Question

Given that \(y\) is inversely proportional to \(x\), state which of the following are true and which are false: (a) when \(x\) is doubled, \(y\) is doubled also (b) \(x\) is inversely proportional to \(y\) (c) when \(x\) is halved, \(y\) is doubled (d) a graph of \(y\) against \(x\) is a straight line with a negative gradient

Short answer

Question: Determine the truth or falsity of the following statements given that y is inversely proportional to x: a) When x is doubled, y is doubled also. b) x is inversely proportional to y. c) When x is halved, y is doubled. d) A graph of y against x is a straight line with a negative gradient. Answer: (a) False, (b) True, (c) True, (d) False

Step-by-step solution

Write the inverse proportionality relationship

We are given that \(y\) is inversely proportional to \(x\). This means that \(y = \dfrac{k}{x}\), where k is the constant of proportionality.

Analyze statement (a)

(a) states that when \(x\) is doubled, \(y\) is doubled also. Let's see if this is true by setting \(x'=2x\). Then, \(y' = \dfrac{k}{x'} = \dfrac{k}{2x}\). In this case, \(y' = \dfrac{1}{2}y\), which indicates that when \(x\) is doubled, \(y\) is actually halved, not doubled. Therefore, statement (a) is false.

Analyze statement (b)

(b) states that \(x\) is inversely proportional to \(y\). We can rewrite the given relationship as \(x = \dfrac{k}{y}\). This shows that \(x\) is indeed inversely proportional to \(y\). Therefore, statement (b) is true.

Analyze statement (c)

(c) states that when \(x\) is halved, \(y\) is doubled. Let's see if this is true by setting \(x' = \dfrac{x}{2}\). Then, \(y' = \dfrac{k}{x'} = \dfrac{k}{\frac{x}{2}} = 2k/x = 2y\). This indicates that when \(x\) is halved, \(y\) is actually doubled. Therefore, statement (c) is true.

Analyze statement (d)

(d) states that a graph of \(y\) against \(x\) is a straight line with a negative gradient. We can rewrite the given relationship as \(y = \dfrac{k}{x}\), which is the equation for a hyperbola, not a straight line. Therefore, statement (d) is false. To summarize: (a) is false (b) is true (c) is true (d) is false

Key concepts

Mathematical Relationships

In mathematics, understanding different types of relationships between variables is fundamental. One such relationship is inverse proportionality, which describes how one variable changes in the opposite direction of another. Specifically, if two quantities are inversely proportional, as one increases, the other decreases at a rate such that their product remains constant.

An inverse relationship can be represented algebraically by an equation of the form \( y = \frac{k}{x} \), where \( y \) and \( x \) are the variables and \( k \) is the constant of proportionality. This concept is crucial for students to grasp because it shows up in many fields, such as physics, economics, and engineering.

To understand inverse proportion better, a helpful exercise is to consider what happens when the value of one variable changes. For example, if the value of \( x \) is doubled, the value of \( y \) is halved, ensuring the product of the two variables remains unchanged, maintaining the constant of proportionality, \( k \). This can be visualized not as a direct one-to-one change but as a 'seesaw' effect, where the increase in one quantity corresponds to a reciprocal decrease in the other.

Constant of Proportionality

In the context of inverse proportionality, the constant of proportionality, represented by \( k \), plays a pivotal role. It is the fixed number that relates two variables inversely. If one variable is equal to the constant divided by the other variable, the two are said to be inversely proportional.

For instance, if we state that \( y \) is inversely proportional to \( x \), and we know the constant of proportionality, we can always find the value of \( y \) for any given \( x \) using the equation \( y = \frac{k}{x} \). This relationship is symmetrical, which means that it also holds true that \( x \) is inversely proportional to \( y \) with the same constant \( k \), as \( x = \frac{k}{y} \).

The constant of proportionality is unique for every relationship and must be determined from known values of the variables. It's what makes proportional relationships predictable and allows us to create equations that model real-world inverse relationships, like the intensity of light decreasing with the square of the distance from the source.

Hyperbola Graph Characteristics

When graphing inverse proportional relationships, the resulting shape is known as a hyperbola. A hyperbola has distinct characteristics that distinguish it from other types of graphs, like straight lines or parabolas. Its most recognizable feature is its two separate curved branches, which open in opposite directions away from each other.

In the case of an inverse proportionality equation \( y = \frac{k}{x} \), the graph will not be a straight line with a negative gradient but rather a curve that approaches the axes asymptotically—meaning it gets closer and closer to the axes but never actually touches them. The constant of proportionality, \( k \), affects the shape and position of the hyperbola on the graph. If \( k \) is positive, the branches of the hyperbola will be in the first and third quadrants; if \( k \) is negative, they will be in the second and fourth quadrants.

The distinct 'inverse' curve of a hyperbola is useful when visualizing the inverse relationship between two variables. This visual representation helps students to better understand the concept of inversely proportional quantities and also confirms why the graph of two inversely proportional variables cannot be a straight line.