Linear functions over unit intervals

Learning Objectives Define a unit interval and identify examples. Calculate the change in a linear function's output (Δy) over any given unit interval. Demonstrate that the change in a linear function over any unit interval is constant and equal to its slope. Determine the slope of a linear function given only its change over a unit interval. Write the equation of a linear function using a single point and the change over a unit interval. Interpret the change over a unit interval on the graph of a linear function. Apply the concept of constant change over unit intervals to solve real-world problems. Ever notice how a car on cruise control covers the exact same distance every single minute? 🚗 That's a linear function in action over a 'unit interval' of...

TermDefinitionExample Linear FunctionA function whose graph is a straight line. It can be written in the form f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept.f(x) = 3x + 2 is a linear function with a slope of 3 and a y-intercept of 2. Unit IntervalAn interval on the number line with a length of exactly 1. It is represented as [a, a+1] for any number 'a'.The interval [4, 5] is a unit interval. So are [-1, 0] and [9.5, 10.5]. Slope (m)The 'steepness' of the line, representing the rate of change. It measures how much the y-value changes for each 1-unit increase in the x-value.In f(x) = -2x + 7, the slope m = -2. This means y decreases by 2 for every 1 unit x increases. Rate of ChangeA ratio that describes how one quantity changes with...

Linear Function Form f(x) = mx + b The standard slope-intercept form for a linear function. 'm' represents the slope, and 'b' represents the y-intercept (the value of f(x) when x=0). Change Over a Unit Interval \Delta y = f(x+1) - f(x) = m This is the core rule for this lesson. For any linear function, the change in its value (Δy) over any interval of length 1 (a unit interval) is always equal to the slope 'm'. Slope Formula m = \frac{y_2 - y_1}{x_2 - x_1} The general formula to calculate the slope between any two distinct points (x₁, y₁) and (x₂, y₂). When applied to a unit interval [x, x+1], this simplifies to m = (f(x+1) - f(x)) / ((x+1) - x) = f(x+1) - f(x).