Digital SAT flashcards

For continuous exponential growth modeled by P(t) = P_0 * e^(kt), the growth rate constant 'k' determines the speed of growth. If given doubling time (T_d), k = ln(2) / T_d.

• e^(kt) is the continuous growth factor.
• Doubling time (T_d) means P(T_d) = 2 * P_0.
• k and T_d are inversely related through ln(2).

Memory trick: Doubling Time T, K is Ln 2 Over T!

A conclusion reached on the basis of evidence and reasoning, rather than explicit statements in the text.

• Requires reading between the lines.
• Must be supported by textual evidence, not just prior knowledge.
• Is a logical deduction from what is stated.

Memory trick: Read, Deduce, Confirm: Is the 'D'eduction solid?

Drawing a logical conclusion from numerical information or statistics, often by identifying patterns or relationships in the data.

• Requires careful interpretation of percentages, trends, or comparisons.
• Must be directly supported by the numbers presented, not assumptions.
• Avoids overgeneralization or drawing conclusions not evident in the data.

Memory trick: Numbers, Notice, Nudge: What do the 'N'umbers hint at?

The distance around the edge of a circle.

• Formula: C = 2πr (where r is the radius)
• Formula: C = πd (where d is the diameter)
• Used to measure the perimeter of circular objects

Memory trick: Circumference is like a fence, around the whole place.

Assessing how new evidence supports, weakens, or refutes different explanations for a phenomenon by comparing their core tenets.

• Requires identifying the central claims of each theory.
• Involves analyzing evidence for direct support or contradiction.
• Often reveals that theories are not mutually exclusive but can be complementary.

Memory trick: Theories, Test, Tally: Do the 'T'ruths align?

Synthesizing information with nuance involves combining multiple pieces of data, including seemingly contradictory ones, to form a comprehensive understanding that acknowledges complexities and limitations.

• Integrate all relevant facts, positive and negative.
• Avoid oversimplification or extreme conclusions.
• Look for a statement that balances different aspects of the evidence.

Memory trick: Balance Both Sides to Build Better Beliefs.

To solve a system of linear equations by substitution, set the expressions for the dependent variable (if both are solved for it) equal to each other, then solve for the independent variable.

• Represents the intersection point of two lines.
• The solution (x, y) satisfies both equations simultaneously.
• Can be used when one or both equations are already solved for a variable.

Memory trick: Where two lines meet, the solution is sweet!

For an inequality of the form |ax + b| < c (where c > 0), it is equivalent to the compound inequality -c < ax + b < c. This means the expression inside the absolute value is between -c and c.

• Translates to a 'sandwich' inequality.
• Solve by isolating the variable in the middle.
• Represents values 'within' a certain distance from a center point.

Memory trick: Less than means 'between' (like a sandwich)!

The mean provides the average value of a dataset, while the standard deviation measures the typical spread or variability of data points around that mean.

• Higher mean indicates a larger average value.
• Lower standard deviation indicates greater consistency or less spread.
• Standard deviation is sensitive to outliers.

Memory trick: Mean is the center, Std Dev is how far points stray.

The amount of space occupied by a three-dimensional cylindrical object.

• Formula: V = πr²h (where r is radius, h is height)
• Units are cubic (e.g., cubic feet, cubic meters).
• Represents the capacity of the cylinder.

Memory trick: Volume is like filling a box: Area of the base times the height.

An absolute value inequality of the form |ax + b| ≤ c (where c > 0) can be rewritten as a compound inequality: -c ≤ ax + b ≤ c. This represents an 'AND' condition.

• For |X| ≤ c, it means -c ≤ X ≤ c.
• For |X| < c, it means -c < X < c.
• The solution is a single interval on the number line.

Memory trick: Sandwich the Inside, then Solve!

For a square root function, the expression under the radical must be non-negative. In real-world contexts, additional restrictions (like distance being non-negative) must also be applied.

• Set the radicand ≥ 0 to find mathematical domain.
• Consider physical constraints of the problem (e.g., time, distance, quantity).
• The final domain is the intersection of mathematical and real-world restrictions.

Memory trick: Radicand non-negative, distance non-negative, combine!

