key term - Y-intercept

Definition

The y-intercept is the point where a line crosses the y-axis on a graph, represented by the coordinate (0, b) where b is the value of the y-intercept. It is a crucial aspect in understanding the equations of lines, especially when determining their positions and relationships to one another, such as in cases of parallelism and perpendicularity.

5 Must Know Facts For Your Next Test

1. The y-intercept can be found directly from the equation of a line in slope-intercept form, y = mx + b, where b is the y-intercept.
2. In a coordinate plane, the y-intercept occurs at x = 0, meaning to find it you can simply substitute 0 for x in the equation.
3. For two lines to be parallel, they must have the same slope but different y-intercepts.
4. Perpendicular lines have slopes that are negative reciprocals of each other; however, they can have any combination of y-intercepts.
5. The value of the y-intercept provides important information about where the line intersects the vertical axis and can help in sketching graphs.

Review Questions

• How can you determine the y-intercept from an equation of a line in slope-intercept form?
  • To find the y-intercept from an equation written in slope-intercept form (y = mx + b), simply identify the value of b. This value represents where the line crosses the y-axis. By setting x to 0 and solving for y, you'll confirm that (0, b) is indeed the point of intersection with the y-axis.
• Discuss how parallel lines relate to their y-intercepts and why this matters in geometric proofs.
  • Parallel lines have identical slopes, which means they rise or fall at the same rate. However, their y-intercepts must differ; otherwise, they would overlap rather than remain distinct. Understanding this relationship allows us to establish properties of parallel lines in geometric proofs and helps us verify whether two given lines are indeed parallel based on their equations.
• Evaluate how changes in the y-intercept affect the overall graph of a linear equation and its relationship with perpendicular lines.
  • When you change the y-intercept of a linear equation while keeping the slope constant, you shift the entire line up or down without altering its angle. This change affects its position relative to other lines, including perpendicular ones. For instance, if one line has a positive slope and its corresponding perpendicular line has a negative reciprocal slope, adjusting their respective y-intercepts can create new intersection points while maintaining their perpendicular relationship. Understanding this concept enhances our ability to manipulate and analyze linear equations geometrically.

Related terms

Slope: The slope of a line measures its steepness and direction, calculated as the change in y divided by the change in x between two points on the line.

Linear Equation: A linear equation is an equation that describes a straight line when graphed, typically in the form y = mx + b, where m is the slope and b is the y-intercept.

Intercept Form: Intercept form is a way of writing a linear equation that highlights both x-intercept and y-intercept, often expressed as \( \frac{x}{a} + \frac{y}{b} = 1 \), where (a, 0) and (0, b) are intercepts.