Algebra II Quizzes: Master Advanced Algebraic Concepts

Question 1

Which graph represents the function f(x) = |x-2|+1?

Accepted Answer

A V-shaped graph with its vertex at (2,1)

Answer

A vertical line at x=2

Answer

A horizontal line at y=1

Answer

A parabola with its vertex at (0,1)

Question 2

Which of the following must be a factor of a polynomial function with three distinct zeros?

Accepted Answer

Product of the zeros

Answer

Sum of the zeros

Answer

Average of the zeros

Answer

Reciprocal of the zeros

Question 3

Which property does NOT apply to logarithms with a positive base?

Accepted Answer

The logarithm of a negative number is always undefined.

Answer

The logarithm of a product is the sum of the logarithms of the factors.

Answer

The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.

Answer

The logarithm of 1 is 0.

Question 4

What is the complex conjugate of the complex number (5 - 2i)?

Accepted Answer

(5 + 2i)

Answer

(5 - 2)

Answer

(2 + 5i)

Answer

(2 - 5i)

Question 5

Solve the absolute value equation: |x| - 2 = 5

Accepted Answer

x = 3 or x = -7

Answer

x = 7

Answer

x = -3

Answer

x = 0

Question 6

What is the domain of the function f(x) = √(x - 2)?

Accepted Answer

x ≥ 2

Answer

x ≤ 2

Answer

x > 2

Answer

x < 2

Question 7

Which option represents a complex number, containing both real and imaginary parts?

Accepted Answer

3 + 4i

Answer

x^2 + 2x + 1

Answer

e^x

Answer

sin(x)

Question 8

Identify the polynomial function:

Accepted Answer

f(x) = x^2 + 2x - 3

Answer

f(x) = x^-1

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 9

Find the equation of the polynomial function whose zeros are -2, 1, and 3.

Accepted Answer

f(x) = (x + 2)(x - 1)(x - 3)

Answer

f(x) = (x - 2)(x + 1)(x + 3)

Answer

f(x) = x^3 - 4x^2 + 5x - 6

Question 10

In the logarithmic function f(x) = a * log_b (x - h) + k, the horizontal asymptote is y = 3. What is the value of k?

Accepted Answer

3

Answer

h

Answer

a

Answer

b

Question 11

Analyze the polynomial function f(x) = x^4 - 2x^2 + 1. Determine how the function behaves as x becomes very large (positive infinity) and very small (negative infinity).

Accepted Answer

As x approaches infinity, f(x) approaches infinity; as x approaches negative infinity, f(x) approaches negative infinity.

Answer

As x approaches infinity, f(x) approaches negative infinity; as x approaches negative infinity, f(x) approaches infinity.

Answer

As x approaches infinity, f(x) approaches 1; as x approaches negative infinity, f(x) approaches -1.

Answer

As x approaches infinity, f(x) approaches 0; as x approaches negative infinity, f(x) approaches 0.

Question 12

Given the rational function f(x) = (x^2 - 4)/(x + 2), determine the values of x for which this function is not defined and explain why.

Accepted Answer

x = 2 because division by zero is undefined.

Answer

x = 4 because it makes the denominator negative, which is not allowed in division.

Answer

x = 0 because it makes the numerator zero, which is not allowed in division.

Answer

x = -4 because it makes the denominator zero, which is undefined.

Question 13

In the exponential function f(x) = ab^x, which factor determines the direction (upward or downward) of the graph?

Accepted Answer

Sign of the base, b

Answer

Value of the constant, a

Answer

Rate of change of the exponent, x

Answer

Domain of the function

Question 14

Given matrix A with dimensions 3x2 and matrix B with dimensions 2x5, what are the dimensions of their product AB?

Accepted Answer

3x5

Answer

5x3

Answer

2x5

Answer

3x2

Question 15

Find the equation of the sine function that models a waveform with an amplitude of 5, a period of π, and a phase shift of π/3 to the left.

Accepted Answer

f(x) = 5sin(x + π/3)

Answer

f(x) = 5cos(x - π/3)

Answer

f(x) = 10sin(2x + π/6)

Answer

f(x) = -5sin(x + 2π/3)

Question 16

Find the degree of the polynomial function: f(x) = 3x^4 - 2x^3 + 5x - 1.

Accepted Answer

4

Answer

1

Answer

3

Answer

5

Question 17

Determine the area enclosed by the curves y = x^2 and y = 4.

Accepted Answer

8

Answer

4

Answer

16

Answer

0

Question 18

Find the real roots of the quadratic equation x² - 5x + 6 = 0:

Accepted Answer

x = 2, x = 3

Answer

x = -2, x = -3

Answer

x = 1, x = 6

Answer

The equation has no real roots

Question 19

Find the inverse of the function f(x) = x² + 1:

Accepted Answer

f^-1(x) = ±√(x - 1)

Answer

f^-1(x) = -√(x + 1)

Answer

f^-1(x) = √(x - 1)

Answer

The function has no inverse

Question 20

Determine the domain of the function f(x) = 1 / (x - 3):

Accepted Answer

All real numbers except x = 3

Answer

All real numbers except x = 0

Answer

All real numbers

Answer

The function has no domain

Question 21

Find the slope and y-intercept of the equation y = 2x - 3.

