13.2 - Some Applications
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Interactive Audio Lesson
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Independent and Dependent Variables
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Today, we're going to learn about independent and dependent variables. Can someone explain what they understand by these terms?
I think the independent variable is the one that can change, right?
Correct! The independent variable is the one we control, while the dependent variable depends on it. Can anyone give me an example?
Like how the more electricity I use, the higher my bill is?
Exactly! In this case, electricity usage is independent, and the bill is dependent. We can visualize this relationship through graphs.
How do we plot this on a graph?
Great question! We’ll plot the independent variable on the x-axis and the dependent variable on the y-axis. This way, we can visualize how changes in one affect the other.
To summarize, independent variables are what we control, while dependent variables are affected by these controls. We will encounter this concept multiple times in our examples.
Graphing Quantity and Cost
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Now let's look at an example involving petrol. If we have the quantity of petrol and its cost, can anyone tell me how we can plot this?
We would put the number of litres on the x-axis and the cost on the y-axis?
Correct! And when we plot the points from the table we have, what do we notice?
It makes a straight line! Does that mean it’s directly proportional?
Yes, excellent observation! This linearity reflects direct variation. In such cases, as one quantity increases, the other does too. Can anyone think of another example where we see a similar relationship?
Simple Interest? The more money we deposit, the higher the interest earned.
Exactly! Great example! Both the quantity of deposit and interest are linear, and we can display this on a graph, too.
To conclude, our lessons on quantity and cost will help us interpret various real-life data through graphing!
Practical Applications of Time and Distance
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Let’s apply everything we’ve discussed to time and distance. If Ajit rides his scooter at a constant speed, how can we represent this relationship graphically?
We plot time on the x-axis and distance on the y-axis, right?
Correct! As time goes on, distance will increase linearly. How can we calculate the distance for a certain amount of time?
We can just multiply the speed by the time!
Right again! So if he rides for 3 hours, what distance does he cover?
He covers 90 kilometers since he goes 30 kilometers per hour!
Fantastic! In summary, we use graphs to visualize relationships between different quantities, enhancing our understanding of everyday situations like traveling or spending.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, we discuss how certain quantities, such as electricity usage and its corresponding bill, exemplify the relationship between independent and dependent variables. The section features examples, graph plotting, and concepts such as linear relationships and direct variation.
Detailed
In this section, we highlight several applications of graphs in everyday situations. The section opens with a discussion on how utility expenses, such as electricity bills, relate to the quantities consumed, identifying the independent variable (amount of electricity used) and the dependent variable (the resulting bill). Through engaging examples, such as the cost of petrol and banking interests, students learn to plot linear graphs representing direct relationships. Each example emphasizes how linear graphs can illustrate data trends, allowing for better comprehension. The section concludes with practice exercises and concepts reinforcing understanding of how to analyze and interpret relationships through graphical data representation.
Youtube Videos
Key Concepts
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Independent Variable: The factor that is controlled or manipulated.
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Dependent Variable: The outcome that depends on the independent variable.
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Linear Relationship: A relationship that can be represented by a straight line on a graph.
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Direct Variation: When one variable directly affects another in a proportional manner.
Examples & Applications
In the relationship between electricity usage and billing, more electricity consumed results in higher bills.
The cost of petrol increases linearly with the number of litres purchased.
In simple interest calculations, the interest earned is directly proportional to the amount deposited.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Independent stays in control, dependent follows the goal.
Stories
Imagine a tree; the sun (independent) determines how much fruit (dependent) it grows.
Memory Tools
DAD - Dependent Affects Dependent.
Acronyms
GRA - Graphs Reflect Applications.
Flash Cards
Glossary
- Independent Variable
The variable that is manipulated or controlled in an experiment.
- Dependent Variable
The variable that is measured or affected in an experiment.
- Linear Graph
A graph that represents a straight line, indicating a direct relationship between two variables.
- Direct Variation
A relationship between two variables where they increase or decrease together.
- Graph
A visual representation of data and relationships between variables.
Reference links
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