뒤로Equations of Lines: Slope, Intercept, and Applications (Chapter 7 Study Notes)
스터디 가이드 - 스마트 노트
자료에 맞춘 맞춤형 노트, 핵심 정의, 예시, 맥락을 확장해 제공합니다.
Graphs, Functions, and Applications
Finding Equations of Lines
This section introduces methods for finding the equation of a line in various scenarios, including when given the slope and y-intercept, the slope and a point, or two points. It also covers applications involving parallel and perpendicular lines, and solving real-world problems using linear functions.
Slope-Intercept Form: The equation of a line with slope m and y-intercept b is given by:
Point-Slope Form: The equation of a line with slope m passing through point is:
Parallel Lines: Lines that have the same slope but different y-intercepts.
Perpendicular Lines: Lines whose slopes are negative reciprocals of each other.
Finding the Equation of a Line: Slope and Y-Intercept Given
When both the slope and y-intercept are known, use the slope-intercept form.
Example: Find the equation of a line with slope and y-intercept . Solution: Substitute and $5by = mx + by = -3.2x + 5$ Application: This form is useful for quickly graphing a line or modeling linear relationships.
Finding the Equation of a Line: Slope and a Point Given
If the slope and a point on the line are given, use the point-slope form to find the equation.
Point-Slope Equation:
Example: Find the equation of the line with slope $3(2, 7)$. Solution: Alternate Method (Slope-Intercept): Substitute , , into : So,
Finding the Equation of a Line: Two Points Given
When two points are given, first calculate the slope, then use either the point-slope or slope-intercept form.
Formula for Slope:
Example: Find the equation of the line containing points and . Solution: Calculate the slope: Use point-slope form with :
Parallel and Perpendicular Lines
Equations of lines parallel or perpendicular to a given line can be found by using the slope of the original line and a given point.
Parallel Lines: Use the same slope as the given line. Example: Find the equation of the line containing point and parallel to . Solution: Slope Use point-slope form:
Perpendicular Lines: Use the negative reciprocal of the original slope. Example: Find the equation of the line containing point and perpendicular to . Solution: Rewrite in slope-intercept form: Slope of perpendicular line: Use point-slope form:
Applications: Linear Functions in Real-World Problems
Linear functions are used to model relationships in various contexts, such as cost, distance, and time.
Example: Calvin Appliance charges a service fee and C(t)t$ is the number of hours. Solution: Graph: The y-intercept is , and the slope is $55$ (rate per hour).
Application: To find the cost of a hour service call: The cost is .
Summary Table: Forms of Linear Equations
Form | Equation | When to Use |
|---|---|---|
Slope-Intercept | When slope and y-intercept are known | |
Point-Slope | When slope and a point are known | |
Standard | General form; can be converted to other forms |
Key Terms: Slope, y-intercept, point-slope form, parallel lines, perpendicular lines, linear function
Additional info: The notes are based on textbook slides for College Algebra, focusing on equations of lines and their applications in real-world contexts.
