Vocabulary Terms
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Slope from a Table
- Definition: The slope from a table is calculated by finding the change in the \(y\) values divided by the change in the \(x\) values between two points. It is typically represented as \(m\), where \(m = \frac{\Delta y}{\Delta x}\).
- Illustration: Consider the following table of values:
| \(x\) | \(y\) | |-------|-------| | 1 | 2 | | 3 | 6 |
To find the slope:
- Choose two points: \((1, 2)\) and \((3, 6)\).
- Calculate the change in \(y\) (Δy) and the change in \(x\) (Δx):
- Δy = \(6 - 2 = 4\)
- Δx = \(3 - 1 = 2\)
- Calculate the slope:
- \(m = \frac{\Delta y}{\Delta x} = \frac{4}{2} = 2\)
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Slope from a Graph
- Definition: The slope from a graph is a visual representation of the steepness or incline of a line. It can be found by selecting two points on the line, determining the rise (vertical change) and run (horizontal change), and then calculating the slope using \(m = \frac{\text{rise}}{\text{run}}\).
- Illustration: Imagine a graph with a line going through the points (2, 3) and (4, 7).
- Rise = \(7 - 3 = 4\)
- Run = \(4 - 2 = 2\)
- Slope \(m = \frac{4}{2} = 2\)
- The line rises by 4 units vertically for every 2 units it runs horizontally. Thus, the slope is 2.
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Slope from an Equation
- Definition: The slope from an equation is derived from the standard linear equation of the form \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. The slope indicates how much \(y\) changes for a one-unit change in \(x\).
- Illustration: Consider the equation \(y = 3x + 1\).
- Here, \(m = 3\), meaning for every increase of 1 unit in \(x\), \(y\) increases by 3 units.
- The slope of this line is steep, indicating a strong positive relationship between \(x\) and \(y\).
Summary
- Slope from a table: Calculate the change in \(y\) and \(x\).
- Slope from a graph: Determine the rise/run between two points on the line.
- Slope from an equation: Extract the coefficient of \(x\) from the linear equation \(y = mx + b\).
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