) where k<0.
• Used to model phenomena that decrease over time or distance.
• Calculators are typically needed for 'e' or non-integer exponents.

Memory trick: Plug in, then calculate, simple as that.

The distance formula calculates the straight-line distance between two points (x1, y1) and (x2, y2) in a two-dimensional Cartesian coordinate system.

• Formula: d = √((x2 - x1)² + (y2 - y1)²).
• Derived from the Pythagorean theorem.
• Always yields a non-negative value.

Memory trick: Distance: difference of Xs squared, plus difference of Ys squared, all under a root.

A mathematical model describing population growth that levels off as it approaches a maximum limit (carrying capacity) due to environmental constraints.

• Formula: P(t) = K / (1 + ae^(-bt)).
• K is the carrying capacity.
• As t → ∞, P(t) → K.

Memory trick: Logistic Limit: As time goes infinite, the top number is the cap!

Two quantities are directly proportional if their ratio is constant. As one quantity increases, the other increases at a constant rate.

• Represented by y = kx, where k is the constant of proportionality.
• The graph is a straight line passing through the origin.
• If x doubles, y also doubles.

Memory trick: Direct means 'D'ivide and 'D'iscover the constant.

A mathematical model describing a quantity that decreases at a constant percentage rate over equal time intervals.

• Formula: V = P(1 - r)^t.
• Used for depreciation, radioactive decay, population decrease.
• The value decreases by a larger absolute amount initially, then less over time.

Memory trick: Decay Formula: Initial price, minus rate, raised to time!

A rule for normal distributions stating that approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.

• Applies specifically to normal (bell-shaped) distributions.
• Provides quick estimates of probabilities.
• Standard deviations are measured from the mean.

Memory trick: Sixty-eight, ninety-five, ninety-nine point seven, for one, two, and three std devs, a data heaven!

For a fixed perimeter, the rectangle with the maximum area is a square. All sides are equal.

• Perimeter = 2(length + width).
• Area = length × width.
• When length = width, the shape is a square, and area is maximized.

Memory trick: Square is 'S'uperior for 'S'urface area with a fixed 'S'ide length when maximizing.

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

• For f(x) = ax² + bx + c, the x-coordinate of the vertex is -b/(2a).
• Substitute the x-coordinate back into the function to find the y-coordinate (max/min value).
• If a > 0, parabola opens up (minimum); if a < 0, parabola opens down (maximum).

Memory trick: Vertex Peak: X is minus B over 2A, then plug X back in!

The formula A = Pe^(rt) calculates the future value (A) of an investment with principal (P), annual interest rate (r), and time (t) in years, compounded continuously.

• Uses the mathematical constant 'e' (Euler's number).
• Represents the theoretical upper limit of compounding frequency.
• Often used for modeling natural growth or decay processes.

Memory trick: Pert for continuous growth, like P-E-R-T.

Analyzing how changes in input variables (percentages) affect the output of a formula, especially with multiplicative/divisive relationships.

• Increase by X% means multiplying by (1 + X/100).
• Decrease by X% means multiplying by (1 - X/100).
• Changes multiply when variables are multiplied/divided.

Memory trick: Variable Shift: Convert percentages, substitute, then simplify ratios!

For a quadratic function, the average rate of change over an interval [a, b] is the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

• Formula: (f(b) - f(a)) / (b - a).
• Represents the average change in the dependent variable per unit change in the independent variable.
• Different from instantaneous rate of change (derivative).

Memory trick: Y2 minus Y1, over X2 minus X1, the average change is done!

Half-life is the time required for a quantity to reduce to half its initial value. It's often used in exponential decay models.

• Amount remaining = Initial Amount * (0.5)^(number of half-lives).
• Number of half-lives = total time / half-life period.
• Can be solved using logarithms or by recognizing powers of 0.5.

Memory trick: Halves 'H'appen in 'H'eaps: count the number of 'H'alves to find the age.

The initial value of an exponential function f(x) = a * b^x is the value of 'a', which represents the output when x = 0.

• In f(x) = a * b^x, 'a' is the initial value.
• It's the y-intercept of the exponential curve.
• Represents the quantity at the start (e.g., time t=0).

Memory trick: Always Start with 'A' for the Amount at the start.

A mathematical model describing population growth that is limited by a carrying capacity, resulting in an S-shaped curve.

• P(t) = K / (1 + Ae^(-kt)) is the standard form.
• K represents the carrying capacity (maximum population).
• As t approaches infinity, P(t) approaches K.

Memory trick: Logistic's K: the ceiling, the limit, the carrying capacity's feeling.

The derivative of an exponential function of the form f(x) = a * e^(kx) is f'(x) = a * k * e^(kx).

• Used to find instantaneous rates of change.
• 'k' is the constant in the exponent.
• 'a' is the coefficient.

Memory trick: The 'K' in 'K'x 'K'omes 'K'icking out front!

A relationship where one quantity is inversely proportional to the square of another quantity.

• Represented as y = k/x^2, where k is the constant of proportionality.
• As x increases, y decreases rapidly.
• Common in physics (e.g., gravitational force, light intensity).

