Writing a Function from Verbal Descriptions
Writing a Function from Verbal Descriptions is a Grade 8 math skill from Reveal Math, Course 3, Module 5: Functions. To write a linear function y = mx + b from a real-world description, identify the rate of change as the slope (m) and the initial value as the y-intercept (b). Keywords like 'per,' 'each,' and 'every' signal the rate of change, while a fixed starting fee or initial amount becomes the y-intercept. For example, a plumber who charges a $50 flat fee plus $75 per hour gives the equation C = 75h + 50. This skill is essential in 8th grade algebra because it bridges the abstract form of a linear function to practical, real-world situations students encounter in science, business, and everyday life.
Key Concepts
Property To write a linear function $y = mx + b$ from a verbal description, you must identify two key pieces of information: 1. Identify the rate of change as the slope (m). 2. Identify the initial value (or starting point) as the y intercept (b).
Examples A plumber charges a $50 initial fee before any work begins, and then $75 for each hour worked. The rate of change is m = 75, and the initial value is b = 50. The equation is C = 75h + 50. A pool drains at a rate of 5 gallons per minute. After 10 minutes, there are 300 gallons left. The rate is decreasing, so m = 5. Use the point (10, 300) to find b: 300 = 5(10) + b, which means b = 350. The equation is y = 5x + 350.
Explanation When a real world situation is modeled by a linear function, the variables have specific, practical meanings. Look for keywords like "per," "each," or "every" to find your rate of change (slope). The y intercept is simply your fixed starting amount or flat fee before the change begins.
Common Questions
How do you write a linear function from a word problem?
Identify the rate of change (slope m) and the initial value (y-intercept b). The slope is the amount that changes 'per unit' and the y-intercept is the fixed starting value. Then write the function as y = mx + b.
What keywords indicate the slope in a word problem?
Words like 'per,' 'each,' 'every,' 'for each,' and 'rate' signal the slope. For example, '$75 per hour' means m = 75. If the value is decreasing, the slope is negative.
What is the y-intercept in a real-world linear function?
The y-intercept is the starting value before any change happens—it is the value of y when x = 0. In a word problem, it might be a flat fee, initial balance, or starting amount, such as a $50 setup fee before hourly charges begin.
How do you find the y-intercept if it is not given directly?
Use a known point (x, y) from the problem and substitute it into y = mx + b along with the known slope. Solve for b. For example, if a pool has 300 gallons after 10 minutes and drains at 5 gallons per minute, then 300 = -5(10) + b gives b = 350.
When do 8th graders learn to write functions from verbal descriptions?
In Grade 8 Reveal Math Course 3, writing linear functions from verbal descriptions is taught in Module 5: Functions, where students connect real-world contexts to formal function notation.
What is the difference between slope and y-intercept in a context problem?
The slope represents how the output changes for each unit increase in input (the rate). The y-intercept represents the output when the input is zero (the starting point). Both are needed to write a complete linear function.