The maximum population size of a biological species that can be sustained indefinitely by a given environment, represented by the constant K in the numerator of the logistic growth function.

• Logistic growth models show initially exponential growth, then slowdown as population approaches carrying capacity.
• The carrying capacity is the horizontal asymptote of the logistic curve.
• It represents the upper limit of the population.

Memory trick: The 'K' on top is where life will stop!

Exponential growth occurs when a quantity increases by a constant factor over equal time intervals, often seen in populations that double or halve.

• Modeled by P(t) = P0 * (growth factor)^(t/T), where T is the doubling period.
• Growth factor for doubling is 2.
• The number of periods is total time divided by the period length.

Memory trick: Initial, then factor to the power of how many times it happens.

To find the intersection points of a quadratic function and a linear function, set their y-expressions equal to each other, forming a quadratic equation. Solve this quadratic equation for x, which may yield zero, one, or two real solutions.

• Set y-values equal: y1 = y2.
• Rearrange into standard quadratic form: ax^2 + bx + c = 0.
• Solve for x using factoring, quadratic formula, or completing the square.
• Each x-solution corresponds to an intersection point.

Memory trick: 'Equal-ize' the paths to find where they 'meet'!

In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

• Only applies to right-angled triangles.
• Formula: a² + b² = c² (where a and b are legs, c is hypotenuse).
• Used to find an unknown side length.

Memory trick: Right triangle? Think 'A-squared plus B-squared equals C-squared' for sides.

Information that reduces the credibility or likelihood of a claim, argument, or hypothesis being true.

• Directly challenges a core assumption or conclusion.
• Can introduce alternative explanations.
• Must be relevant to the specific claim being made.

Memory trick: Hypothesis, Contradict, Undermine: Does it 'C'rack the foundation?

For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex is the point (h, k) where h = -b/(2a) and k = f(h). It represents the maximum or minimum value of the function.

• If a > 0, the parabola opens upwards, and the vertex is a minimum.
• If a < 0, the parabola opens downwards, and the vertex is a maximum.
• The x-coordinate of the vertex gives the input value at which the max/min occurs.

Memory trick: Vertex is the 'top' or 'bottom' of the 'V' or 'A' shape.

Ratios express the relative size of two or more values. They can be used to find unknown quantities when one quantity and the ratio are known.

• Ratios can be written as a:b, a/b, or 'a to b'.
• To solve for an unknown, set up a proportion.
• Ensure units and corresponding parts of the ratio are consistent.

Memory trick: Ratios Relate Amounts, Find the Missing Piece.

The ability to identify and analyze evidence within a text, and to determine what evidence best supports a given claim or hypothesis.

• Involves evaluating the relevance and strength of evidence.
• Requires understanding the relationship between claims and supporting details.
• Can involve both textual and quantitative evidence.

Memory trick: Claim, Check, Connect: Does the evidence 'C'ement the claim?

For a logistic growth function P(t) = K / (1 + Ce^(-rt)), as t approaches infinity, the term e^(-rt) approaches 0, and the function approaches the carrying capacity K.

• Represents the long-term stable population size.
• The limit is always the numerator K.
• Assumes r > 0 for growth towards K.

Memory trick: Time flies, 'e' shrinks, K capacity it thinks.

When finding the limit of a rational function as the variable approaches positive or negative infinity, if the degree of the numerator is equal to the degree of the denominator, the limit is the ratio of their leading coefficients.

• Degree of numerator = Degree of denominator.
• Limit = (leading coefficient of numerator) / (leading coefficient of denominator).
• This determines the horizontal asymptote of the function.

Memory trick: Degrees Equal? 'Ratio' the Leaders!

Substituting a specific value for the independent variable into an exponential function to find the corresponding dependent variable value.

• Involves a base raised to a power.
• Used to model growth or decay over time.
• Direct substitution is the primary method.

Memory trick: Exponent's path, value in its grasp.

To find the limit of a rational function as x approaches infinity (or negative infinity), compare the degrees of the numerator and denominator.

• If deg(num) < deg(den), limit is 0.
• If deg(num) > deg(den), limit is ±∞.
• If deg(num) = deg(den), limit is the ratio of leading coefficients.
• This limit represents the horizontal asymptote of the function.