Memory trick: Inverse square, strength decreases with distance, beware!

Substituting a given value for a variable into a linear equation to find the corresponding value of another variable.

• Linear equations represent a straight line.
• They often model real-world relationships.
• Evaluation involves a direct substitution and calculation.

Memory trick: Line's Value: Plug it in, find the number!

The rate at which a quantity increases over time, expressed as a percentage, in an exponential growth model.

• Model form: A(t) = P(1 + r)^t, where r is the growth rate.
• To find percentage, convert decimal 'r' to percent (r * 100%).
• The base of the exponent (1+r) is the growth factor.

Memory trick: Growth factor's 'r' is key, convert to percent, you'll see!

Calculating a percentage based on a value derived from a linear equation relative to a total quantity.

• Evaluate the linear function to find the count of interest.
• Percentage = (Part / Whole) * 100%.
• Ensure units are consistent before calculating percentage.

Memory trick: Linear output first, then divide by whole, for a percentage burst!

Dilution calculations involve determining the amount of solvent needed to change the concentration of a solution while keeping the amount of solute constant.

• Amount of solute (e.g., acid) remains constant during dilution.
• Concentration is typically expressed as a percentage or molarity.
• Total volume changes with added solvent.

Memory trick: Solute stays same, Volume changes, Concentration shifts.

The algebraic representation of a circle centered at the origin (0,0) with a specific radius 'r', given by x² + y² = r².

• All points (x,y) on the circle are equidistant from the origin.
• The distance is the radius 'r'.
• Based on the Pythagorean theorem.

Memory trick: Circle's Center: Point gives radius-squared, plug it in!

The ability to understand and use ratios to solve problems where quantities are related by a constant factor.

• Ratios compare two quantities.
• Proportions state that two ratios are equal.
• Cross-multiplication is a common method to solve proportions.

Memory trick: Ratio's balance, cross-multiply to advance.

The amount of three-dimensional space occupied by a rectangular prism, calculated by multiplying its length, width, and height.

• Units are always cubic (e.g., cubic feet, cubic meters).
• Applies to boxes, rooms, slabs, etc.
• Formula: V = l × w × h.

Memory trick: Box's Space: Length, Width, Height, Multiply!

For an exponential decay term e^(-kt) where k>0, as t approaches infinity, the term approaches 0. This often reveals a horizontal asymptote.

• e^(-kt) approaches 0 as t -> ∞.
• The limit of a sum/difference is the sum/difference of the limits.
• Reveals the long-term behavior or carrying capacity in models.

Memory trick: Negative 'N'umbers in the 'N'umerator make 'N'othing appear at infinity for 'e'.

In a logistic growth model, the carrying capacity (K) is the maximum population size that the environment can sustain indefinitely, represented by the numerator of the logistic function.

• The population approaches K as time approaches infinity.
• It's the upper horizontal asymptote of the logistic curve.
• Limits resources and environmental factors determine K.

Memory trick: The 'K'ap is 'K'ing at the 'K'arriage 'K'apacity.

Solving a linear equation involves isolating the unknown variable using inverse operations (addition/subtraction, multiplication/division) while maintaining equality.

• The goal is to get the variable by itself on one side of the equation.
• Perform the same operation on both sides of the equation.
• Often involves multiple steps following order of operations in reverse.

Memory trick: Balance the scale! Do unto one side as you do unto the other.

Estimating a value within two known values on a linear scale.

• Assumes a constant rate of change.
• Requires at least two data points.
• Used to predict intermediate values.

Memory trick: Slope's path, point's start, equation's heart.

The correlation coefficient (r) measures the strength and direction of a linear relationship between two quantitative variables. It ranges from -1 to +1.

• Sign indicates direction: positive (+) or negative (-).
• Absolute value indicates strength: closer to 1 is strong, closer to 0 is weak.
• Does NOT imply causation.

Memory trick: Sign for 'S'lope, 'S'ize for 'S'trength.

Evaluating a quadratic function involves substituting a specific value for the independent variable (x) into the function's equation and then performing the arithmetic operations to find the corresponding output (y).

• Quadratic functions are typically of the form ax² + bx + c.
• Follow the order of operations (PEMDAS/BODMAS).
• Careful with squaring negative numbers (if applicable).

Memory trick: Plug in, square, multiply, then add it all up.

Calculating a specified percentage of a given total quantity, often involving finding the complement percentage.

• Percentages are parts per hundred.
• To find 'x%' of a number, multiply the number by x/100.
• The 'complement' percentage is 100% minus the given percentage.

Memory trick: Percent Part: Find the complement, then multiply by total!

For a quadratic function f(x) = ax^2 + bx + c with a < 0, the vertex represents the maximum value of the function.

• The x-coordinate of the vertex is given by x = -b / (2a).
• The maximum value is f(-b / (2a)).
• The parabola opens downwards when 'a' is negative.

Memory trick: Vertex's peak, -b/2a, then plug it back, no delay!

Words or phrases used to introduce an opposing viewpoint or concession in argumentative writing, signaling a shift to a different perspective.

• Precedes a point that goes against the main argument.
• Prepares the reader for a refutation.
• Helps maintain logical flow in debate.

