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Questions and Answers
What is the correct factorization of the expression $x^2 - 4$?
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?
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?
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?
What is the role of the x-coordinate in an ordered pair?
Who is credited with the discovery of the Cartesian coordinate system?
Who is credited with the discovery of the Cartesian coordinate system?
What is the result of adding a positive number and a negative number with different signs?
What is the result of adding a positive number and a negative number with different signs?
Which of the following expressions accurately represents the product of two binomials?
Which of the following expressions accurately represents the product of two binomials?
Which multiplication statement between integers is correct?
Which multiplication statement between integers is correct?
What is the final result of the operation: 5 + (−5)?
What is the final result of the operation: 5 + (−5)?
Which of the following is an example illustrating the concept of inverse functions?
Which of the following is an example illustrating the concept of inverse functions?
How do you evaluate the expression f(x) when f(x) is defined as f(x) = 2x + 3?
How do you evaluate the expression f(x) when f(x) is defined as f(x) = 2x + 3?
What is the correct rule for subtracting two integers with different signs?
What is the correct rule for subtracting two integers with different signs?
In functional notation, what does it mean for g(f(x)) to represent?
In functional notation, what does it mean for g(f(x)) to represent?
In which quadrant do both the abscissa and the ordinate have positive values?
In which quadrant do both the abscissa and the ordinate have positive values?
What is the domain of a relation?
What is the domain of a relation?
Which type of relation is characterized by every element of X being paired with a unique element of Y?
Which type of relation is characterized by every element of X being paired with a unique element of Y?
What defines a dependent variable in a relation?
What defines a dependent variable in a relation?
Who formally introduced the term 'function' in mathematics?
Who formally introduced the term 'function' in mathematics?
Which of the following best describes a 'Many to One' relation?
Which of the following best describes a 'Many to One' relation?
What is the range of a relation?
What is the range of a relation?
If an element of X is paired with more than one element of Y, what type of relation is this?
If an element of X is paired with more than one element of Y, what type of relation is this?
What distinguishes a function from a mere relation?
What distinguishes a function from a mere relation?
What does the vertical line test determine?
What does the vertical line test determine?
Which of the following describes a constant function?
Which of the following describes a constant function?
Which function type is represented by the equation f(x) = mx + b?
Which function type is represented by the equation f(x) = mx + b?
How is function notation typically represented?
How is function notation typically represented?
What is true about polynomial functions?
What is true about polynomial functions?
What does the Absolute Value Function produce?
What does the Absolute Value Function produce?
A quadratic function is characterized by which highest exponent?
A quadratic function is characterized by which highest exponent?
What is the result of simplifying the expression $-2x + 5x$?
What is the result of simplifying the expression $-2x + 5x$?
How do you factor the trinomial $x^2 + 5x + 6$?
How do you factor the trinomial $x^2 + 5x + 6$?
What is the square of the binomial $(x + 4)$?
What is the square of the binomial $(x + 4)$?
When factoring the expression $(x + 3)(x - 3)$, what type of product does it represent?
When factoring the expression $(x + 3)(x - 3)$, what type of product does it represent?
What are the factors of the term $x^2 - 6x + 9$?
What are the factors of the term $x^2 - 6x + 9$?
Which equation correctly represents the square of the binomial $(a - b)$?
Which equation correctly represents the square of the binomial $(a - b)$?
What is the product of the sum and difference of two terms, represented by the expression $(x + 5)(x - 5)$?
What is the product of the sum and difference of two terms, represented by the expression $(x + 5)(x - 5)$?
When factoring a quadratic trinomial $1x^2 + 7x + 10$, which factors are correct?
When factoring a quadratic trinomial $1x^2 + 7x + 10$, which factors are correct?
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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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