Comprehensive Algebra & Trigonometry Review

A set of 60 question-and-answer flashcards covering key algebra, geometry, trigonometry, probability, and statistics concepts from the lecture notes.

What is the algebraic form of direct proportion between variables x and y?

y = kx, where k is the constant of proportionality.

How is inverse proportion written algebraically for variables x and y?

y = k/x, where k is the constant of proportionality.

How many possible solution counts exist for a 2×2 system of linear equations?

One solution, no solution, or infinitely many solutions.

In a linear inequality graph, when do you draw a dashed boundary line?

When the inequality uses < or > (strict inequality).

Which point is commonly tested to determine the solution region of a 2-variable inequality?

(0, 0), unless the line itself passes through that point.

What is the value of i² in complex numbers?

i² = −1

How do you rationalize a complex denominator a+bi?

Multiply numerator and denominator by the conjugate (a−bi).

On the complex plane, which axis represents imaginary numbers?

The y-axis (vertical axis).

What shape is the graph of y = ax² + bx + c?

A parabola.

When does a parabola open upward?

When the leading coefficient a > 0.

Give the formula for the axis of symmetry of a quadratic y = ax² + bx + c.

x = −b/(2a).

State the quadratic formula.

x = [-b ± √(b² − 4ac)] / (2a).

What does a discriminant b² − 4ac < 0 indicate?

No real roots (the roots are complex).

Sum of the roots of ax² + bx + c = 0 equals .

−b/a

Product of the roots of ax² + bx + c = 0 equals .

c/a

Write the vertex form of a quadratic.

y = a(x − h)² + k, with vertex (h, k).

What is the basic V-shaped graph in algebra?

The graph of an absolute-value function.

Rewrite |x| < 12 as a compound inequality.

−12 < x < 12

State the negative-exponent rule for x⁻ᵃ.

x⁻ᵃ = 1 / xᵃ

What is the formula for the nth term of an arithmetic sequence?

aₙ = a₁ + d(n − 1)

How do you calculate Sn, the sum of n terms of an arithmetic sequence?

Sₙ = n/2 (a₁ + aₙ)

Write the nth-term formula for a geometric sequence.

aₙ = a₁ r^(n−1)

Give the sum of the first n terms of a geometric sequence (r ≠ 1).

Sₙ = a₁(1 − rⁿ)/(1 − r)

What is the slope-intercept form of a line?

y = mx + c (or b), where m is slope, c is y-intercept.

Formula for slope between two points (x₁,y₁) and (x₂,y₂).

m = (y₂ − y₁)/(x₂ − x₁)

What is the slope of a horizontal line?

0

What type of slopes do perpendicular lines have?

Negative reciprocals of each other (m₁ · m₂ = −1).

State the distance formula between two points in the plane.

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

What is the triangle inequality?

The sum of any two sides of a triangle is greater than or equal to the third side (a + b ≥ c, etc.).

List the side ratios of a 30-60-90 right triangle.

1 : √3 : 2

Standard equation of a circle with center (h,k) and radius r.

(x − h)² + (y − k)² = r²

What angle is formed by a tangent line and the radius drawn to the point of tangency?

A right angle (90°).

Sum of interior angles of an n-gon?

(n − 2)180°

What fundamental trig identity relates sine and cosine squares?

sin²θ + cos²θ = 1

Complementary angle identity for sine?

sin(90° − θ) = cos θ

When are two triangles similar? Name one criterion.

When corresponding angles are equal (AA similarity) or corresponding sides are proportional (SSS or SAS in ratio).

How many permutations of n distinct items exist?

n! (n factorial)

Formula for permutations of n items taken r at a time (order matters).

n! / (n − r)!

Combination formula for choosing r items from n (order not important).

n! / [r!(n − r)!]

Define simple probability P(E).

P(E) = (number of desired outcomes)/(total number of possible outcomes).

What statistic measures the ‘middle’ value of an ordered data set?

The median.

What does a positive association between two variables mean?

As one variable increases, the other tends to increase.

In a rational function, what causes a vertical asymptote?

A factor remaining in the denominator after simplification (where the denominator = 0).

How do you find the y-intercept of a rational function?

Set x = 0 and evaluate y.

When deg(p) < deg(q) in p(x)/q(x), what is the horizontal asymptote?

y = 0

Describe the end behavior of an even-degree polynomial with positive leading coefficient.

Rises to +∞ on both ends (up-up).

General exponential growth/decay model.

y = ab^x

Which condition on b causes exponential decay?

0 < b < 1

Give the change-of-base formula for logarithms.

logb x = (loga x) / (log_a b)

State the product rule of logarithms.

logb (xy) = logb x + log_b y

What transformation moves a graph upward by 3 units?

Add 3 to the function: f(x) → f(x) + 3.

When is matrix multiplication AB defined (in terms of dimensions)?

When the number of columns in A equals the number of rows in B.

Determinant of 2×2 matrix [a b; c d]?

|A| = ad − bc

Factor a³ − b³.

(a − b)(a² + ab + b²)

For x = ay² + by + c, when does the horizontal parabola open right?

When a > 0.

What is the length of the major axis of an ellipse with semi-major axis a?

2a

Standard equation of a vertical ellipse centered at (h,k).

(x − h)² / a² + (y − k)² / b² = 1 (where a² > b² and major axis is vertical).

Reference angle in Quadrant II for an angle θ?

180° − θ

Give the general form of a sine function with transformations.

Y = A sin[B(x + C)] + D

How is the period of y = A sin(Bx) calculated?

Period = 2π / |B|

State the Law of Sines.

sin A / a = sin B / b = sin C / c

Write one form of the Law of Cosines.

a² = b² + c² − 2bc cos A

What does the vector 〈6,2〉 equal in i-j notation?

6i + 2j

Define polar coordinates (r, θ).

r is the distance from origin to the point; θ is the angle from the positive x-axis to the radius vector.