key term - Point-Slope Form

Definition

The point-slope form is a way to represent the equation of a linear line using the slope of the line and a single point on the line. It allows you to easily graph linear equations and find the equation of a line given specific information.

5 Must Know Facts For Your Next Test

1. The point-slope form of a linear equation is written as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope of the line.
2. Using the point-slope form allows you to easily graph a linear equation by plotting the given point and then using the slope to determine the direction and steepness of the line.
3. The point-slope form is particularly useful when you are given the slope of a line and a single point, as it provides a straightforward way to find the equation of the line.
4. Transforming an equation from point-slope form to slope-intercept form, which is $y = mx + b$, can be done by rearranging the terms to isolate the $y$ variable.
5. Understanding point-slope form is essential for graphing linear equations, determining the equation of a line, and analyzing the relationships between points and slopes in the coordinate plane.

Review Questions

• Explain how the point-slope form of a linear equation is used to graph a line.
  • The point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, provides the necessary information to graph a line. By knowing a single point $(x_1, y_1)$ on the line and the slope $m$, you can plot the given point and then use the slope to determine the direction and steepness of the line. This allows you to draw the line through the known point with the correct orientation and slope, effectively graphing the linear equation.
• Describe the relationship between the point-slope form and the slope-intercept form of a linear equation.
  • The point-slope form, $y - y_1 = m(x - x_1)$, and the slope-intercept form, $y = mx + b$, are two different ways of representing the same linear equation. The point-slope form uses a known point $(x_1, y_1)$ and the slope $m$ to define the line, while the slope-intercept form uses the slope $m$ and the $y$-intercept $b$ to define the line. By rearranging the terms in the point-slope form, you can convert it to the slope-intercept form, which may be more convenient for certain applications, such as graphing the line or determining its characteristics.
• Analyze how the point-slope form can be used to find the equation of a line given specific information about the line.
  • The point-slope form, $y - y_1 = m(x - x_1)$, is particularly useful when you are given a single point $(x_1, y_1)$ on the line and the slope $m$ of the line. By substituting the known values into the point-slope equation, you can solve for the equation of the line in the standard form $y = mx + b$. This allows you to determine the complete equation of the line, including the $y$-intercept $b$, which can be useful for further analysis and applications, such as graphing the line, predicting values, or understanding the line's properties.

Related terms

Slope:

The slope of a line is a measure of its steepness, calculated as the ratio of the vertical change to the horizontal change between any two points on the line.

Linear Equation:

A linear equation is an equation that represents a straight line, where the variables are raised to the first power and there are no nonlinear terms like squares or square roots.

Coordinate Plane:

The coordinate plane is a two-dimensional grid used to graph linear equations, where the horizontal axis is the x-axis and the vertical axis is the y-axis.