Memory trick: Argumentative writing needs phrases to bridge between claims, evidence, and counterclaims.

A writing style characterized by formality, impartiality, and evidence-based reasoning, avoiding personal opinions, emotional language, and colloquialisms.

• Focuses on facts and evidence.
• Avoids first-person pronouns (I, me, my).
• Maintains a neutral and unbiased perspective.

Memory trick: Academic writing is like a judge: impartial, evidence-based, and formal.

The opening greeting in a formal email, designed to show respect and professionalism to the recipient, especially in academic or professional contexts.

• Should be formal and respectful.
• Includes title and last name.
• Avoids informal greetings or first names.

Memory trick: Formal emails are like dressing up: start with a proper greeting, clear body, and polite closing.

The repetition of a chosen grammatical form within a sentence. By making sure all compared or listed items are in the same grammatical structure, parallel structure creates a sense of rhythm and balance.

• Applies to words, phrases, or clauses in a series.
• Ensures grammatical consistency.
• Enhances clarity and readability.

Memory trick: Parallelism is like a balanced scale: all parts must match weight and form.

To find the percentage of a quantity from a linear model, first calculate the quantity using the model, then divide by the total and multiply by 100%.

• Evaluate the linear model for the given input.
• Divide the output by the total number of items.
• Multiply the result by 100 to express as a percentage.

Memory trick: Follow the 'Linear Path' to 'Calculate' and then 'Percent-age'.

Words or phrases that signal a contrast, contradiction, or concession, allowing a writer to acknowledge an opposing point before presenting a counter-argument or a different perspective.

• Introduces opposing ideas.
• Helps maintain logical flow.
• Examples: 'however,' 'although,' 'despite,' 'nevertheless.'

Memory trick: Contrasting ideas need a Clear Crossroad.

For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex represents the maximum point if a < 0 (parabola opens down) or the minimum point if a > 0 (parabola opens up).

• x-coordinate of vertex: -b / (2a).
• y-coordinate of vertex: substitute x-coordinate into the function.
• If 'a' is negative, the vertex is a maximum; if 'a' is positive, it's a minimum.

Memory trick: The 'Vertex' is the 'Peak' or 'Valley' of the 'Parabola's Path'.

Using specific and accurate vocabulary to describe the functions and processes of artificial intelligence, particularly its ability to learn and change.

• Accurate word choice enhances clarity.
• Technical fields require precise terminology.
• Misused words can lead to misunderstanding.

Memory trick: AI Always Adapts to Accumulate Acurate Answers.

A form of active camouflage where an animal produces light to match the intensity and wavelength of ambient light, making it difficult to see from below.

• Common in deep-sea organisms.
• Light is typically emitted from the ventral (underside) surface.
• Helps blend with faint downwelling light.

Memory trick: Light helps fish find, hide, or fight.

The initial value of a logistic function P(t) = K / (1 + Ae^(-kt)) is found by setting t=0, which simplifies the exponential term to 1.

• Set t=0 to find the initial value.
• e^0 = 1, simplifying the calculation.
• The initial value is always K / (1 + A) for the standard form.

Memory trick: Start at 'Zero Time' to see the 'Initial Climb'.

Words or phrases that connect ideas by adding more information, examples, or elaboration to a previous point.

• Used to build upon arguments.
• Creates flow and coherence.
• Examples: 'in addition,' 'moreover,' 'also.'

Memory trick: Additive phrases Always Add Another Angle.

A moment in a play or other work when a character makes a critical discovery or recognition, often related to their own identity or the true nature of their circumstances.

• Often associated with tragedy, leading to a character's downfall or profound sorrow.
• A shift from ignorance to knowledge.
• Can occur alongside or as a result of peripeteia (reversal of fortune).

Memory trick: Tragedy's key: Reversal, Recognition, Release.

Combined proportionality describes how one variable relates directly, inversely, or jointly to two or more other variables, often involving powers.

• Direct proportionality: y = kx.
• Inverse proportionality: y = k/x.
• Combined: y = k * (direct_vars) / (inverse_vars).

Memory trick: Balance the 'Constants' with 'Variables' to find the 'Relationship'.

The technique of merging multiple short, simple sentences into longer, more complex ones to improve flow, reduce choppiness, and enhance reader engagement.

• Uses conjunctions, relative clauses, appositives.
• Reduces repetition and improves coherence.
• Adds sophistication to writing style.

Memory trick: Vary your sentences, or your writing will be Very Vexing.

Words and phrases that appeal to the sense of hearing, creating vivid and immersive descriptions of sounds in writing.

• Uses strong verbs for sounds (e.g., roared, whispered, clanged).
• Employs similes or metaphors to compare sounds.
• Helps readers 'hear' the scene.

Memory trick: Sound descriptions should Sing, not be Silent.

A persuasive appeal that relies on logic and reason, often through evidence, facts, statistics, and rational arguments, to convince an audience.

• Appeals to the audience's intellect.
• Uses evidence and logical reasoning.
• Aims for a rational response.

Memory trick: Every Public Speaker Loves Logic.