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Identifying Linear Relationships
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Today we're going to talk about how to identify linear relationships in word problems. Can anyone tell me what a linear function is?
Is it something that graphs as a straight line?
Exactly! A linear function graphs as a straight line and can be expressed in the form `y = mx + c`. Now, what about some real-life examples where we might see these linear functions?
Like how a taxi charges a flat fee and then per mile?
Great example! If the taxi charges $5 for the first mile and $2 for each additional mile, we can express this as a linear function. Can you represent this mathematically?
It would be `C(x) = 2x + 5` where `x` is the number of miles.
Perfect! Always remember to identify the constant and variable in such scenarios. Now, let's summarize: identifying keywords in problems can give us clues about the variables involved.
Translating Scenarios to Equations
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Now that we've identified linear functions, let's practice translating word problems into equations. For example, 'A plant grows 3 cm per day. At day 0, it’s 5 cm tall.' How can we turn this into a linear function?
I think we would express height as `H(x) = 3x + 5`?
That's correct! Here, `x` represents days, and the plant starts at 5 cm, growing 3 cm each day. Any questions about how to identify the variables here?
What if there were more variables, like multiple plants growing at different rates?
Excellent question! In that case, you’d set equations for each plant separately but still follow the same principle. Now, let’s recap: identifying key phrases is crucial to understanding and translating the problems correctly.
Problem-Solving with Linear Equations
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Now we will solve some word problems! Here’s one: 'A mobile plan costs $10 per month plus $0.05 per text. Can we write a function for total cost?'
I think it’s `C(x) = 0.05x + 10`, where `x` is the number of texts.
Exactly! Now, if a user sends 200 texts in a month, what is the total cost?
So, we calculate it as `C(200) = 0.05 * 200 + 10`, which equals $20.
What if someone sends more texts? Does the cost change linearly?
Yes! That’s the beauty of linear functions—they model these kinds of relationships consistently. Let’s summarize: writing down equations helps us systematically find solutions.
Application and Practice
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Let's review everything we've learned! We talked about how to identify, translate, and solve word problems using linear functions. Can anyone provide an example or practice problem?
How about a situation like, 'A car rental company charges a base fee plus extra per mile.'
Great! We can represent that with `C(x) = base fee + rate * x`. What is the base fee and rate in your scenario?
Let’s say it’s $50 per day plus $0.25 per mile.
Exactly! So, `C(x) = 0.25x + 50`. Now, let’s recap what we’ve learned: identifying scenarios, creating equations, and solving them step by step!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The Word Problems section covers the identification of linear relationships in real-life scenarios and how to translate these relationships into mathematical expressions, leading to problem-solving through linear equations.
Detailed
Word Problems in Linear Functions
In this section, we delve into the concept of word problems that can be modeled with linear functions. A word problem often describes a real-life scenario where relationships between variables can be expressed linearly. By transforming these scenarios into equations, we can solve for unknowns systematically.
Key Concepts:
- Linear Function Representation: Understand how to identify and express situations in the format of linear functions, such as
y = mx + c. - Real-Life Applications: Familiarize yourself with common occurrences where linear functions apply, including growth rates and costs tied to usage, as exemplified by plant growth rates and mobile plans.
- Translating Word Problems: Learn techniques for converting text-based descriptions into mathematical representations. This section will enhance your ability to think mathematically when faced with day-to-day challenges.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Example 1: Plant Growth
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Example 1:
"A plant grows 3 cm per day. On day 0, it is 5 cm tall."
Height = 3𝑥 + 5
Where 𝑥 is the number of days.
Detailed Explanation
In this example, we are modeling the height of a plant as a linear function. The height of the plant increases by 3 cm each day. This means for every day that passes, we add 3 cm to the initial height of 5 cm. The function is represented as 'Height = 3𝑥 + 5', where '𝑥' represents the number of days. So if you want to know how tall the plant is after 5 days, you substitute '5' for '𝑥': Height = 3(5) + 5 = 15 cm.
Examples & Analogies
Imagine you have a houseplant that you are measuring every day. When you first bought the plant, it was 5 cm tall. Each day you notice that it grows a little taller, by 3 cm each time. If you keep track of how tall it is each day using this linear function, you can see how quickly your plant grows over the week or month!
Example 2: Mobile Plan Costs
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Example 2:
"A mobile plan costs $10/month plus $0.05 per text. Write a linear function for total cost."
𝐶(𝑥) = 0.05𝑥 + 10
Detailed Explanation
This example focuses on creating a linear function to model the total cost of a mobile plan based on two variables: a fixed cost of $10 per month and a variable cost of $0.05 for each text message sent. Here, '𝐶(𝑥)' represents the total cost, where '𝑥' is the number of texts sent. The equation '𝐶(𝑥) = 0.05𝑥 + 10' means that for every text you send, you will add $0.05 to the initial $10 monthly fee, allowing you to calculate how much your total bill will be each month.
Examples & Analogies
Think of your monthly phone bill as you combine both a flat rate and variable usage. You have a plan that charges you $10 for the basic service and an additional charge for each text you send. If you send a few texts, your bill isn't just the flat fee—it's that fee plus a little extra for every message you've sent. This helps you understand why keeping track of your texts is essential if you want to anticipate your total monthly bill!
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Linear Function Representation: Understand how to identify and express situations in the format of linear functions, such as
y = mx + c. -
Real-Life Applications: Familiarize yourself with common occurrences where linear functions apply, including growth rates and costs tied to usage, as exemplified by plant growth rates and mobile plans.
-
Translating Word Problems: Learn techniques for converting text-based descriptions into mathematical representations. This section will enhance your ability to think mathematically when faced with day-to-day challenges.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
If a car costs $2000 to rent plus $0.20 per mile, the function can be expressed as
C(x) = 0.20x + 2000. -
For a bank that charges $5 monthly plus $0.50 for every additional transaction, the function can be expressed as
C(x) = 0.50x + 5.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
For cost and growth we need to know, / Just write the function and let it flow.
📖 Fascinating Stories
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Imagine a gardener keeping track of his plants; each day, they grow taller. If he starts at 5 cm and each day adds more, he can easily predict their height using a simple equation.
🧠 Other Memory Gems
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F.L.O.W. – Find the linear function, Label the variables, Organize the equation, Write the solution.
🎯 Super Acronyms
C.A.R.E. – Costs And Rates Equate; helps us remember how to build equations from costs.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Linear Function
Definition:
A function whose graph is a straight line, expressed in the form
y = mx + c. -
Term: Variables
Definition:
Symbols used to represent numbers in equations, typically
xfor independent andyfor dependent. -
Term: Intercepts
Definition:
Points where the line crosses the axes; y-intercept is where x=0, and x-intercept is where y=0.
-
Term: Cost Function
Definition:
A type of linear function that models the total cost related to a specific activity or usage.
-
Term: Growth Rate
Definition:
The speed at which a quantity, such as a plant's height, increases over time.