The Main Idea

1. Forms of Linear Equations
   • Slope-intercept form: [latex]y = mx + b[/latex]
   • Point-slope form: [latex]y - y_1 = m(x - x_1)[/latex]
2. Essential Components
   • Slope (m): Rate of change
   • y-intercept (b): Point where the line crosses the y-axis
3. Determining the Equation
   • From a graph
   • From two points
   • From a point and slope

Graphical Approach

1. Identify two points on the line
2. Calculate the slope: [latex]m = \frac{y_2 - y_1}{x_2 - x_1}[/latex]
3. Find the y-intercept visually or by calculation
4. Formulate the equation: [latex]y = mx + b[/latex]

Algebraic Approach

1. Given two points: [latex](x_1, y_1)[/latex] and [latex](x_2, y_2)[/latex]
2. Calculate slope: [latex]m = \frac{y_2 - y_1}{x_2 - x_1}[/latex]
3. Use point-slope form: [latex]y - y_1 = m(x - x_1)[/latex]
4. Rearrange to slope-intercept form if needed

The Main Idea

• Horizontal Lines
  • Equation: [latex]y = b[/latex] or [latex]f(x) = b[/latex], where [latex]b[/latex] is constant
  • Slope: [latex]m = 0[/latex]
  • [latex]Y[/latex]-intercept = [latex](0, b)[/latex]
  • No [latex]x[/latex]-intercept (unless [latex]b = 0[/latex])
  • Represent constant output ([latex]y[/latex]-value)
  • Parallel to [latex]x[/latex]-axis
• Vertical Lines
  • Equation: [latex]x = a[/latex], where [latex]a[/latex] is constant
  • Slope: Undefined
  • No [latex]y[/latex]-intercept (unless [latex]a = 0[/latex])
  • [latex]X[/latex]-intercept = [latex](a, 0)[/latex]
  • Represent constant input ([latex]x[/latex]-value)
  • Not a function
  • Parallel to [latex]y[/latex]-axis

The Main Idea

• Three primary methods for graphing linear functions:
  • Plotting points
  • Using y-intercept and slope
  • Applying transformations to [latex]f(x) = x[/latex]
• Linear function equation: [latex]f(x) = mx + b[/latex]
  • [latex]m[/latex]: slope (rate of change)
  • [latex]b[/latex]: [latex]y[/latex]-intercept (where the line crosses the [latex]y[/latex]-axis)
• Slope-intercept form provides direct access to key graph characteristics
• Transformations of the identity function offer insights into function behavior
• Each method provides unique perspectives on the function’s properties

Method 1: Plotting Points

Steps:

1. Choose at least two input values ([latex]x[/latex]-coordinates)
2. Calculate corresponding output values ([latex]y[/latex]-coordinates)
3. Plot the points on a coordinate plane
4. Draw a line through the points

Method 2: Using [latex]y[/latex]-intercept and Slope

Key Concepts:

• y-intercept ([latex]b[/latex]): Point where the line crosses the y-axis [latex](0, b)[/latex]
• Slope ([latex]m[/latex]): Rate of change, [latex]m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}[/latex]

Steps:

1. Identify the y-intercept ([latex]b[/latex])
2. Identify the slope ([latex]m[/latex])
3. Plot the [latex]y[/latex]-intercept
4. Use the slope to plot additional points
5. Draw the line through the points

Method 3: Transformations of [latex]f(x) = x[/latex]

Key Transformations:

1. Vertical stretch/compression: [latex]f(x) = mx[/latex]
2. Vertical shift: [latex]f(x) = x + b[/latex]

Steps:

1. Start with [latex]f(x) = x[/latex]
2. Apply vertical stretch/compression by factor [latex]m[/latex]
3. Apply vertical shift by [latex]b[/latex] units

The Main Idea

• Parallel lines never intersect and have the same slope.

  • Key characteristics:

    • Same slope ([latex]m[/latex])
    • Different y-intercept ([latex]b[/latex])
    • Equation form: [latex]y = mx + b_1[/latex] and [latex]y = mx + b_2[/latex], where [latex]b_1 \neq b_2[/latex]
• Perpendicular lines intersect at a 90-degree angle and have slopes that are negative reciprocals of each other.

  • Key characteristics:

    • Slopes are negative reciprocals: [latex]m_1 = -\frac{1}{m_2}[/latex]
    • Product of slopes equals [latex]-1[/latex]: [latex]m_1 \cdot m_2 = -1[/latex]
    • Equation form: [latex]y = m_1x + b_1[/latex] and [latex]y = -\frac{1}{m_1}x + b_2[/latex]
• The slope-intercept form of a line ([latex]y = mx + b[/latex]) is key to identifying parallel and perpendicular lines.

The Main Idea

• Parallel lines have the same slope but different [latex]y[/latex]-intercepts.
• Perpendicular lines have slopes that are negative reciprocals of each other.
• Point-slope form and slope-intercept form are key tools for writing these equations.
• Given a line and a point, we can write equations of parallel or perpendicular lines passing through that point.
• The process involves manipulating slopes and using known points to determine y-intercepts.

Writing Equations of Parallel Lines

Given a line [latex]f(x) = mx + b[/latex] and a point [latex](x_0, y_0)[/latex], to write the equation of a parallel line through the point:

1. Identify the slope [latex]m[/latex] of the original line.
2. Use point-slope form with the new point: [latex]y - y_0 = m(x - x_0)[/latex]
3. Simplify to slope-intercept form: [latex]y = mx + (y_0 - mx_0)[/latex]

Writing Equations of Perpendicular Lines

Given a line [latex]f(x) = mx + b[/latex] and a point [latex](x_0, y_0)[/latex], to write the equation of a perpendicular line through the point:

1. Find the negative reciprocal of the slope: [latex]m_{\perp} = -\frac{1}{m}[/latex]
2. Use point-slope form with the new point: [latex]y - y_0 = m_{\perp}(x - x_0)[/latex]
3. Simplify to slope-intercept form: [latex]y = m_{\perp}x + (y_0 - m_{\perp}x_0)[/latex]