Absolute Value Concepts

Questions and Answers

What is the abscissa of the point (3, 4)?

-3

3 (correct)

If a point has coordinates (0, -2), where is it located?

On the x-axis

In Quadrant II

On the y-axis (correct)

In Quadrant III

Which ordered pair corresponds to a point in Quadrant I?

(−1, −4)

(2, 5) (correct)

(0, −3)

(−3, 1)

The origin corresponds to which ordered pair?

(0, 0) Signup and view all the answers

Points with coordinates of the form (a, 0) are located on which axis?

Horizontal axis Signup and view all the answers

Where are points with coordinates (−3, −5) found?

In Quadrant III Signup and view all the answers

If a point lies directly above the origin, what can be said about its coordinates?

The y-coordinate is positive. Signup and view all the answers

Which of the following coordinates represents a point in Quadrant II?

(-1, 2) Signup and view all the answers

What condition must be satisfied for points A, B, and C to be collinear using the distance formula?

AB + BC = AC Signup and view all the answers

Which of the following describes how to show that a triangle is isosceles?

Show that two sides are equal using the distance formula. Signup and view all the answers

In proving that a triangle is a right triangle, which theorem must the lengths of the sides satisfy?

Pythagorean Theorem Signup and view all the answers

What is one way to show that a given quadrilateral is a parallelogram?

Show that the opposite sides are parallel using the slope formula. Signup and view all the answers

Which test shows that a rectangle is actually a square?

Show that all sides are equal. Signup and view all the answers

How can you find the x-intercept of the equation y = 3x - 4?

Set y = 0 and solve for x. Signup and view all the answers

What must be established to show that two sides of a triangle are perpendicular?

The slopes of the sides must be the negatives of each other. Signup and view all the answers

In showing that diagonals bisect each other in a parallelogram, what is the necessary condition?

They must have a common midpoint. Signup and view all the answers

What is the absolute value of a negative number?

The positive of the number Signup and view all the answers

Which of the following properties of absolute value indicates that |a| is always non-negative?

|a| ≥ 0 Signup and view all the answers

If |x| < 3, which of the following inequalities must hold true?

-3 < x < 3 Signup and view all the answers

Which of the following describes the relationship represented by |x| ≥ 2?

x ≥ 2 or x ≤ -2 Signup and view all the answers

What does the property |a + b| ≤ |a| + |b| represent?

Triangle inequality in absolute values Signup and view all the answers

In a two-dimensional coordinate system, what does the point (0, 0) represent?

The origin Signup and view all the answers

What does the inequality |x| > 5 imply?

x > 5 or x < -5 Signup and view all the answers

If |a| = 3, which of the following values could a be?

-3 or 3 Signup and view all the answers

Which of the following represents the general form of the equation of a line?

Ax + By + C = 0 Signup and view all the answers

What is the slope-intercept form of a line with slope $m$ and y-intercept $b$?

y = mx + b Signup and view all the answers

What is the domain of a function?

The set of all independent variables in a function Signup and view all the answers

In the equation y = f(x), which symbol represents the dependent variable?

y Signup and view all the answers

Which equation represents the point-slope form of a line passing through the point $(x1, y1)$ with slope $m$?

y - y1 = m(x - x1) Signup and view all the answers

How is the slope of two nonvertical lines defined if they are parallel?

m1 = m2 Signup and view all the answers

When evaluating the function f at a given number, what do you do with that number?

Substitute it for x Signup and view all the answers

Which condition must be satisfied when finding the domain of a function involving a fraction?

The denominator must be non-zero Signup and view all the answers

What is the intercept form of the equation of a line with x-intercept $a$ and y-intercept $b$?

x/a + y/b = 1 Signup and view all the answers

What is the range of a function?

The set composed of the second components of all ordered pairs Signup and view all the answers

If two lines are perpendicular, how is the slope of one line related to the other?

m2 = -m1 Signup and view all the answers

Given the points (3, -2) and (1, 6), which equation represents the two-point form of a line?

y + 2 = (8/2)(x - 3) Signup and view all the answers

What must be true for the expression $ ext{a}^{1/n}$ when n is even?

a must be greater than or equal to zero Signup and view all the answers

What is the correct expression for the slope between the points (x1, y1) and (x2, y2)?

m = (y2 - y1)/(x2 - x1) Signup and view all the answers

In the graph of a function, which of the following is true about the ordered pairs (x, y)?

They are pairs formed by both independent and dependent variables Signup and view all the answers

If f(x) = 2x^2 + x - 1, what is f(2)?

9 Signup and view all the answers

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.

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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.