key term - $m = \frac{y_2 - y_1}{x_2 - x_1}$

Definition

This term represents the slope of a linear function, which is the measure of the steepness or rate of change of the line. It describes the change in the dependent variable (y) with respect to the change in the independent variable (x). The slope is a crucial concept in modeling linear relationships between variables.

5 Must Know Facts For Your Next Test

1. The slope formula, $m = \frac{y_2 - y_1}{x_2 - x_1}$, allows you to calculate the slope of a line given two points on the line.
2. The slope of a line represents the rate of change between the dependent and independent variables. A positive slope indicates an increasing relationship, while a negative slope indicates a decreasing relationship.
3. The slope of a line can be used to determine the direction and steepness of the line. A steeper line has a larger absolute value of the slope, while a flatter line has a smaller absolute value of the slope.
4. The slope formula can be used to model real-world linear relationships, such as the rate of change in a person's height over time or the relationship between the price and quantity of a product.
5. Understanding the slope of a line is essential for interpreting the behavior of linear functions and making predictions based on the rate of change between variables.

Review Questions

• Explain how the slope formula, $m = \frac{y_2 - y_1}{x_2 - x_1}$, can be used to determine the direction and steepness of a linear relationship.
  • The slope formula, $m = \frac{y_2 - y_1}{x_2 - x_1}$, can be used to determine the direction and steepness of a linear relationship. The sign of the slope indicates the direction of the relationship: a positive slope suggests an increasing relationship, while a negative slope suggests a decreasing relationship. The absolute value of the slope represents the steepness of the line, with a larger absolute value indicating a steeper line and a smaller absolute value indicating a flatter line. This information can be used to interpret the behavior of linear functions and make predictions about the rate of change between the dependent and independent variables.
• Describe how the slope formula can be used to model real-world linear relationships, such as the rate of change in a person's height over time or the relationship between the price and quantity of a product.
  • The slope formula, $m = \frac{y_2 - y_1}{x_2 - x_1}$, can be used to model real-world linear relationships by representing the rate of change between two variables. For example, to model the rate of change in a person's height over time, you could use the slope formula to calculate the change in height (y-coordinate) divided by the change in time (x-coordinate) between two points. This would give you the average rate of growth over that time period. Similarly, the slope formula could be used to model the relationship between the price and quantity of a product, where the slope would represent the rate of change in price as the quantity changes. By understanding the slope of these linear relationships, you can make predictions and draw conclusions about the behavior of the variables in the real-world context.
• Analyze how the slope formula, $m = \frac{y_2 - y_1}{x_2 - x_1}$, is connected to the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, and explain how this relationship can be used to model and interpret linear functions.
  • The slope formula, $m = \frac{y_2 - y_1}{x_2 - x_1}$, is directly connected to the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$. The slope formula allows you to calculate the slope, $m$, given two points on the line, $(x_1, y_1)$ and $(x_2, y_2)$. This slope can then be substituted into the point-slope form to write the equation of the line passing through those two points. By understanding this relationship, you can use the slope formula to model linear functions and interpret their behavior. For example, if you know the slope of a line and a point on the line, you can use the point-slope form to write the equation of the line and make predictions about the dependent variable (y) for different values of the independent variable (x). This connection between the slope formula and the point-slope form is essential for analyzing and understanding the properties of linear functions in real-world applications.

Related terms

Linear Function:

A linear function is a function where the relationship between the independent and dependent variables can be represented by a straight line. The equation for a linear function is typically written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Rise Over Run:

The rise over run is another way to express the slope of a line. It represents the change in the y-coordinate (rise) divided by the change in the x-coordinate (run) between two points on the line.

Point-Slope Form:

The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope of the line.