Accepted Answer

Slope: 2, y-intercept: -3

Answer

Slope: -3, y-intercept: 2

Answer

Slope: 3, y-intercept: -2

Answer

Slope: -2, y-intercept: 3

Question 22

Find the slope of the line that connects the points (-1, 2) and (3, 8).

Accepted Answer

1.5

Answer

3

Answer

1

Answer

0

Question 23

What is the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...?

Accepted Answer

2

Answer

1

Answer

0

Question 24

Solve for x: 2(3x - 5) = 4(x + 2)

Accepted Answer

x = 3

Answer

x = 5

Answer

x = -3

Answer

x = -5

Question 25

Factor the expression: x² - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x - 3)(x + 3)

Answer

(x + 9)(x - 1)

Answer

(x - 9)(x + 1)

Question 26

Simplify the expression: (x^a/x^b)

Accepted Answer

x^(a-b)

Answer

x^(a+b)

Answer

x^(b-a)

Answer

x^(a/b)

Question 27

Simplify the expression: (2x + 3)²

Accepted Answer

4x² + 12x + 9

Answer

4x² + 6x + 9

Answer

2x² + 6x + 9

Answer

2x² + 12x + 3

Question 28

Find the inverse of the linear function f(x) = 2x + 1.

Accepted Answer

f^-1(x) = (x - 1)/2

Answer

f^-1(x) = 2x - 1

Answer

f^-1(x) = 1/2x + 1

Answer

f^-1(x) = x/2 + 1

Question 29

Identify the function type that is NOT a polynomial:

Accepted Answer

Exponential

Answer

Linear

Answer

Quadratic

Answer

Cubic

Question 30

Solve the inequality and represent the solution on a number line: 2x - 5 is greater than 11

Accepted Answer

x is greater than 8

Answer

x is less than 8

Answer

x is less than -8

Question 31

Solve the system of equations using elimination: x + y = 5, x - y = 1

Accepted Answer

x = 3, y = 2

Answer

x = 2, y = 3

Answer

x = 1, y = 4

Answer

x = 4, y = 1

Question 32

Find the domain and range of the function f(x) = √(x - 2) + 1.

Accepted Answer

Domain: [2, ∞), Range: [1, ∞)

Answer

Domain: (2, ∞), Range: [0, ∞)

Answer

Domain: (-∞, 2), Range: (1, ∞)

Answer

Domain: [-1, ∞), Range: [0, ∞)

Question 33

Simplify the following expression: (x^3 + 2x^2 - 5) - (x^2 - 3x + 1)

Accepted Answer

x^3 + x^2 - 6

Answer

x^3 + 5x - 4

Answer

x^3 + 3x - 4

Answer

x^3 - x^2 - 4

Question 34

Factor the following quadratic equation: x^2 - 9

Accepted Answer

(x - 3)(x + 3)

Answer

(x - 9)(x + 1)

Answer

(x + 3)^2

Answer

(x - 3)^2

Question 35

Solve the following rational equation: (x - 1)/(x + 2) = 2/3

Accepted Answer

x = 5

Answer

x = -5

Answer

x = 1/5

Answer

x = -1/5

Question 36

Simplify the following exponential expression: (x^2y)^3

Accepted Answer

x^6y^3

Answer

x^5y^2

Answer

x^5y^3

Answer

x^6y^2

Question 37

What is the equation of the line that passes through the points (1, 2) and (3, 6)?

Accepted Answer

y = 2x + 1

Answer

y = x + 1

Answer

y = 2x

Answer

y = x + 2

Question 38

Which matrix transformation leaves the shape of a figure unchanged?

Accepted Answer

Identity

Answer

Rotation

Answer

Dilation

Answer

Reflection

Question 39

What is the degree of the polynomial p(x) = 3x^4 - 2x^3 + x - 1?

Accepted Answer

4

Answer

1

Answer

3

Answer

5

Question 40

What is the least common denominator (LCD) of the rational expressions 1/(x-2) and 1/(x+3)?

Accepted Answer

(x-2)(x+3)

Answer

(x-3)(x+2)

Answer

x-2

Answer

x+3

Question 41

Simplify the expression (2x^2y)^3:

Accepted Answer

16x^6y^3

Answer

6x^5y^3

Answer

8x^6y^3

Answer

8x^5y^9

Question 42

Simplify the expression (2x - 3)^2:

Accepted Answer

4x^2 - 12x + 9

Answer

4x^2 - 9

Answer

2x^2 - 6x + 9

Answer

x^2 - 6x + 9

Question 43

What is the domain of the function f(x) = 1/x?

