Advanced Algebra Grading Reviewer

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Questions and Answers

What is the correct factorization of the expression $x^2 - 4$?

(x - 4)(x + 4)

(x + 2)(x - 2) (correct)

(x + 2)(x + 2)

(x - 2)(x - 2)

Which of the following statements is true for coplanar lines?

They lie on the same plane. (correct)

They lie in different planes.

They can cross at any angle.

They always intersect at a right angle.

In the Cartesian coordinate system, what are the coordinates of the origin?

(1, 0)

(1, 1)

(0, 0) (correct)

(0, 1)

What is the role of the x-coordinate in an ordered pair?

It corresponds to a real number on the x-axis. Signup and view all the answers

Who is credited with the discovery of the Cartesian coordinate system?

René Descartes Signup and view all the answers

What is the result of adding a positive number and a negative number with different signs?

You subtract the numbers and take the sign of the greater number. Signup and view all the answers

Which of the following expressions accurately represents the product of two binomials?

(𝑎x + b)(𝑐x + d) = acx² + (ad + bc)x + bd Signup and view all the answers

Which multiplication statement between integers is correct?

(−12)(−3) = 36 Signup and view all the answers

What is the final result of the operation: 5 + (−5)?

0 Signup and view all the answers

Which of the following is an example illustrating the concept of inverse functions?

If f(x) = 3x then f^{-1}(y) = y/3 Signup and view all the answers

How do you evaluate the expression f(x) when f(x) is defined as f(x) = 2x + 3?

Substitute x into the equation and simplify. Signup and view all the answers

What is the correct rule for subtracting two integers with different signs?

Add the absolute values and keep the sign of the larger absolute value. Signup and view all the answers

In functional notation, what does it mean for g(f(x)) to represent?

The composition of function g with function f. Signup and view all the answers

In which quadrant do both the abscissa and the ordinate have positive values?

Quadrant 1 Signup and view all the answers

What is the domain of a relation?

Set of all input (x) values Signup and view all the answers

Which type of relation is characterized by every element of X being paired with a unique element of Y?

One to One Signup and view all the answers

What defines a dependent variable in a relation?

Its value is influenced by changes in the independent variable. Signup and view all the answers

Who formally introduced the term 'function' in mathematics?

Gottfried Wilhelm Leibniz Signup and view all the answers

Which of the following best describes a 'Many to One' relation?

An element of Y is paired with multiple elements of X. Signup and view all the answers

What is the range of a relation?

Set of all output (y) values Signup and view all the answers

If an element of X is paired with more than one element of Y, what type of relation is this?

One to Many Signup and view all the answers

What distinguishes a function from a mere relation?

Each x-value has exactly one corresponding y-value. Signup and view all the answers

What does the vertical line test determine?

Whether a graph represents a function. Signup and view all the answers

Which of the following describes a constant function?

A function that has a constant output regardless of input. Signup and view all the answers

Which function type is represented by the equation f(x) = mx + b?

Linear Function Signup and view all the answers

How is function notation typically represented?

By writing f(x) = x^2 + 1 Signup and view all the answers

What is true about polynomial functions?

They can only have variables raised to integer exponents. Signup and view all the answers

What does the Absolute Value Function produce?

Non-negative outputs regardless of input. Signup and view all the answers

A quadratic function is characterized by which highest exponent?

2 Signup and view all the answers

What is the result of simplifying the expression $-2x + 5x$?

$3x$ Signup and view all the answers

How do you factor the trinomial $x^2 + 5x + 6$?

$(x + 2)(x + 3)$ Signup and view all the answers

What is the square of the binomial $(x + 4)$?

$x^2 + 8x + 16$ Signup and view all the answers

When factoring the expression $(x + 3)(x - 3)$, what type of product does it represent?

Difference of Squares Signup and view all the answers

What are the factors of the term $x^2 - 6x + 9$?

$(x - 3)(x - 3)$ Signup and view all the answers

Which equation correctly represents the square of the binomial $(a - b)$?

$a^2 - 2ab + b^2$ Signup and view all the answers

What is the product of the sum and difference of two terms, represented by the expression $(x + 5)(x - 5)$?

$x^2 - 25$ Signup and view all the answers

When factoring a quadratic trinomial $1x^2 + 7x + 10$, which factors are correct?

$(x + 5)(x + 2)$ Signup and view all the answers

Study Notes

Linear Equations and Functions

Cartesian Coordinate System consists of two perpendicular number lines: x-axis (horizontal) and y-axis (vertical).
Relations: Associations between two quantities forming ordered pairs (inputs vs. outputs).
Functions: Special type of relation with a unique output for each input; can be tested using the vertical line test.
Function notation expresses functions concisely, written as f(x) where f represents the function name, and x is the input.

Special Products

Product of Two Binomials: (ax + b)(cx + d) = acx² + (ad + bc)x + bd; involves multiplying first, outer, inner, and last terms.
Square of a Binomial: (a ± b)² = a² ± 2ab + b²; square first term, then double the product of the two terms, followed by squaring the last term.
Difference of Two Squares: (a + b)(a - b) = a² - b²; involves squaring the first term and subtracting the square of the last term.

Factoring

Factoring Quadratic Trinomials: acx² + (ad + bc)x + bd = (ax + b)(cx + d); involves finding factors of the first and last terms that add up to the middle term.
Perfect Square Trinomial: a² ± 2ab + b² = (a ± b)²; determined by taking square roots of the first and last terms.
Difference of Two Squares: a² - b² = (a + b)(a - b); involves square roots of both terms.

The Cartesian Coordinate System

Origin: Intersection of x and y axes at (0,0).
Coordinates: Written as ordered pairs (x, y).
Quadrants: Four regions of the plane:
- Quadrant 1: (+, +)
- Quadrant 2: (-, +)
- Quadrant 3: (-, -)
- Quadrant 4: (+, -)

Relations

Independent Variable: Can change freely; acts as the cause.
Dependent Variable: Value influenced by the independent variable; represents the effect.
Domain: Set of all x-values in a relation.
Range: Set of all y-values in a relation.
Types of Relations:
- One-to-One: Unique pairing for every element.
- One-to-Many: One element pairs with multiple.
- Many-to-One: Multiple elements pair with one.

Functions

Functions: Introduced by Leibniz, are relations where every input has one output; all functions are relations, but not vice versa.
Examples of Functions: { (1,1), (2,8), (3,27) }
Vertical Line Test: Determines if a relation is a function by checking intersection points.
Special Types of Functions:
- Polynomial Function: Non-negative integer exponents (e.g., quadratic).
- Constant Function: Same output for various inputs; horizontal graph.
- Linear Function: Straight line, expressed as f(x) = mx + b.
- Absolute Value Function: Outputs non-negative values, even if input is negative, expressed as f(x) = |x|.
- Quadratic Function: Highest exponent of 2 indicates a polynomial with a parabolic graph.

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Description

Prepare for your advanced algebra assessments with this comprehensive review covering linear equations, the Cartesian coordinate system, and various types of functions. This quiz will test your understanding of relations, functions, and functional notation. Hone your skills and boost your confidence before the exam!