Memory trick: Degree 'D'etermines 'D'irection at 'D'istant ends.

The distance from the vertex to the focus (or directrix) of a parabola.

• For y = (1/(4p))x² (vertex at origin, opens up/down), 'p' is the focal length.
• The focus is a key point in optical and acoustic applications (e.g., satellite dishes).
• Determines the 'width' or 'depth' of the parabolic curve.

Memory trick: A parabola's focus is where parallel rays meet, like a spotlight.

The derivative of an exponential function f(x) = ae^(kx) is f'(x) = ake^(kx), which represents the rate at which the function's output changes at a specific instant.

• Uses the chain rule for differentiation.
• For e^(kx), the derivative is k*e^(kx).
• Calculated by evaluating the derivative function at a specific point.

Memory trick: Derive, then plug in; the instant's change you'll win.

To find the remaining portion of a whole after several percentages have been allocated, subtract the sum of the allocated percentages from 100%. Then, calculate that remaining percentage of the total quantity.

• The sum of all parts of a whole is 100%.
• Convert percentages to decimals for calculations (e.g., 45% = 0.45).
• Multiply the decimal by the total quantity to find the part.

Memory trick: Portions accounted for, remaining space found.

If direct substitution into a rational function results in the indeterminate form 0/0, it often indicates a 'hole' in the graph. The limit can be found by factoring and canceling common factors.

• 0/0 is an indeterminate form, suggesting further analysis is needed.
• Factoring the numerator and denominator can reveal common factors.
• Canceling common factors simplifies the expression, allowing direct substitution for the limit.
• A hole exists at the x-value where the canceled factor was zero.

Memory trick: 0/0 means 'Go Go' simplify!

For a quadratic function f(x) = ax^2 + bx + c with a < 0, the x-coordinate of the vertex, given by -b/(2a), represents the input value at which the function reaches its maximum output.

• Occurs at the vertex of the parabola.
• If 'a' is negative, the parabola opens downward, indicating a maximum.
• The formula -b/(2a) finds the specific input (e.g., units, time) for the maximum.

Memory trick: X marks the spot for the 'top' of the profit hill!

The average cost per unit is calculated by dividing the total cost of production by the number of units produced.

• Average Cost (AC) = Total Cost (TC) / Quantity (Q).
• If TC is C(x) and Q is x, then AC = C(x) / x.
• Inequalities like 'no more than' translate to '≤'.

Memory trick: Total Cost Over Units, Less Than Limit!

The roots (or x-intercepts, or zeros) of a quadratic equation ax^2 + bx + c = 0 are the values of x for which the function equals zero. They can be found by factoring, completing the square, or using the quadratic formula.

• Quadratic formula: x = [-b ± √(b^2 - 4ac)] / (2a).
• Factoring is possible if roots are rational.
• The discriminant (b^2 - 4ac) determines the nature of the roots.

Memory trick: Set Function to Zero, Solve for X!

Combining several individual inferences drawn from different pieces of data or textual evidence into a single, more comprehensive and nuanced conclusion.

• Requires identifying all relevant implicit meanings.
• Looks for overarching patterns or explanations across multiple data points.
• Often leads to a more complex understanding than any single inference alone.

Memory trick: Data, Deduce, Double-Check: Do all 'D'etails fit?

An inequality of the form |x| > a (where a > 0) is equivalent to x > a or x < -a, representing values outside a certain range.

• Transforms into two separate 'or' inequalities.
• The solution set is two disjoint intervals.
• Often represents values outside a tolerance or unacceptable range.

Memory trick: Greater than, outside the bounds, two solutions are found.

For an exponential function in the form f(x) = a * b^x, the initial value (when x=0) is 'a'.

• The 'a' term represents the starting quantity.
• Any non-zero number raised to the power of 0 is 1.
• The initial value is found by setting the independent variable to zero.

Memory trick: Start with 'A' for 'Amount at first' in 'A times B to the X'.

The maximum population size of a biological species that can be sustained indefinitely by the environment, given the available resources.

• Represented by 'K' in logistic growth models.
• It's the upper limit a population reaches.
• Determined by environmental factors like food, water, and space.

Memory trick: K is for 'Kapacity' – the top limit!

