Linear functions

Learning Objectives Derive the equation of a linear function given various conditions, including parallel and perpendicular constraints. Analyze and interpret the slope as a constant rate of change in mathematical and real-world contexts. Relate the concept of the slope of a linear function to the first derivative. Model real-world scenarios using linear functions and use these models to make predictions. Transform linear equations between slope-intercept, point-slope, and general forms. Determine the geometric relationship between two lines (parallel, perpendicular, or intersecting) by analyzing their equations. If a car travels at a constant 60 mph, its distance is a simple function of time. But how does this relate to the tangent line problems you're solving in calcu...

TermDefinitionExample Linear FunctionA function of the form f(x) = mx + b, where 'm' and 'b' are real constants. Its graph is a straight line.f(x) = -2x + 5 is a linear function where the slope is -2 and the y-intercept is 5. Slope (m)The measure of the steepness of a line, representing the constant rate of change. For any two distinct points (x₁, y₁) and (x₂, y₂), the slope is m = (y₂ - y₁) / (x₂ - x₁).For the points (1, 3) and (3, 7), the slope m = (7 - 3) / (3 - 1) = 4 / 2 = 2. This means for every 1 unit increase in x, y increases by 2 units. y-intercept (b)The point where the graph of the function crosses the y-axis. It is the value of the function when x = 0.In the function y = 3x - 4, the y-intercept is -4. The graph passes through the point (0, -4). Parallel Li...

Slope-Intercept Form y = mx + b Most useful for quickly identifying the slope (m) and y-intercept (b) of a line. It is the standard form for writing the final equation of a function. Point-Slope Form y - y₁ = m(x - x₁) Used when you know the slope (m) and a single point (x₁, y₁) on the line. It is the most direct way to write the equation of a line from this information before simplifying to slope-intercept form. General Form Ax + By + C = 0 A form where A, B, and C are integers, and A is typically non-negative. It is useful for certain applications like finding the distance from a point to a line, but requires rearrangement to easily find the slope.