3.2 Slope of a Line

Slope and graphing lines are essential concepts in algebra. They help us understand how lines behave on a coordinate plane. By learning to calculate slope and use different equation forms, we can easily plot lines and analyze their relationships.

These skills are crucial for solving real-world problems. We can model situations like car rentals or savings accounts using linear equations. Understanding parallel and perpendicular lines also helps us grasp more complex geometric relationships in mathematics.

Slope and Graphing Lines

Slope calculation using points

• Calculates the steepness and direction of a line by comparing the vertical change ($\Delta y$) to the horizontal change ($\Delta x$) between any two distinct points on the line
• Uses the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of two different points on the line
• Positive slope indicates the line rises from left to right (uphill)
• Negative slope means the line falls from left to right (downhill)
• Zero slope represents a horizontal line (flat)
• Undefined slope occurs with vertical lines due to division by zero (infinite steepness)

Point-slope form for graphing

• Utilizes the point-slope form equation $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a known point on the line
• To graph using point-slope form:
  1. Identify the slope ($m$) and a point $(x_1, y_1)$ that lies on the line
  2. Plug the values into the point-slope form equation
  3. Select a few x-values and calculate the corresponding y-values
  4. Plot the points and connect them with a straight line
• This form is particularly useful when working with linear equations in the coordinate plane

Slope-intercept form for plotting

• Employs the slope-intercept form equation $y = mx + b$, where $m$ represents the slope and $b$ is the y-intercept (the point where the line crosses the y-axis when $x = 0$)
• To graph using slope-intercept form:
  1. Determine the slope ($m$) and y-intercept ($b$)
  2. Plot the y-intercept point $(0, b)$
  3. Use the slope to locate another point on the line (for example, if $m = \frac{2}{3}$, move 3 units right and 2 units up from the y-intercept)
  4. Draw a straight line connecting the two points

Efficient line graphing methods

• When given the slope and y-intercept, employ slope-intercept form for quick graphing
• When given the slope and a point, utilize point-slope form to plot the line
• When given two points, first calculate the slope using the slope formula, then use point-slope form to graph
• When given an equation in standard form ($Ax + By = C$), convert it to slope-intercept form by solving for $y$ before graphing
• A graphing calculator can be used to quickly plot and visualize linear equations

Applications and Relationships

Real-world slope-intercept applications

• In a linear model, the slope signifies the rate of change, while the y-intercept represents the initial value or starting point
• Examples:
  • A car rental company charges a base fee of $50 plus$0.25 per mile driven (base fee is y-intercept, per-mile charge is slope)
  • A person's savings account balance increases by $100 per month (initial balance is y-intercept, monthly deposit is slope)

Parallel vs perpendicular line slopes

• Parallel lines share the same slope but have different y-intercepts
  • If $y = m_1x + b_1$ and $y = m_2x + b_2$ are parallel lines, then $m_1 = m_2$
• Perpendicular lines have slopes that are negative reciprocals of each other
  • If $y = m_1x + b_1$ and $y = m_2x + b_2$ are perpendicular lines, then $m_1 = -\frac{1}{m_2}$ or $m_1m_2 = -1$
• To find the equation of a line parallel or perpendicular to a given line, use the appropriate slope relationship and a point on the desired line

Functions and Linear Equations

• A linear equation represents a straight line and is a type of function
• Functions describe a relationship between variables, where each input (x-value) corresponds to exactly one output (y-value)
• Linear functions can be represented in various forms, including slope-intercept and point-slope forms