Linear Functions, Slope, and Initial Value
Students construct linear functions and interpret slope and initial value from equations, tables, graphs, and real-world contexts.
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Functions and Input-Output Rules
A function assigns exactly one output to each input. An input-output rule tells how to calculate the output, often using variables such as x for the input and y for the output. For example, the rule y = 3x + 2 means multiply the input by 3, then add 2. If x = 0, then y = 2. If x = 1, then y = 5, and if x = 2, then y = 8. These ordered pairs, (0, 2), (1, 5), and (2, 8), can be placed in a table or plotted on a graph. Because every x-value has only one y-value and the outputs change at a constant rate, this rule defines a linear function. A function may be represented by a description, equation, table, graph, or mapping diagram.
Rate of Change
The rate of change describes how much the output changes for each one-unit change in the input. For a linear function, this rate is constant. It can be found from a table by dividing the change in y by the corresponding change in x. Suppose a water tank contains 20 gallons and fills at 5 gallons per minute. After 0, 1, 2, and 3 minutes, it contains 20, 25, 30, and 35 gallons. Each time the input increases by 1 minute, the output increases by 5 gallons. The rate of change is 5 gallons per minute. If the amount decreased by 5 gallons each minute instead, the rate would be negative. Units are important because they explain what the rate means in the situation.
Finding Slope
Slope is the rate of change of a line. Between two points, slope is calculated as m = change in y divided by change in x. The vertical change is called the rise, and the horizontal change is called the run. Consider the points (1, 2) and (4, 8). Moving from the first point to the second gives a rise of 8 − 2 = 6 and a run of 4 − 1 = 3. Therefore, m = 6 ÷ 3 = 2. A positive slope rises from left to right, while a negative slope falls from left to right. A horizontal line has slope 0. A vertical line has undefined slope because its run is 0, and division by zero is not defined. Always subtract coordinates in the same order.
Initial Value and Intercepts
The initial value of a linear function is the output when the input is 0. In the equation y = mx + b, the initial value is b. It is also the y-intercept, the point where the graph crosses the y-axis. For example, y = −2x + 6 has an initial value of 6 because y = 6 when x = 0. Its y-intercept is (0, 6). In a real-world situation, this value represents the starting amount. If a candle is 6 inches tall and burns 2 inches per hour, 6 is its initial height and −2 is its hourly rate of change. The x-intercept has a different meaning: it occurs where y = 0. For this candle model, the x-intercept is (3, 0), meaning the modeled height reaches 0 after 3 hours.
Building Linear Models
To build a linear model, identify the constant rate of change and the initial value, then write the equation y = mx + b. Suppose a bicycle rental costs an initial fee of $8 plus $4 for each hour. The input x is the number of hours, and the output y is the total cost in dollars. The hourly rate is 4, so m = 4. The cost at 0 hours is $8, so b = 8. The model is y = 4x + 8. For a 3-hour rental, y = 4(3) + 8 = 20, so the total cost is $20. This model is linear because equal increases in rental time cause equal increases in cost. The model should be used only for reasonable input values, such as nonnegative rental times.
Comparing Representations
The same linear function can be shown with words, a table, a graph, or an equation. To compare representations, identify the slope and initial value in each one. Consider Function A, given by y = 2x + 5. Its slope is 2 and its initial value is 5. Function B is shown in a table with points (0, 1), (2, 7), and (4, 13). Its initial value is 1. Its slope is (7 − 1) divided by (2 − 0), which equals 3. Function B starts lower but grows faster because 3 is greater than 2. On a graph, Function B would be steeper. Comparing both values matters: the function with the greater slope does not always have the greater output because its initial value may be smaller.