Graphs can show two types of relationships: direct proportion and inverse proportion. Let’s break them down:

Direct Proportion

• What It Means: In direct proportion, when one thing goes up, the other thing goes up too. You can think of it like this: if you have more of something, you also get more of the other thing. The way to write this is $y = kx$, where $k$ is a number that stays the same.

• How It Looks on a Graph:

  • Shape: The graph is a straight line that starts at the point (0,0).
  • Slope: The steepness of this line tells you the value of $k$.

• Example: If $k = 3$, this means for every time $x$ increases by 1, $y$ goes up by 3.

Inverse Proportion

• What It Means: In inverse proportion, when one thing goes up, the other thing goes down. It's like saying that if you have more of one, you have less of the other. You can write it like this: $y = \frac{k}{x}$.

• How It Looks on a Graph:

  • Shape: The graph forms a curve that gets closer to the axes but never actually touches them.
  • Asymptotic Behavior: This means the graph will keep getting closer to the axes but won’t ever meet them.

• Example: If $k = 12$, when $x$ goes up to 3, then $y$ goes down to 4.

In summary, direct proportion means both things increase together, while inverse proportion means one goes up and the other goes down. The graphs help us see these relationships more clearly!