MYP Standard Mathematics 2.3 Proportion Notes

Proportion Models Real Relationships

In mathematics, many situations are described by how two variables change together.

A proportional relationship is a particularly useful type of relationship because it has a simple algebraic form, a recognizable graph, and predictable scaling.

In a relationship, one variable is often the independent variable (the input you choose or control) and the other is the dependent variable (the output that changes as a result).

Direct Proportion Means A Constant Ratio

1. Two variables $x$ and $y$ are in direct proportion if, and only if, their ratio is constant for all values (with $x\neq 0$): $$\frac{y}{x}=k$$
2. This is written $y\propto x$ and means $$y=kx \quad (k\neq 0)$$
3. The function $y=kx$ is called a linear variation function.

How To Recognize Direct Proportion

There are three equivalent ways to identify direct proportion.

1. Table test (constant ratio): compute $y/x$ for several pairs; it should always equal the same number $k$.
2. Equation form: it can be written in the form $y=kx$ with no extra constant added.
3. Graph test: it is a straight line that passes through the origin $(0,0)$.

Meaning Of $k$ On The Graph

1. On a graph of $y=kx$, the constant $k$ is the gradient (slope): $$k=\frac{\Delta y}{\Delta x}$$
2. So $k$ tells you how many units $y$ changes for each 1 unit change in $x$.

Solving Direct Proportion Problems

Most direct proportion questions follow the same method.

1. Write $y=kx$.
2. Substitute a known pair $(x,y)$ to find $k$.
3. Use the equation to find the unknown value.

Scaling Rules In Direct Proportion

If $y=kx$, then multiplying $x$ by a factor multiplies $y$ by the same factor.

Inverse Proportion Means A Constant Product

1. Two variables $x$ and $y$ are inversely proportional if multiplying one by a non-zero number results in the other being divided by the same number.
2. This is written $y\propto \tfrac{1}{x}$ and means $$y=\frac{k}{x} \quad (k\neq 0,\ x\neq 0)$$
3. A key equivalent condition is that the product is constant: $$xy=k$$
4. Also, if $y$ is inversely proportional to $x$, then $y$ is directly proportional to $\tfrac{1}{x}$.

How The Graph Looks

1. For $k>0$, the graph of $y=\tfrac{k}{x}$ is a curved reciprocal relationship (a rectangular hyperbola).
2. The curve approaches the axes but does not touch them (the axes are asymptotes).
3. In many real situations, you only use positive values, so you focus on the first-quadrant branch.

Direct Non-Linear Proportion Uses Powers

Sometimes a variable is proportional to a power of another variable.

If $y=kx^n$ and you multiply $x$ by a constant factor $c$, then $y$ is multiplied by $c^n$.

What The Exponent Changes

1. If $n=2$, then doubling $x$ multiplies $y$ by $2^2=4$.
2. If $n=3$, then tripling $x$ multiplies $y$ by $3^3=27$.
3. The graph shape (for $k>0$) depends on whether $n$ is even or odd:
   1. even $n$ gives a U-shaped curve (symmetric about the $y$-axis)
   2. odd $n$ gives a curve through the origin (symmetric about the origin)

Negative Proportion Versus Inverse Proportion

1. Sometimes two variables move in opposite directions in a linear way.
2. This is often described as negative proportionality.
3. Negative proportion is still "direct" in structure because the ratio $\frac{y}{x$ is constant, but that constant is negative.

Key Differences To Remember

1. Negative proportion: $y=-kx$, straight line through the origin, constant ratio $\frac{y}{x}=-k$.
2. Inverse proportion: $y=\tfrac{k}{x}$, curved graph, constant product $xy=k$.

Using Data To Verify Proportion

When you are given a table of values, you can test for proportionality and find $k$.

Direct Proportion From A Table

Check whether $\frac{y}{x}$ is constant.

Inverse Proportion From A Table

1. Check whether $xy$ is constant.
2. If the products are the same (or very close, allowing for rounding), then $y\propto \tfrac{1}{x}$.

Proportion And Function Notation

1. You will often see proportionality expressed using function notation, which highlights input and output clearly.
2. For example, writing $P(l)=1.50l$ emphasizes that price $P$ depends on liters $l$.
3. In any proportional model:
   1. the dependent variable is the function value (output)
   2. the independent variable is the input variable