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What is the abscissa of the point (3, 4)?
What is the abscissa of the point (3, 4)?
If a point has coordinates (0, -2), where is it located?
If a point has coordinates (0, -2), where is it located?
Which ordered pair corresponds to a point in Quadrant I?
Which ordered pair corresponds to a point in Quadrant I?
The origin corresponds to which ordered pair?
The origin corresponds to which ordered pair?
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Points with coordinates of the form (a, 0) are located on which axis?
Points with coordinates of the form (a, 0) are located on which axis?
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Where are points with coordinates (−3, −5) found?
Where are points with coordinates (−3, −5) found?
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If a point lies directly above the origin, what can be said about its coordinates?
If a point lies directly above the origin, what can be said about its coordinates?
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Which of the following coordinates represents a point in Quadrant II?
Which of the following coordinates represents a point in Quadrant II?
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What condition must be satisfied for points A, B, and C to be collinear using the distance formula?
What condition must be satisfied for points A, B, and C to be collinear using the distance formula?
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Which of the following describes how to show that a triangle is isosceles?
Which of the following describes how to show that a triangle is isosceles?
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In proving that a triangle is a right triangle, which theorem must the lengths of the sides satisfy?
In proving that a triangle is a right triangle, which theorem must the lengths of the sides satisfy?
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What is one way to show that a given quadrilateral is a parallelogram?
What is one way to show that a given quadrilateral is a parallelogram?
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Which test shows that a rectangle is actually a square?
Which test shows that a rectangle is actually a square?
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How can you find the x-intercept of the equation y = 3x - 4?
How can you find the x-intercept of the equation y = 3x - 4?
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What must be established to show that two sides of a triangle are perpendicular?
What must be established to show that two sides of a triangle are perpendicular?
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In showing that diagonals bisect each other in a parallelogram, what is the necessary condition?
In showing that diagonals bisect each other in a parallelogram, what is the necessary condition?
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What is the absolute value of a negative number?
What is the absolute value of a negative number?
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Which of the following properties of absolute value indicates that |a| is always non-negative?
Which of the following properties of absolute value indicates that |a| is always non-negative?
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If |x| < 3, which of the following inequalities must hold true?
If |x| < 3, which of the following inequalities must hold true?
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Which of the following describes the relationship represented by |x| ≥ 2?
Which of the following describes the relationship represented by |x| ≥ 2?
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What does the property |a + b| ≤ |a| + |b| represent?
What does the property |a + b| ≤ |a| + |b| represent?
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In a two-dimensional coordinate system, what does the point (0, 0) represent?
In a two-dimensional coordinate system, what does the point (0, 0) represent?
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What does the inequality |x| > 5 imply?
What does the inequality |x| > 5 imply?
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If |a| = 3, which of the following values could a be?
If |a| = 3, which of the following values could a be?
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Which of the following represents the general form of the equation of a line?
Which of the following represents the general form of the equation of a line?
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What is the slope-intercept form of a line with slope $m$ and y-intercept $b$?
What is the slope-intercept form of a line with slope $m$ and y-intercept $b$?
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What is the domain of a function?
What is the domain of a function?
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In the equation y = f(x), which symbol represents the dependent variable?
In the equation y = f(x), which symbol represents the dependent variable?
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Which equation represents the point-slope form of a line passing through the point $(x1, y1)$ with slope $m$?
Which equation represents the point-slope form of a line passing through the point $(x1, y1)$ with slope $m$?
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How is the slope of two nonvertical lines defined if they are parallel?
How is the slope of two nonvertical lines defined if they are parallel?
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When evaluating the function f at a given number, what do you do with that number?
When evaluating the function f at a given number, what do you do with that number?
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Which condition must be satisfied when finding the domain of a function involving a fraction?
Which condition must be satisfied when finding the domain of a function involving a fraction?
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What is the intercept form of the equation of a line with x-intercept $a$ and y-intercept $b$?
What is the intercept form of the equation of a line with x-intercept $a$ and y-intercept $b$?
