Тест: Mastering Algebra and Trigonometry Fundamentals — 8 въпроса

1. A quadratic equation has discriminant equal to 0. What does this tell you about the graph of the quadratic?

A. It has no axis of symmetry

B. It never intersects the x-axis

C. It crosses the x-axis at two distinct points

D. It touches the x-axis at exactly one point

Answer: (D)

Обяснение

A discriminant of 0 means the quadratic has one repeated real root, so the parabola touches the x-axis at exactly one point. Two distinct x-intercepts would require a positive discriminant.

2. What is the effect of a reciprocal function having a denominator of 0 at a particular x-value?

A. The function is undefined there and may have asymptotic behavior near that x-value

B. The function crosses the x-axis at that x-value

C. The function becomes a constant value for all nearby x-values

D. The function has a removable discontinuity that can be filled in with one point

Answer: (A)

Обяснение

When the denominator is 0, the reciprocal function is undefined at that x-value, which leads to vertical asymptotic behavior nearby. The other choices describe different types of graph behavior that do not follow from a zero denominator.

3. How many radians complete one full turn on the unit circle?

A. π radians

B. 3π radians

C. 2π radians

D. π/2 radians

Answer: (C)

Обяснение

A full turn corresponds to 2π radians. π radians is only half a turn, while π/2 radians is a quarter turn.

4. Which feature best describes a binomial distribution?

A. It measures the probability of two events happening at the same time when they overlap

B. It counts the number of successes in a fixed number of independent trials with the same success probability

C. It describes how outputs change as inputs vary over a continuous interval

D. It gives the number of ways to arrange objects when order matters

Answer: (B)

Обяснение

A binomial distribution models a fixed number of independent trials where each trial has the same probability of success. The other options describe overlapping events, functions, or permutations rather than a binomial setting.

5. How does the discriminant differ from completing the square when working with a quadratic expression?

A. The discriminant rewrites the quadratic to reveal its vertex form, while completing the square predicts the number of real roots

B. The discriminant predicts the number of real roots, while completing the square rewrites the quadratic to reveal its vertex form

C. The discriminant identifies perfect-square factors, while completing the square determines whether the quadratic has rational coefficients

D. The discriminant simplifies surds, while completing the square finds the domain of the expression

Answer: (B)

Обяснение

The discriminant is used to determine how many real solutions a quadratic has, based on the value of b² − 4ac. Completing the square instead rewrites the quadratic in squared form so key graph features like the vertex can be identified.

6. What is the main purpose of a piecewise function?

A. To define different rules on different parts of the domain

B. To assign one output to every input using a single formula

C. To describe probabilities for repeated independent trials

D. To show how many real roots a quadratic has

Answer: (A)

Обяснение

A piecewise function uses separate formulas on different intervals of the domain. The other options describe a general function, a quadratic feature, or a probability model.

7. On the unit circle, which trigonometric ratio corresponds to tangent?

A. The y-coordinate divided by the x-coordinate

B. The arc length divided by the radius

C. The opposite side divided by the hypotenuse

D. The x-coordinate divided by the y-coordinate

Answer: (A)

Обяснение

Tangent is given by y/x on the unit circle, matching the ratio of the y-coordinate to the x-coordinate. The arc length divided by the radius describes radians, not tangent.

8. In probability and counting, what does a combination represent?

A. A probability based on two dependent events

B. A selection of items where order changes the count

C. A sequence of repeated trials with fixed success probability

D. A selection of items where order does not matter

Answer: (D)

Обяснение

A combination counts how many groups can be formed without regard to order. The other choices describe permutations, conditional probability, or a binomial setting instead.