Δy, or delta y, represents the change in the dependent variable y between two points on a graph or function. It is a fundamental concept in understanding the slope of a line, as the slope is defined as the ratio of the change in y to the change in x between two points.
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Δy represents the vertical change or rise between two points on a graph or function.
The slope of a line is calculated as the ratio of Δy to Δx, where Δx represents the change in the independent variable.
Δy is a crucial component in determining the rate of change or steepness of a line, which is a fundamental concept in understanding linear functions.
The sign of Δy (positive or negative) indicates the direction of the change in the dependent variable, and is used to determine the orientation of the line on the coordinate plane.
Δy is an essential element in the equation of a line, $y = mx + b$, where the slope, $m$, is equal to Δy/Δx.
Review Questions
Explain how Δy is used to calculate the slope of a line.
The slope of a line is defined as the ratio of the change in the y-coordinate (Δy) to the change in the x-coordinate (Δx) between two points on the line. This can be expressed mathematically as $m = Δy/Δx$, where $m$ represents the slope. By finding the change in y-values (Δy) between two points, and dividing it by the change in x-values (Δx), you can determine the slope of the line, which represents the rate of change or steepness of the line.
Describe the relationship between Δy and the orientation of a line on the coordinate plane.
The sign of Δy (positive or negative) indicates the direction of the change in the dependent variable, and is used to determine the orientation of the line on the coordinate plane. If Δy is positive, the line is sloping upward from left to right, indicating a positive slope. If Δy is negative, the line is sloping downward from left to right, indicating a negative slope. The magnitude of Δy determines the steepness or rate of change of the line, with larger absolute values of Δy corresponding to steeper lines.
Explain how Δy is used in the equation of a line and how it relates to the slope-intercept form.
In the equation of a line, $y = mx + b$, the slope, $m$, is equal to Δy/Δx. This means that Δy is a crucial component in determining the slope of the line, which represents the rate of change or steepness of the line. The slope-intercept form of the line equation, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept, directly incorporates Δy through the slope term, $m = Δy/Δx$. Understanding the relationship between Δy and the slope of a line is essential for interpreting and working with linear functions.
The dependent variable is the variable that changes in response to changes in the independent variable. In the context of a line, the dependent variable is typically represented by the y-coordinate.
The independent variable is the variable that is manipulated or changed to observe the effect on the dependent variable. In the context of a line, the independent variable is typically represented by the x-coordinate.