Straight Line Through Origin

Definition

A straight line through the origin is a linear function that passes through the point (0, 0) on the coordinate plane. This type of line has a constant rate of change, or slope, that represents the relationship between the independent and dependent variables.

5 Must Know Facts For Your Next Test

1. When a straight line passes through the origin, the $y$-intercept is zero, and the equation of the line can be written in the form $y = mx$, where $m$ is the slope.
2. The slope of a straight line through the origin is the ratio of the $y$-coordinate to the $x$-coordinate of any point on the line.
3. Straight lines through the origin represent direct variation, where the dependent variable is proportional to the independent variable.
4. The equation of a straight line through the origin can be written in the form $y = kx$, where $k$ is the constant of proportionality.
5. Straight lines through the origin are often used to model relationships where one variable is directly proportional to another, such as in the study of direct and inverse variation.

Review Questions

• Explain how the equation of a straight line through the origin differs from the general linear function equation.
  • The equation of a straight line through the origin is simplified compared to the general linear function equation. While the general linear function has the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept, the equation of a straight line through the origin has the form $y = mx$, where the $y$-intercept $b$ is equal to zero. This is because a straight line through the origin passes through the point (0, 0), making the $y$-intercept zero.
• Describe the relationship between the slope of a straight line through the origin and the concept of direct variation.
  • The slope of a straight line through the origin is directly related to the concept of direct variation. In a direct variation, the dependent variable is proportional to the independent variable, meaning that as one variable increases, the other variable increases at a constant rate. This constant rate of change is represented by the slope of the straight line through the origin. The equation of the line, $y = mx$, where $m$ is the slope, demonstrates the direct proportional relationship between the two variables.
• Analyze how the properties of a straight line through the origin can be used to model real-world situations involving direct and inverse variation.
  • Straight lines through the origin are commonly used to model real-world situations involving direct and inverse variation. The properties of these lines, such as the constant slope and the fact that they pass through the origin, allow for the representation of proportional relationships between variables. For example, in direct variation, the graph of the relationship between two variables is a straight line through the origin, with the slope representing the constant of proportionality. Conversely, in inverse variation, the graph is a hyperbola that passes through the origin, and the properties of the straight line through the origin can be used to analyze the inverse relationship between the variables.

Related terms

Linear Function: A linear function is a polynomial function of degree one, where the graph is a straight line. It has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Slope: The slope of a line is the constant rate of change, or the ratio of the change in the $y$-value to the change in the $x$-value between any two points on the line.

Direct Variation: Direct variation is a relationship between two variables where one variable is proportional to the other. The graph of a direct variation is a straight line through the origin.