反比：曲线、求斜率

Inversely Proportional Graphs

Let's take a look at some directly proportional graphs when $k=1$, $k=5$, and $k=10$.

These graphs are not linear. Instead they're hyperbolic. These graphs get infinitesimally close to the $x$ and $y$ axes, but they never touch them. If a graph gets close but never touches a line, we call that an asymptote. Inversely proportional equations will always have an asymptote at $x=0$ and $y=0$.

Notice also that the $x$ and $y$ values of each point multiply to equal $k$.

If we write the above equations out, we get: $$y=\frac{{\color{#bf4f4c}{1}}}{x}\quad\quad y=\frac{{\color{#4079b2}{5}}}{x}\quad\quad y=\frac{{\color{#419d51}{10}}}{x}$$

Example

The following point is located on an inversely proportional graph.

Are you able to draw the rest of the graph?

Yes! We're told that the graph is inversely proportional so it has the form: $$y=\frac kx$$ We can rearrange this equation to find $k$. $$k=xy$$ So, $k$ can be found by multiplying the $x$- and $y$-coordinate. $$\begin{align} k&=(13)(6)\\ k&=78 \end{align}$$

We now know the equation of the graph. $$y=\frac{78}x$$ We can either use a graphing calculator, or plot points. Our graph will come out to be:

Practice Problem

Below is a portion of an inversely proportional graph.

Solution

We're told the graph is inversely proportional, so we know its equation is of the form: $$y=\frac kx$$

Recall that $k$ is the product of the $x$- and $y$-coordinate. Using the obvious point in the graph above, we see $k$ is: $$\begin{align} k&=xy\\ k&=(30)(4)\\ k&=120 \end{align}$$

The correct answer will be the coordinates whose product is also $120$.

• $10\times12=120$ This is the correct answer!
• $0\times0=0$ This is not on our graph. Inversely proportional graphs never reach $0$ on any axis.
• $40\times30=1200$ This is not on our graph.
• $6\times15=90$ This is not on our graph.

Plot of the answer choices along with a zoomed out version of the graph:

To review how to graph functions, check out this lesson. For more practice with continuous data and graphs, check out this lesson. For more practice with evaluation functions, check out this lesson.

(Video) Inverse Variation

by HCCMathHelp

This video by HCCMathHelp goes over how to solve inverse variation problems.

Summary

Inverse variation has a lot in common with direct variation, and the formula looks pretty similar.

Here's the formula for inverse variation:

$$y=k/x$$

, where $k$ is the constant of variation or constant of proportionality.

You can read this formula like: $y$ varies inversely as $x$ or $y$ is inversely proportional to $x$.

Example 1: If $y$ varies inversely as $x$ and $y=4$ when $x=6$, find $y$ if $x=16$.

The example problem says varies inversely, so we know it's an inverse variation problem.

We now that the inverse variation formula is $y=k/x$. If we plug in the first set of known values,

$$4 = k/6$$

Therefore, $$24=k$$

Let's plug in the $24$ for $k$ back in the general formula:

$$y=24/x$$

The problem is asking us to find $y$ when $x=16$, so:

$$y=24/16$$

Therefore, $$y=3/2 = 1.5$$

Now you try an inverse variation problem on your own!

Example 2: If $y$ is inversely proportional to $x^3$ and $y=1/2$ when $x=2$, find $y=3$.

Inverse Variation

Inverse variation is another way of describing a relationship between two variables. While direct variation shows a direct relationship, inverse variation shows an inverse variation. That means that as one variable increases, the other one decreases.

What are some examples of inverse variation problems?

Perhaps we could say that as the number of hours of TV we watch increases, the number of hours we spend doing homework decreases. This is an example of inverse variation!