Free AEPA Mathematics Exam Flashcards

Estimate with nearby friendly numbers, compare the size of the result to the context, and check units. A correct procedure can still produce an impossible answer if the scale or unit is wrong.

The second percent is applied to a new base. Multiplying by 1.20 and then by 0.80 gives 0.96, so the final value is 96% of the original.

Convert each ratio to an equivalent rate, fraction, percent, or unit rate before comparing. Raw counts can mislead when the group sizes are different.

It exposes the building blocks of an integer, making GCF, LCM, divisibility, perfect powers, and many proof arguments easier to see.

For positive integers a and b, GCF(a,b) * LCM(a,b) = a * b. Use this only after finding the GCF correctly.

It removes the imaginary part: (a + bi)(a - bi) = a^2 + b^2. This is useful for division because it makes the denominator real.

The contrapositive is equivalent: if p then q becomes if not q then not p. The converse, if q then p, is not guaranteed by the original statement.

Only when p is true and q is false. If the hypothesis is false, the implication is considered true in formal logic.

A single valid counterexample disproves a universal claim. Many confirming examples can suggest a pattern, but they do not prove it for all cases.

Terms such as average rate, slope, factor, solution, and root carry specific meanings. Clear language shows whether the reasoning is valid, not just whether the arithmetic was lucky.

A function assigns each input exactly one output. On a graph, the vertical line test detects whether any x-value has more than one y-value.

The domain excludes inputs that break the rule, such as zero denominators, even roots of negative numbers in real-valued functions, or nonpositive logarithm arguments.

Write y = f(x), interchange x and y, solve for y, and state any domain or range restrictions. The original function must be one-to-one for the inverse to be a function.

Use g(x) as the input to f. The domain must include only x-values where g is defined and g(x) lies in the domain of f.

In f(x) = a(x - h)^2 + k, the vertex is (h,k), the axis of symmetry is x = h, and the sign of a tells whether the parabola opens up or down.

A zero is an input where f(x) = 0. It may appear as an x-intercept, a factor of the polynomial, or a solution of the related equation.

Critical numbers split the number line into intervals. Testing each interval shows where the expression is positive, negative, zero, or undefined.

Vertical asymptotes come from denominator zeros that do not cancel. Horizontal asymptotes come from end behavior, usually by comparing the degrees of numerator and denominator.

A logarithm answers the exponent question. log_b(a) = c means b^c = a, with b > 0, b not equal to 1, and a > 0.

Outside changes move or stretch vertically; inside changes move or stretch horizontally. For example, f(x - h) shifts right h, while f(x) + k shifts up k.

Squaring and multiplying by expressions can introduce values that do not satisfy the original equation. Always substitute proposed solutions back into the original form.

It represents the overlap of all half-plane solution regions. Boundary type matters: solid lines include equality; dashed lines do not.

Congruent figures have the same shape and size. Similar figures have the same shape but possibly different size, so side lengths scale by a common factor while angle measures stay equal.

If lengths scale by k, areas scale by k^2 and volumes scale by k^3. This is a common trap when moving from linear dimensions to area or volume.

It measures straight-line distance between two points using the Pythagorean theorem: horizontal change and vertical change are the legs of a right triangle.

Nonvertical parallel lines have equal slopes. Nonvertical perpendicular lines have slopes whose product is -1, meaning they are negative reciprocals.

A result cannot be more meaningful than the measurements used to create it. Rounding, significant digits, and percent error should reflect instrument precision.

After completing the square, a circle has equal positive coefficients on x^2 and y^2 and can be written (x - h)^2 + (y - k)^2 = r^2.

Reflections, rotations, and translations preserve distances and angle measures. They can change position or orientation but not size or shape.

An axiom is accepted without proof inside a system and used to prove theorems. Changing axioms, such as the parallel postulate, changes the geometry that follows.

A cross section is the shape made by slicing a solid with a plane. The same solid can produce different cross sections depending on the angle and location of the slice.

For an angle in standard position on the unit circle, cosine is the x-coordinate and sine is the y-coordinate of the point on the circle.

The period is the horizontal length of one complete cycle. For y = a sin(bx) or y = a cos(bx), the period is 2pi/|b|.

It comes from the unit circle and the Pythagorean theorem: x^2 + y^2 = 1 with x = cos x and y = sin x.

A limit describes the value a function approaches as the input approaches a point. The function does not have to equal that value at the point, or even be defined there.

The function is defined at the point, the limit exists there, and the function value equals the limit. A break in any one condition means discontinuity.

The derivative gives instantaneous rate of change. Geometrically, it is the slope of the tangent line to the graph at a point.

If f'(x) is positive, f is increasing; if f'(x) is negative, f is decreasing. Changes in sign around critical points help identify local maxima and minima.

A definite integral gives signed accumulation over an interval. On a graph, it is net area: area above the x-axis counts positive and below counts negative.

It connects derivatives and integrals: accumulation functions have derivatives, and definite integrals can be evaluated with antiderivatives.

The median is less affected by extreme values because it depends on order, not magnitude. Use it when a distribution is skewed or has outliers.

It measures typical spread around the mean. Larger standard deviation means data values are more dispersed, but it does not by itself identify the shape of the distribution.

A biased sample systematically overrepresents or underrepresents part of the population. Increasing sample size does not fix a flawed sampling method.

P(not A) = 1 - P(A). It is useful when finding the probability of at least one event is easier by subtracting the probability of none.

Use multiplication for an AND situation. If events are independent, multiply P(A)P(B); if dependent, multiply P(A) by P(B given A).

Ask whether order matters. If different orders count as different outcomes, use a permutation; if only the selected group matters, use a combination.

It prevents double-counting overlap. For two sets, n(A union B) = n(A) + n(B) - n(A intersect B).

A recursive definition gives starting value(s) and a rule for finding later terms from earlier terms. Without the starting value, the rule is incomplete.

The matrices must have the same dimensions. Addition happens entry by entry, unlike matrix multiplication, which has a different dimension rule.

A vector represents magnitude and direction. It can be described by components, added componentwise, and used to model displacement, force, or movement.