Accepted Answer

All real numbers except 0

Answer

All real numbers including 0

Answer

All positive real numbers

Answer

All negative real numbers

Question 44

Identify the polynomial that is in factored form:

Accepted Answer

(x + 2)(x - 3)

Answer

2x² - 5x + 3

Answer

x³  - 2x²  + x

Answer

x²  + 5x + 6

Question 45

Simplify the rational expression:

Accepted Answer

1/(x - 2)

Answer

(x + 2)/(x - 2)

Answer

(x - 2)/(x + 2)

Answer

x/(x - 2)

Question 46

Factor the expression: x^2 - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 3)(x + 3)

Answer

Not factorable

Answer

(x - 3)(x - 3)

Question 47

State the domain of the function: f(x) = 1 / (x - 2)

Accepted Answer

All real numbers except 2

Answer

x < 2

Answer

All real numbers

Answer

x > 2

Question 48

Identify the correct factorization of the quadratic expression x^2 - 9:

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 1)(x - 1)

Answer

(x + 9)(x - 9)

Answer

(x + 6)(x - 6)

Question 49

Determine the standard equation of a circle with center (h, k) and radius r:

Accepted Answer

(x - h)^2 + (y - k)^2 = r^2

Answer

(x - h)^2 - (y - k)^2 = r^2

Answer

(x + h)^2 + (y + k)^2 = r^2

Answer

(x + h)^2 + (y - k)^2 = r^2

Question 50

Which of the following sequences converges to 0?

Accepted Answer

a_n = (1/2)^n

Answer

a_n = n^2

Answer

a_n = 2^n

Answer

a_n = n

Question 51

Identify the correct polynomial function among the following options:

Accepted Answer

f(x) = 2x^3 - 5x^2 + 7x - 3

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Answer

f(x) = 2^(x+1)

Question 52

Identify the quadratic equation:

Accepted Answer

y = 2x^2 - 5x + 3

Answer

y = √x + 5

Answer

y = x^3 + 2x^2 - 1

Answer

y = 1 / x

Question 53

Which graph represents the inequality: y > x^2 - 4?

Accepted Answer

Image of a parabola opening upwards, with the region above it shaded

Answer

Image of a line with a positive slope, with the region above it shaded

Answer

Image of a parabola opening downwards, with the region below it shaded

Question 54

What is the discriminant of the quadratic equation: 2x^2 + 5x - 3 = 0?

Accepted Answer

37

Answer

-37

Answer

11

Answer

-11

Question 55

The inverse operation of squaring a number is:

Accepted Answer

Taking the square root

Answer

Adding 1

Answer

Multiplying by 2

Answer

Dividing by itself

Question 56

Which function is a rational function?

Accepted Answer

f(x) = (x^2 + 2x) / (x - 1)

Answer

f(x) = x^2 + 2x

Answer

f(x) = e^x

Answer

f(x) = √x + 1

Question 57

Identify the logarithmic equation:

Accepted Answer

log₂(x + 3) = 5

Answer

y = 2x³ - 5x² + 3x - 1

Answer

x² - 5x + 6 = 0

Answer

e^x = 10

Question 58

Solve the quadratic equation: 2x^2 + 5x - 12 = 0

Accepted Answer

x = 3 or x = -2

Answer

x = 4 or x = -6

Answer

x = 2 or x = -3

Question 59

Identify the shape of the graph of the equation: y = 2x^2 - 5x + 3

Accepted Answer

A parabola opening upwards

Answer

A straight line with a positive slope

Answer

A straight line with a negative slope

Answer

A parabola opening downwards

Question 60

Find the domain of the function: f(x) = sqrt(x - 2)

Accepted Answer

x ≥ 2

Answer

x < 2

Answer

x ≤ 2

Answer

All real numbers

Question 61

Which of the following is a factor of (x^2 - 4)?

Accepted Answer

$$(x - 2)(x + 2)$$

Answer

$$(x-1)(x + 1)$$

Answer

$$(x-3)(x + 3)$$

Answer

$$(x - 4)(x + 4)$$

Question 62

Graph the inequality: $$y < 2x - 1$$

Accepted Answer

A shaded half-plane below a dashed line with slope 2 and y-intercept -1.

Answer

A shaded half-plane below a solid line with slope 2 and y-intercept -1.

Answer

A shaded half-plane above a dashed line with slope 2 and y-intercept -1.

Question 63

Find the inverse of the function: $$f(x) = 2x + 3$$

Accepted Answer

$$f^{-1}(x) = rac{x - 3}{2}$$

Answer

$$f^{-1}(x) = 3x - 2$$

Answer

$$f^{-1}(x) = x - 3/2$$

Answer

$$f^{-1}(x) = rac{3 - x}{2}$$

Question 64

Solve for x in the equation 3x + 5 = 14.

Accepted Answer

3

Answer

-5

Answer

-3

Answer

5