Praxis (Praxis Core and Praxis Subject Assessments) Praxis Subject Assessment: Mathematics: Content Knowledge (5165) Syllabus

What are the domain, range, and asymptote of the logarithmic parent function f(x) = log_b x (b > 1)?

Domain: x > 0; range: all real numbers; vertical asymptote x = 0; increasing with x-intercept (1, 0).

How do the transformations f(x) + k and f(x − h) affect a graph?

f(x) + k shifts the graph vertically by k units; f(x − h) shifts it horizontally by h units (right for positive h).

What distinguishes exponential growth from linear growth in a real-world model?

Linear growth adds a constant amount per unit time (constant difference); exponential growth multiplies by a constant factor per unit time (constant ratio), producing rapidly accelerating change.

In the exponential model A = P(1 + r)ᵗ, what does each variable represent?

A is the final amount, P the initial amount (principal), r the growth rate per period (decimal), and t the number of periods.

What is the explicit formula for the nth term of an arithmetic sequence with first term a₁ and common difference d?

aₙ = a₁ + (n − 1)d.

What is the formula for the sum of the first n terms of an arithmetic series?

Sₙ = n(a₁ + aₙ)/2, or equivalently Sₙ = (n/2)[2a₁ + (n − 1)d].

What is the explicit formula for the nth term of a geometric sequence with first term a₁ and common ratio r?

aₙ = a₁ · r^(n−1).

When does an infinite geometric series converge, and what is its sum?

It converges when |r| < 1, and the sum is S = a₁/(1 − r).

What is the formal idea of the limit of f(x) as x approaches a?

It is the single value L that f(x) gets arbitrarily close to as x gets arbitrarily close to a (from both sides), regardless of the value f(a) itself.

What three conditions must hold for a function f to be continuous at x = a?

f(a) is defined, the limit of f(x) as x→a exists, and that limit equals f(a).

How can L'Hôpital's Rule be used to evaluate a limit of indeterminate form 0/0 or ∞/∞?

Replace the limit of f(x)/g(x) with the limit of f′(x)/g′(x), provided that new limit exists; repeat if still indeterminate.

What is the limit definition of the derivative f′(x)?

f′(x) = lim_(h→0) [f(x + h) − f(x)] / h.

State the power rule for derivatives.

d/dx [xⁿ] = n·x^(n−1).

State the product rule and quotient rule for derivatives.

Product: (fg)′ = f′g + fg′. Quotient: (f/g)′ = (f′g − fg′)/g².

State the chain rule for the derivative of a composite function f(g(x)).

d/dx f(g(x)) = f′(g(x)) · g′(x).

How do you use the first derivative to find and classify relative extrema?

Find critical points where f′(x) = 0 or is undefined; if f′ changes from positive to negative there is a relative maximum, and from negative to positive a relative minimum (first derivative test).

What does the second derivative tell you about concavity and inflection points?

f″ > 0 means concave up, f″ < 0 means concave down; an inflection point occurs where concavity changes (f″ changes sign).

State the power rule for integration of xⁿ (n ≠ −1).

∫ xⁿ dx = x^(n+1)/(n+1) + C.

State the Fundamental Theorem of Calculus (evaluation part) for a definite integral.

If F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

How is the definite integral ∫ₐᵇ f(x) dx interpreted geometrically?

It is the net (signed) area between the curve y = f(x) and the x-axis from x = a to x = b; area above the axis is positive, below is negative.

How many edges does a complete graph Kₙ have, and what is the degree of each vertex?

Kₙ has n(n−1)/2 edges, and every vertex has degree n − 1.

What distinguishes a permutation from a combination, and what are their formulas?

Permutations count ordered arrangements: nPr = n!/(n−r)!. Combinations count unordered selections: nCr = n!/[r!(n−r)!]. Order matters for permutations, not combinations.

State the Pythagorean Theorem and its converse.

In a right triangle with legs a, b and hypotenuse c: a² + b² = c². Converse: if a² + b² = c² for a triangle's sides, then the triangle is right-angled.

What are the triangle congruence criteria?

SSS, SAS, ASA, AAS, and HL (hypotenuse-leg for right triangles). SSA and AAA are not valid congruence criteria.

What are the formulas for the distance and midpoint between points (x₁,y₁) and (x₂,y₂)?

Distance = √[(x₂−x₁)² + (y₂−y₁)²]; Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

What is the relationship between the slopes of parallel lines and of perpendicular lines?

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (their product is −1).

Which rigid transformations (isometries) preserve distance, and which transformation does not?

Translations, rotations, and reflections are isometries that preserve distance and shape; a dilation is not an isometry (it changes size but preserves shape).

State the law of sines and the law of cosines for a triangle.

Law of Sines: a/sin A = b/sin B = c/sin C. Law of Cosines: c² = a² + b² − 2ab·cos C.

What is the addition rule for the probability of A or B, and how does it simplify for mutually exclusive events?

P(A or B) = P(A) + P(B) − P(A and B). If A and B are mutually exclusive, P(A and B) = 0, so P(A or B) = P(A) + P(B).

What is the formula for conditional probability P(A | B), and when are two events independent?

P(A | B) = P(A and B)/P(B). Events are independent when P(A and B) = P(A)·P(B), equivalently P(A | B) = P(A).

How are mean, median, and mode each affected by an extreme outlier in a data set?

The mean is strongly pulled toward the outlier; the median is resistant and changes little; the mode is unaffected unless the outlier creates a new most-frequent value.

In a normal distribution, what does the 68-95-99.7 (empirical) rule state?

About 68% of data lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3 standard deviations.