Graphs and Functions

The Slope of a Line

The concept of slope is fundamental in algebra and analytic geometry, describing the steepness and direction of a line on the coordinate plane. Understanding slope allows us to analyze, compare, and interpret linear relationships in various contexts.

Finding the Slope of a Line Given Two Points

• Definition: The slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

• Key Point: The numerator represents the change in y (vertical change), and the denominator represents the change in x (horizontal change).

• Helpful Hint: It does not matter which point is labeled as 1 or 2, as long as the order is consistent in both numerator and denominator.

• Example: Find the slope of the line through (0, 3) and (2, 5):

• Example: Find the slope of the line through (4, -3) and (2, 2):

Finding Slope Given an Equation

• Slope-Intercept Form: A linear equation can be written as , where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).

• To find the slope from a standard form equation (e.g., ), solve for y to rewrite in slope-intercept form.

• Example: For :

  • Solve for y:

  • Slope is , y-intercept is .

• Example: For :

  • Solve for y:

  • Slope is .

Interpreting Slope-Intercept Form

The slope-intercept form allows us to interpret real-world situations where one variable depends linearly on another.

• Example: The equation models the price y of a Disney World pass, where x is years since 2006. The slope (3.53) represents the annual increase in price, and the y-intercept (67.39) is the price in 2006.

• Application: To predict the price in 2026 ():

Finding Slopes of Horizontal and Vertical Lines

• Vertical Lines: Equations of the form are vertical lines. Their slope is undefined because the change in x is zero (division by zero is undefined).

  • Example: The line has an undefined slope.

• Horizontal Lines: Equations of the form are horizontal lines. Their slope is 0 because the change in y is zero.

  • Example: The line has a slope of 0.

Appearance of Lines with Given Slopes

• Positive Slope: Lines with rise as increases (upward to the right).

• Negative Slope: Lines with fall as increases (downward to the right).

Compare the Slopes of Parallel and Perpendicular Lines

• Parallel Lines: Two lines are parallel if they have the same slope (). Vertical lines are also parallel to each other.

  • Example: The lines and are parallel if their slopes are equal.

• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is (). Their slopes are negative reciprocals.

  • Example: The lines and are perpendicular if their slopes are negative reciprocals.

• Horizontal and Vertical Lines: These are always perpendicular to each other.

Summary Table: Types of Lines and Their Slopes