Mathematics
Grade 8
15 min
Write linear functions: word problems
Write linear functions: word problems
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1
Introduction & Learning Objectives
Learning Objectives
Identify independent and dependent variables in a word problem.
Determine the rate of change (slope) from a word problem.
Identify the initial value (y-intercept) from a word problem.
Write a linear function in slope-intercept form ($y = mx + b$) from a given word problem.
Interpret the meaning of the slope and y-intercept in the context of a word problem.
Use a linear function to make predictions or solve related questions from a word problem.
Ever wonder how companies predict their profits or how much gas you'll use on a road trip? 🚗💨 Linear functions help us model these real-world situations!
In this lesson, you'll learn how to translate everyday scenarios described in word problems into mathematical equations called linear functions. This...
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Key Concepts & Vocabulary
TermDefinitionExample
Linear FunctionA relationship between two variables where a constant change in one variable results in a constant change in the other. When graphed, it forms a straight line.The total cost of a gym membership is $20 per month plus a $50 sign-up fee. The cost increases linearly with each month.
Independent VariableThe variable whose value can be chosen or changes independently; it causes a change in another variable. It is typically represented by 'x' and plotted on the horizontal axis.In the gym membership example, the 'number of months' is the independent variable.
Dependent VariableThe variable whose value depends on the independent variable. It is typically represented by 'y' and plotted on the vertical axis.In the gym membership exam...
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Core Formulas
Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Used to calculate the slope (rate of change) of a line when given two specific data points $(x_1, y_1)$ and $(x_2, y_2)$ from a word problem. This helps determine 'm' when the rate isn't directly stated.
Slope-Intercept Form
$y = mx + b$
This is the standard form for writing linear functions from word problems. 'm' represents the slope (rate of change), and 'b' represents the y-intercept (initial value). Your goal is to identify 'm' and 'b' from the problem and substitute them into this equation.
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Challenging
A city's population was 25,200 in 2005 and grew to 28,000 in 2015. Assuming a constant rate of growth, write a linear function P(t) for the population, where t is the number of years since 2000, and use it to predict the population in 2025.
A.30,800
B.30,100
C.31,500
D.32,200
Challenging
A pot of water is cooling. After 4 minutes, its temperature is 84°C. The water cools at a constant rate of 3°C per minute. What was the initial temperature of the water when cooling began (at 0 minutes)?
A.72°C
B.87°C
C.96°C
D.100°C
Challenging
A full 30-gallon hot tub is draining at a rate of 2.5 gallons per minute. The function W = -2.5m + 30 models the water level. What does the x-intercept of this function's graph represent in the context of the problem?
A.The initial amount of water in the tub.
B.The rate at which the water is draining.
C.The time it takes for the tub to be completely empty.
D.The amount of water left after 1 minute.
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