Slope

Definition

Slope is a measure of the steepness or incline of a line. It represents the rate of change between two points on a line, indicating how much the y-value changes for a given change in the x-value.

5 Must Know Facts For Your Next Test

1. The slope of a line is calculated as the ratio of the rise (change in y-value) to the run (change in x-value) between any two points on the line.
2. A positive slope indicates that the line is sloping upward from left to right, while a negative slope indicates a downward sloping line.
3. A slope of zero means the line is horizontal, and a slope of infinity means the line is vertical.
4. The slope of a line can be used to determine the angle of the line relative to the x-axis, with a slope of 1 representing a 45-degree angle.
5. Slope is a key characteristic of a linear equation and is essential for understanding the behavior and properties of linear functions.

Review Questions

• How can the slope of a line be used to determine the angle of the line relative to the x-axis?
  • The slope of a line is directly related to the angle of the line relative to the x-axis. A slope of 1 represents a 45-degree angle, while a slope greater than 1 represents an angle greater than 45 degrees, and a slope less than 1 represents an angle less than 45 degrees. The specific angle can be calculated using the inverse tangent function, where the angle is equal to the arctangent of the slope.
• Explain how the slope of a line is used in the slope-intercept form of a linear equation.
  • The slope-intercept form of a linear equation is written as $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the y-intercept. The slope, $m$, determines the rate of change between the x-value and the y-value, indicating how much the y-value changes for a given change in the x-value. This slope information is crucial for understanding the behavior and properties of the linear function, such as its direction, steepness, and rate of change.
• Describe how the slope of a line can be used to classify the type of linear relationship between two variables.
  • The slope of a line can be used to determine the type of linear relationship between two variables. A positive slope indicates a positive linear relationship, where the two variables increase or decrease together. A negative slope indicates a negative linear relationship, where one variable increases as the other decreases. A slope of zero indicates no linear relationship, as the y-value does not change with changes in the x-value. The magnitude of the slope also provides information about the strength of the linear relationship, with steeper slopes (larger absolute values of the slope) indicating a stronger linear relationship between the variables.

Related terms

Rise: The change in the y-coordinate between two points on a line.

Run: The change in the x-coordinate between two points on a line.

Slope-Intercept Form: The equation of a line written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.