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What is the range of a function?
What is the range of a function?
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If two lines are perpendicular, how is the slope of one line related to the other?
If two lines are perpendicular, how is the slope of one line related to the other?
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Given the points (3, -2) and (1, 6), which equation represents the two-point form of a line?
Given the points (3, -2) and (1, 6), which equation represents the two-point form of a line?
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What must be true for the expression $ ext{a}^{1/n}$ when n is even?
What must be true for the expression $ ext{a}^{1/n}$ when n is even?
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What is the correct expression for the slope between the points (x1, y1) and (x2, y2)?
What is the correct expression for the slope between the points (x1, y1) and (x2, y2)?
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In the graph of a function, which of the following is true about the ordered pairs (x, y)?
In the graph of a function, which of the following is true about the ordered pairs (x, y)?
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If f(x) = 2x^2 + x - 1, what is f(2)?
If f(x) = 2x^2 + x - 1, what is f(2)?
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Study Notes
Absolute Value
- Absolute value, denoted as |a|, represents the distance from a to 0 on the real number line; |a| is always non-negative (|a| ≥ 0).
- Definition:
- |a| = a if a ≥ 0
- |a| = -a if a < 0
- Properties of Absolute Value:
- Non-negativity: |a| ≥ 0
- Symmetry: |a| = |-a|
- Multiplication: |ab| = |a||b|
- Triangle inequality: |a + b| ≤ |a| + |b|
- Difference relation: |a| - |b| ≤ |a - b|
Inequalities involving Absolute Value
- Properties:
- |x| < a ⟷ -a < x < a
- |x| ≤ a ⟷ -a ≤ x ≤ a
- |x| > a ⟷ x > a or x < -a
- |x| ≥ a ⟷ x ≥ a or x ≤ -a
Two-Dimensional Coordinate System
- Comprised of two perpendicular axes: x-axis (horizontal) and y-axis (vertical); intersection at (0, 0) is the origin.
- Points represented by ordered pairs (x, y), establishing one-to-one correspondence between points and elements in R × R.
- Quadrants:
- Quadrant I: (+, +)
- Quadrant II: (−, +)
- Quadrant III: (−, −)
- Quadrant IV: (+, −)
Distance, Midpoint, and Slope Formulas
- Distance formula: Determines distance between points P1(x1, y1) and P2(x2, y2).
- Slope formula: Compares slopes for collinearity and demonstrates relationships between points.
- Collinearity conditions:
- Points A, B, C are collinear if AB + BC = AC.
- Slopes must be equal: mAB = mBC = mAC.
Forms of Equations of Lines
- General Form: Ax + By + C = 0 where A, B, and C are real numbers.
- Slope-Intercept Form: y = mx + b where m is slope, b is y-intercept.
- Point-Slope Form: y - y1 = m(x - x1) for a line through (x1, y1) with slope m.
- Two-Point Form: y - y1 = (y2 - y1)/(x2 - x1)(x - x1).
- Intercept Form: x/a + y/b = 1 where a and b are x- and y-intercepts respectively.
Relations and Functions
- Function notation: y = f(x), where f is the function; domain is the set of input values (independent variable) and range is the set of outputs (dependent variable).
Domain and Range
- Domain: Set of first components in ordered pairs of a function.
- Range: Set of second components in ordered pairs of a function.
- Restrictions include:
- Denominator cannot be zero in a fraction.
- Even roots require the radicand to be non-negative.
Evaluating Functions
- Substitute the input value into the function's formula to find f(x).
Graph of a Function
- The graph consists of all points (x, y) in the plane where (x, y) belongs to the function f.
Types of Functions
- Polynomial functions are among several types to be explored further; focus on their defining characteristics and behaviors.
Studying That Suits You
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Description
Explore the fundamental concepts of absolute value in mathematics. This quiz will cover the definition and properties of absolute value, including its representation on the real number line. Perfect for students wanting to solidify their understanding of this key topic.
