Questions and Answers
What is the slope (unit rate/rate of change)?
The ratio of the vertical change to the horizontal change of any two points on a line, and is constant.
What describes a negative slope?
A line that moves downward from left to right has a negative slope.
What describes a positive slope?
A line that moves upward from left to right has a positive slope.
What does zero slope represent?
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What does undefined slope represent?
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What is slope-intercept form?
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What is the rate of change (slope)?
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What is a linear equation?
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Find the slope of the line through the points (1, 1) and (3, 4).
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What is the slope through the points (5, 7) and (10, 14)?
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What is the slope represented by the equation y = -2x + (-3)?
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What is the slope represented by the equation y = 1x + 3?
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Find the slope when the equation is given as y = 1/5x + (-5).
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What is the slope for the line through the points (-2, 6) and (5, -3)?
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What is the slope represented by the coordinates (2, 5) and (6, 10)?
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What is the slope when given the formula rise/run?
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What is the general form of an equation of a straight line?
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How would you describe a line with a slope of 0?
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How would you define a slope of undefined?
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Study Notes
Slope Definitions and Concepts
- Slope: The ratio of vertical change to horizontal change between two points on a line; remains constant across the line.
- Positive Slope: Indicates a line moving upward from left to right.
- Negative Slope: Represents a line moving downward from left to right.
- Zero Slope: Describes a horizontal line; characterized by no vertical change (0/#).
- Undefined Slope: Refers to a vertical line where the horizontal change is zero (#/0), making slope calculation impossible.
- Slope-Intercept Form: Expressed as y = mx + b, where m is the slope and b is the y-intercept.
Applications and Relationships
- Rate of Change (Slope): Demonstrates how two quantities change in relation to each other, expressed as the change in the dependent variable over the change in the independent variable.
- Linear Equation: An equation that results in a straight line when graphed, typically in the form y = mx + b.
Graph Interpretations
- Constant Rates: In scenarios such as temperature regulation over time, a slope of zero indicates stability, while a constant decrease reflects a negative slope.
- Volume Relationships: In geometric comparisons (like a rectangular prism vs. a pyramid), slopes can represent volume ratios.
Calculation of Slope
- Slope can be calculated using the formula rise/run, which measures vertical change over horizontal change between two points on a line.
- Various examples express slope values based on given points, illustrating calculations resulting in both positive and negative slopes, and even zero values.
Practical Examples
- Balloon Artist: Graphing the number of balloons remaining after creating hats showcases a negative slope, meaning as hats are created, balloons decrease.
- Loans and Payments: A line representing loan balance over time typically has a negative slope, indicating regular payments reducing total debt.
Summary Calculations
- Examples of calculations yield various slopes such as:
- (2, 5) to (6, 10) results in 5/4.
- (1, 1) to (3, 4) yields 3/2.
- Geometric examples demonstrate applying formulas to achieve consistent results.
Additional Key Slope Values
- Specifically noted slopes include:
- Slope of 4/3 from points (0, 0) and (3, 4).
- Slope of 4/9 from points (1, 1) and (10, 5).
- Slope of -1/6 from points (-7, 2) and (-4, 1).
Noteworthy Characteristics
- The steeper a line, the larger the absolute value of its slope.
- An undefined slope indicates a perfect vertical line, while a slope of zero represents a perfect horizontal line, both exhibiting unique graphical properties.
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Description
Test your understanding of the concept of slope in mathematics with these flashcards. From defining slope to understanding positive, negative, and zero slopes, these cards cover essential terminology. Perfect for students looking to reinforce their knowledge of linear equations and graphing.