The angle of elevation is the angle formed by the horizontal line of sight and the line of sight upwards to an object. It can be found using the tangent ratio (opposite/adjacent) and its inverse function.

• Tangent relates the opposite and adjacent sides of a right triangle to an angle.
• arctan (or tan⁻¹) is used to find the angle when the ratio is known.
• The 'opposite' side is the rise, and the 'adjacent' side is the horizontal distance for angle of elevation.

Memory trick: SOH CAH TOA: Tan for Opposite and Adjacent!

Combining information from multiple sources or different data points, especially when they present partially aligned or contradictory findings, to form a comprehensive understanding.

• Requires identifying points of agreement and disagreement.
• Involves weighing the strength of different pieces of evidence.
• Aims to form a nuanced conclusion that incorporates all relevant data.

Memory trick: Compare, Connect, Conclude: What's the 'C'omplete picture?

Drawing a conclusion that integrates multiple pieces of information, often involving nuanced interpretations of implications and potential trade-offs.

• Requires careful reading to identify subtle cues and implications.
• Often involves synthesizing both positive and negative aspects.
• Goes beyond simple cause-and-effect to understand broader impacts.

Memory trick: Details, Duality, Deduction: What's the 'D'eeper truth?

The inverse trigonometric function that finds the angle whose tangent is a given ratio.

• Denoted as tan⁻¹ or arctan.
• Used to find an unknown angle in a right triangle when opposite and adjacent sides are known.
• Result is typically in degrees or radians.

Memory trick: To find an angle, you need the 'inverse' function – like unwrapping a present.

A technique to solve exponential equations where both sides of the equation can be expressed with the same base. Once the bases are equal, the exponents can be set equal to each other and solved.

• Requires rewriting numbers as powers of a common base.
• If b^x = b^y, then x = y.
• Useful when logarithms are not directly applicable or simpler.

Memory trick: Make the bases 'match' then cut off the 'tops'!

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

• One equation usually represents the total quantity of the mixture.
• Another equation represents the total amount of a specific component (e.g., protein, concentration).
• Careful conversion of percentages or rates to decimals is crucial.

Memory trick: Total amount, total quality, two equations to find the mix.

To convert an angle from radians to degrees, multiply the radian value by the conversion factor 180°/π.

• π radians is equivalent to 180 degrees.
• Radians are often used in advanced mathematics and physics.
• Degrees are more commonly used in everyday applications and geometry.

Memory trick: Radians to degrees, multiply by 180 over pi!

The Principle of Inclusion-Exclusion is used to find the probability of the union of two events by summing individual probabilities and subtracting the probability of their intersection to avoid double-counting.

• P(A or B) = P(A) + P(B) - P(A and B).
• Used when events are not mutually exclusive.
• The sum of P(A or B) and P(neither A nor B) is 1 (or 100%).

Memory trick: Add the circles, then take out the middle overlap, then subtract from total.

The percentage of a whole represents a part of a total quantity expressed as a fraction of 100.

• All percentages of parts must sum to 100% for the whole.
• To find the quantity, multiply the percentage (as a decimal) by the total.
• Useful for breaking down populations or distributions.

Memory trick: Total is 100%, subtract what you know to find the rest.

To draw an inference from data means to conclude something not explicitly stated but strongly suggested by the evidence provided. It involves interpreting patterns and relationships within the given information.

• Look for patterns or correlations in the data.
• Avoid making assumptions not supported by the text.
• Distinguish between what is stated and what is implied.

Memory trick: Data Patterns Point to Plausible Conclusions.

A system of linear equations involves two or more linear equations with the same variables, used to model and solve real-world problems with multiple unknown quantities.

• Each unknown quantity needs a variable.
• Each piece of information (total, cost, etc.) forms an equation.
• Common solution methods include substitution or elimination.

Memory trick: Two unknowns, two equations, find the intersection.

A set of three positive integers a, b, and c, such that a² + b² = c².

• Common triples include (3, 4, 5) and its multiples (e.g., 6, 8, 10).
• Useful for quickly solving right triangle problems on exams.
• Indicates a right-angled triangle.

Memory trick: For 'as the crow flies', think of a right triangle.

