Linear Functions and Graphing

• A linear function is a function whose graph is a straight line. It can be represented by the equation y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

Key Characteristics of Linear Functions

• The slope (m) represents the rate of change of y with respect to x. A positive slope indicates an upward trend, a negative slope a downward trend, and a zero slope a horizontal line.

• The y-intercept (b) is the point where the line crosses the y-axis. It represents the value of y when x is zero.

• The equation is of the first degree (the highest power of the variable is 1).

Graphing Linear Functions

• To graph a linear function, two points are sufficient. Find two points that satisfy the equation (x, y) and plot them on a coordinate plane. Draw a straight line passing through these two points.

• Alternatively, you can use the slope-intercept form (y = mx + b) directly: plot the y-intercept (0, b), then use the slope (m) to find a second point. Rise over run (m = rise/run) determines the change in y for a given change in x.

Finding the Equation of a Line

• Given two points (x₁, y₁) and (x₂, y₂), the slope can be calculated as m = (y₂ - y₁)/(x₂ - x₁)

• After finding the slope, use the point-slope form, y - y₁ = m(x - x₁), to find the equation. Substitute one of the points and the slope into this form.

Special Cases of Linear Functions

• Horizontal lines: These have a slope of zero, and their equation is of the form y = b.

• Vertical lines: These have an undefined slope (division by zero) and their equation is of the form x = a.

Applications of Linear Functions

• Many real-world situations involve linear relationships. Examples include:
  • Calculating total cost (if there's a constant rate like cost per item).
  • Determining distance traveled if speed is constant.
  • Modeling growth or decay at a fixed rate.

Determining the Slope and Y-Intercept

• Given an equation in the form y = mx + b: the slope is 'm', and the y-intercept is 'b'.

• Sometimes an equation is given in a different form and needs to be rearranged to the slope-intercept form. For example, if you are given 2x + 3y = 6, rearrange to isolate 'y'.

Example Problems (Illustrative)

• Find the slope and y-intercept of the line described by the equation 2x - 4y = 8.

  • Rearrange the equation to slope-intercept form (y = mx + b):

    • 2x - 8 = 4y
    • y = (1/2)x -2

  • The slope is 1/2 and the y-intercept is -2.

• Find the equation of the line passing through the points (3, 1) and (-1, 5).

  • Calculate the slope: m = (5 - 1) / (-1 - 3) = -1.
  • Use the point-slope form: y - 1 = -1(x - 3)
    • Simplify to get the equation: y = -x + 4