Finding and Comparing Rates of Change (Slope)
Grade 7 students in Big Ideas Math Advanced 2 (Chapter 6: Functions) learn to find and compare rates of change (slope) from graphs and equations. A larger absolute value of slope indicates a steeper line and faster rate of change, allowing students to compare how quickly different functions increase or decrease.
Key Concepts
Property The rate of change (or slope, m) describes how much the output (y) changes for every one unit increase in the input (x). From a Graph: Calculate the "rise over run" between any two points. Comparing Slopes: The absolute value of the slope indicates how quickly the function values change. A larger absolute value means a steeper line and a faster rate of change.
Examples Comparing Speeds: Compare f(x) = 2x + 1 and g(x) = 1/3x + 1. Since 2 is greater than 1/3, function f changes more rapidly than function g. Comparing Decreases: Compare h(x) = 3x + 5 and k(x) = x + 5. Since the absolute value of 3 is greater than the absolute value of 1, function h decreases more rapidly than function k.
Explanation The slope is the "speed limit" of your function. It tells you the rate at which a quantity is changing, such as cost per hour or distance per minute. When comparing two functions, the one with the steeper slope (larger absolute value, whether positive or negative) is the one changing the fastest.
Common Questions
What is the rate of change in a linear function?
The rate of change (slope, m) describes how much the output y changes for every one-unit increase in input x. It is calculated as rise over run between any two points on the line.
How do you compare rates of change between two functions?
Compare the absolute values of their slopes. The function with a larger absolute value of slope changes more rapidly. For example, f(x) = 2x changes faster than g(x) = (1/3)x.
What does a negative rate of change mean?
A negative rate of change means the function is decreasing: as x increases, y decreases. The absolute value still tells you how steeply it decreases.
What chapter in Big Ideas Math Advanced 2 covers rates of change?
Chapter 6: Functions in Big Ideas Math Advanced 2 (Grade 7) covers finding and comparing rates of change (slope).
How do you find slope from a graph?
Pick two clear points on the line. Calculate rise over run: (change in y) divided by (change in x). This ratio is the slope or rate of change.