The main point or message an author wants to convey about a topic, often requiring synthesis of multiple details.

• Not just a topic, but a specific argument about the topic.
• Requires identifying recurring themes and key details.
• Often stated implicitly and needs to be inferred.

Memory trick: Read, Recall, Relate: What's the 'R'eal story?

The central idea (or main idea) is the overarching message or point that the author is trying to convey. It is the core concept around which all other details revolve.

• Look for recurring themes and dominant messages.
• Distinguish between main ideas and supporting details.
• Consider the author's purpose in conveying the information.

Memory trick: Main Message Makes Most Meaning.

The area of a region bounded by two radii and the intercepted arc of a circle.

• Formula (degrees): A = (θ/360°) * πr²
• Formula (radians): A = (1/2)r²θ
• Represents a 'slice' of the circle.

Memory trick: A sector is a pizza slice – its area is a fraction of the whole pizza.

A straight line that touches a circle at exactly one point.

• The tangent line is always perpendicular to the radius at the point of tangency.
• Forms a right angle, useful for Pythagorean theorem and trigonometry.
• Important concept in geometry and calculus.

Memory trick: Tangent means 'touching just once,' and it's always 'right' with the radius.

Inferring quality of life involves drawing conclusions about the well-being and characteristics of an environment or population based on demographic, infrastructural, and social indicators, rather than direct statements.

• Connect observed characteristics to common implications for daily living.
• Consider how different elements of an environment interact.
• Avoid making subjective judgments not supported by objective indicators.

Memory trick: Details Describe Daily Delights, Distinguishing Different Dwellings.

The process of changing an angle's measurement from degrees to radians.

• Conversion factor: π radians = 180°.
• Multiply degrees by (π/180) to get radians.
• Radians are a unit of angular measurement based on the radius of a circle.

Memory trick: Degrees to Radians: 'Pi over 180' is your conversion key.

For a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex, given by -b/(2a), represents the time or input value at which the function reaches its maximum or minimum output.

• If 'a' is positive, the vertex is a minimum point.
• If 'a' is negative, the vertex is a maximum point.
• The formula -b/(2a) finds the input (e.g., time) for the extremum.

Memory trick: Time to find the 'tip-top' or 'bottom-out' of the curve!

The Pythagorean theorem (a^2 + b^2 = c^2) is used to find an unknown side of a right-angled triangle when two sides are known.

• Applies only to right-angled triangles.
• c always represents the hypotenuse (the longest side, opposite the right angle).
• a and b represent the other two sides (legs).

Memory trick: Right angle, right answer, sides squared, sum it up!

Rates describe how one quantity changes in relation to another, often requiring conversion of units to ensure consistency for calculations.

• Ensure units are consistent (e.g., all in minutes or hours).
• A rate is a ratio of two quantities with different units.
• Multiply the rate by the quantity of the independent variable to find the total.

Memory trick: Match units, find rate, then multiply for the new total time.

Binomial probability calculates the likelihood of getting exactly 'k' successes in 'n' independent Bernoulli trials, where each trial has only two possible outcomes (success/failure).

• Requires fixed number of trials (n).
• Each trial has only two outcomes (success/failure).
• Probability of success (p) is constant for each trial.
• Trials are independent.

Memory trick: How many ways to get successes, times probability of one way.

A proportional relationship exists between two quantities when their ratio is constant. This means as one quantity changes, the other changes by a constant multiple.

• Can be expressed as a/b = c/d
• Graph is a straight line through the origin
• Constant of proportionality is the ratio y/x

Memory trick: Ratios Remain Equal, Always Balanced.

Relative frequency is the proportion or fraction of times an event occurs in a dataset, calculated by dividing the frequency of the specific event by the total number of observations.

• Expressed as a decimal, fraction, or percentage.
• Sum of all relative frequencies for all categories is 1 (or 100%).
• Used to understand the distribution of data.

Memory trick: Part over Whole, always a fraction of the total.

The deeper, often abstract meaning or representation a concrete object, character, or event holds within a text.

• Symbols often recur and evolve in meaning.
• Requires interpretation beyond the literal meaning.
• Connects to the text's themes or character development.

Memory trick: Object, Observe, Outcome: What's the 'O'verarching message?