≡Mathematics EN 9th grade: translate this: 1. Grandezas inversamente proporcionais
1. Inversely Proportional Quantities
Two quantities are inversely proportional when:
One increases → the other decreases
The product between them is constant
👉 Mathematically:
If 𝑥 and 𝑦 are inversely proportional, then:
\( x \cdot y = k \)
where 𝑘 is a constant.
✔ This means that:
\( y = \frac{k}{x} \)
2. Inverse Proportionality Function
It is a function of the type:
\( y = \frac{k}{x} \)
Where:
𝑥 ≠ 0
𝑘 is a constant (fixed value)
𝑦 depends on 𝑥
📌 Main characteristics:
The graph is a curve (hyperbola)
It does not touch the 𝑥 and 𝑦 axes
The larger 𝑥, the smaller 𝑦
3. Practical Example 1
A quantity y is inversely proportional to x, with k = 12.
Function:
\( y = \frac{12}{x} \)
Examples:
When x = 1, then \( y = \frac{12}{1} = 12
When x = 2, then \( y = \frac{12}{2} = 6
When x = 3, then \( y = \frac{12}{3} = 4
When x = 4, then \( y = \frac{12}{4} = 3
✔ It is observed that:
When x increases, y decreases
The product x ⋅ y = 12
4. Practical Example 2 (real context)
A car travels a fixed distance. Speed and time are inversely proportional:
\( \text{speed} \cdot \text{time} = \text{distance} \)
If the distance is constant, then:
\( t = \frac{k}{v} \)
👉 If the speed increases, the time decreases.
Example:
If a car goes at 60 km/h it takes 2 hours
If it increases to 120 km/h, it takes 1 hour
5. How to identify it on the exam
✔ Says inversely proportional
✔ A formula like \( y = \frac{k}{x} \) appears
✔ Product between variables is constant
✔ Graph in the form of a curve
6. Quick summary (for review)
Function: \( y = \frac{k}{x} \)
Constant product: 𝑥 ⋅ 𝑦 = 𝑘
When one increases, the other decreases
Graph: curve (hyperbola)
Very common in problems with time, speed and work
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