Questions and Answers
What are whole numbers?
Which of the following lists shows the integers?
What is a rational number?
Any number that can be written as a fraction
What defines an irrational number?
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What are opposites?
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What does absolute value represent?
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What is a function?
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What is the zero of a function?
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What are the domain and range of a function?
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What is the vertical line test?
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What is slope-intercept form?
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What is the standard form of a linear equation?
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What is point-slope form of an equation?
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What is an x-intercept?
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What is a y-intercept?
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What defines a horizontal line?
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What defines a vertical line?
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What are parallel lines?
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What are perpendicular lines?
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What is the slope formula?
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What is direct variation?
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What do you need to do when solving an inequality?
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How do you graph inequalities on a number line?
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What are absolute value equations?
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How do you interpret absolute value inequalities?
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Study Notes
Whole Numbers
- Defined as 0, 1, 2, 3, and so on.
Integers
- Include negative and positive whole numbers: -3, -2, -1, 0, 1, 2, 3, etc.
Rational Numbers
- Can be expressed as a fraction (e.g., proper and improper fractions, mixed numbers, terminating and repeating decimals).
- Includes integers and whole numbers.
Irrational Numbers
- Cannot be represented as a fraction.
- Example includes decimal numbers that repeat without a pattern.
Opposites
- For any number A, its opposite is -A (e.g., -5 is the opposite of 5).
Absolute Value
- Represents the distance of a number from zero; always a positive value.
Functions
- Each input corresponds to exactly one output, ensuring no repeated input values.
Zero of a Function
- The x-intercept found when y (or f(x)) is equal to zero.
Domain
- Represents the set of input values (or x-values).
Range
- Represents the set of output values (or y-values).
Vertical Line Test
- Determines if a graphed relation is a function by checking if a vertical line passes through more than one point.
Slope-Intercept Form
- Expressed as y = mx + b, where m is the slope and b is the y-intercept.
Standard Form
- Written as Ax + By = C, where A, B, and C are integers.
Point-Slope Form
- Defined as y - y₁ = m(x - x₁); m is the slope and (x₁, y₁) is a point on the line.
X-Intercept
- The point where a graph crosses the x-axis, represented as (x, 0).
- Found by setting y = 0.
Y-Intercept
- The point where a graph crosses the y-axis, represented as (0, y).
- Found by setting x = 0.
Horizontal Line
- Defined by the equation y = b, where all points have the same y-value.
Vertical Line
- Defined by the equation x = a, where all points have the same x-value.
Parallel Lines
- Two lines with identical slopes, indicating they will never intersect.
Perpendicular Lines
- Two lines with slopes that are opposite reciprocals (e.g., a slope of 3 has a perpendicular slope of -1/3).
Slope Formula
- Calculated as Δy/Δx or (y₂ - y₁)/(x₂ - x₁).
Direct Variation
- Describes a relationship y = kx, where k is the constant of variation (slope).
- Graphically represented as a line that always passes through the origin.
Solving an Inequality
- When multiplying or dividing both sides by a negative number, the inequality sign must be reversed.
Graphing Inequalities on a Number Line
- Use a dashed line for "less than" or "greater than" inequalities and a solid line for "less than or equal to"/"greater than or equal to."
- Determine which side to shade with a test point; overlapping regions signify the solution in linear systems.
Absolute Value Equations
- E.g., |x| = 4 results in x = 4 or x = -4. If |x| = no solution, the equation has no valid x-value.
Absolute Value Inequalities
- "Less than" creates an "and" compound inequality: |x| < k leads to two inequalities: -k < x < k.
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