Rise

Definition

The term 'rise' refers to the vertical change or increase in the value of a variable, particularly in the context of graphing linear equations and understanding the concept of slope. It is a fundamental concept that underpins the understanding of the relationship between variables and the direction of a line on a coordinate plane.

5 Must Know Facts For Your Next Test

1. The rise of a line is the vertical change or increase in the $y$-value as you move from one point to another on the line.
2. The rise, along with the run (the horizontal change), determines the slope of a line, which is a key characteristic of a linear equation.
3. In the slope-intercept form of a linear equation, $y = mx + b$, the slope $m$ represents the ratio of the rise to the run, or the rate of change between the variables.
4. When graphing a linear equation, the rise of the line is represented by the vertical distance between two points on the line, and it is an important factor in determining the direction and steepness of the line.
5. Understanding the concept of rise is crucial in interpreting the meaning of the slope of a line and the relationship between the variables in a linear equation.

Review Questions

• Explain how the rise of a line is related to the slope of the line.
  • The rise of a line is directly related to the slope of the line. The slope of a line is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. The rise represents the vertical change, and it is a key component in calculating the slope of a line. The greater the rise, the steeper the slope of the line, and vice versa. Understanding the relationship between rise and slope is essential in graphing linear equations and interpreting the meaning of the slope in the context of the problem.
• Describe how the rise of a line is represented in the slope-intercept form of a linear equation.
  • In the slope-intercept form of a linear equation, $y = mx + b$, the slope $m$ represents the ratio of the rise to the run. The rise is the vertical change in the $y$-value as you move from one point to another on the line. The slope $m$ captures this vertical change, or rise, in relation to the horizontal change, or run. Therefore, the rise of the line is directly reflected in the slope term of the slope-intercept equation, and understanding the meaning of the rise is crucial in interpreting the slope and the overall relationship between the variables in the linear equation.
• Analyze how the rise of a line affects the direction and steepness of the line when graphed on a coordinate plane.
  • The rise of a line has a direct impact on the direction and steepness of the line when graphed on a coordinate plane. A positive rise indicates that the line is sloping upward from left to right, while a negative rise indicates a downward sloping line. The magnitude of the rise also determines the steepness of the line, with a larger rise resulting in a steeper line and a smaller rise resulting in a more gradual line. This relationship between the rise and the direction and steepness of the line is crucial in understanding the behavior of linear equations and interpreting their graphical representations on the coordinate plane.

Related terms

Slope: The slope of a line represents the rate of change or the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

Coordinate Plane: A two-dimensional plane where the vertical axis (y-axis) represents the rise or fall of a variable, and the horizontal axis (x-axis) represents the change in another variable.

Slope-Intercept Form: The equation of a line in the form $y = mx + b$, where $m$ represents the slope (the ratio of rise to run) and $b$ represents the $y$-intercept (the point where the line crosses the $y$-axis).