Point-Slope Form

Definition

The point-slope form is a way of expressing the equation of a linear function. It provides a convenient method to write the equation of a line when given a point on the line and the slope of the line.

5 Must Know Facts For Your Next Test

1. 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.
2. The point-slope form is useful when you know a point on the line and the slope, but don't know the $y$-intercept.
3. To convert a point-slope equation to slope-intercept form, you can solve for $y$ by rearranging the equation to $y = mx + b$.
4. The derivative of a linear function in point-slope form is a constant function, where the derivative is equal to the slope of the original line.
5. The point-slope form can be used to find the equation of a tangent line to a function at a specific point, which is useful in the context of derivatives.

Review Questions

• Explain how the point-slope form of a linear equation is used to write the equation of a line.
  • The point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, allows you to write the equation of a line when you know a point $(x_1, y_1)$ on the line and the slope $m$ of the line. By substituting the known point and slope into this formula, you can solve for the equation of the line in the standard linear form $y = mx + b$. This is particularly useful when you don't know the $y$-intercept $b$ but have the necessary information about a point on the line and the slope.
• Describe how the point-slope form of a linear equation is related to the concept of derivatives in the context of 12.4 Derivatives.
  • In the context of 12.4 Derivatives, the point-slope form of a linear equation is useful because the derivative of a linear function is a constant function, where the derivative is equal to the slope of the original line. When finding the equation of the tangent line to a function at a specific point, the point-slope form can be used to write the equation of the tangent line, as the slope of the tangent line is equal to the derivative of the function at that point. This connection between the point-slope form and derivatives allows you to apply your understanding of linear functions to the study of derivatives and tangent lines.
• Analyze how the point-slope form of a linear equation can be used to model and understand real-world situations involving linear relationships, such as those discussed in 2.1 Linear Functions.
  • $$ The point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, can be used to model and understand real-world situations involving linear relationships, as discussed in 2.1 Linear Functions. By identifying a known point $(x_1, y_1)$ on the line and the slope $m$, you can use the point-slope form to write the equation of the line that represents the linear relationship. This is useful for analyzing and making predictions about the behavior of the linear function, such as its rate of change, the $y$-intercept, and how the dependent variable $y$ changes in relation to the independent variable $x$. The point-slope form provides a flexible and intuitive way to work with linear functions in various real-world contexts. $$

Related terms

Slope-Intercept Form: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope of the line and $b$ is the $y$-intercept.

Slope: The slope of a line is a measure of the steepness of the line, and it represents the rate of change between the $x$ and $y$ variables.

Linear Function: A linear function is a function that can be expressed in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.