Inverse Proportionality Function
To travel the roughly 240 km from Mexico City to Querétaro, the faster you go, the less time you take: at 60 km/h, 4 hours; at 80 km/h, 3 hours; at 120 km/h, 2 hours. The product speed × time is always 240. This relationship, which you already know from inverse proportionality in Secondary Year 2, has a very distinctive graph: the hyperbola.
The function $y = \frac{k}{x}$
In a table, you can recognize it because multiplying each $x$ by its $y$ always gives the same result: if $x$ doubles, $y$ is cut in half.
| speed $x$ (km/h) | 60 | 80 | 120 |
|---|---|---|---|
| time $y$ (h) | 4 | 3 | 2 |
| $x \times y$ | 240 | 240 | 240 |
Change $k$, even to negative values:
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If $k > 0$: both branches are in quadrants I and III, and the function decreases on each branch.
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If $k < 0$: the branches are in quadrants II and IV, and the function increases on each branch.
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The greater $|k|$ is, the farther from the origin the branches lie.
Consider the function $y = \frac{12}{x}$. a) What is the value of the constant of inverse proportion $k$? b) Copy and complete the table in your notebook for $x = 1$,…
Consider the function $y = \frac{12}{x}$.
- a) What is the value of the constant of inverse proportion $k$?
- b) Copy and complete the table in your notebook for $x = 1$, $2$, $3$, $4$, $6$ and $12$.
- c) What happens to the values of $y$ as $x$ gets larger?
Without plotting them, state which quadrants the branches of each of these hyperbolas lie in and reason why: a) $y = \frac{7}{x}$ b) $y = \frac{-4}{x}$ c)…
Without plotting them, state which quadrants the branches of each of these hyperbolas lie in and reason why:
- a) $y = \frac{7}{x}$
- b) $y = \frac{-4}{x}$
- c) $y = \frac{1}{5x}$
What determines which quadrants a hyperbola $y = \frac{k}{x}$ occupies?
The figure shows the hyperbolas $y = \frac{3}{x}$ and $y = \frac{12}{x}$ for positive values of $x$, as shown below. a) Match each curve with its equation. b) What…
The figure shows the hyperbolas $y = \frac{3}{x}$ and $y = \frac{12}{x}$ for positive values of $x$, as shown below.
- a) Match each curve with its equation.
- b) What effect does the value of $k$ have on the position of the hyperbola relative to the origin?
- c) Where would you draw $y = \frac{6}{x}$? Justify your answer.
Asymptotes
The hyperbola gets closer and closer to the axes, but never touches them:
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When $x$ gets close to 0, $y$ becomes huge (positive or negative).
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When $x$ becomes very large, $y$ gets close to 0.
Those lines that the graph hugs without ever touching are called asymptotes. For $y = \frac{k}{x}$, they are the two axes.
Consider the function $y = \frac{8}{x}$. a) Does its graph intersect the $X$-axis? And the $Y$-axis? Justify your answers using the equation. b) What values does $y$…
Consider the function $y = \frac{8}{x}$.
- a) Does its graph intersect the $X$-axis? And the $Y$-axis? Justify your answers using the equation.
- b) What values does $y$ take when $x$ approaches $0$ from the right ($x = 1$, $0.1$, $0.01$…)? And when $x$ gets very large?
- c) Explain in your own words what it means for the coordinate axes to be asymptotes of the hyperbola.
Problems with the hyperbola
If you know it is an inverse proportionality function and you know one point, you already know $k$: $k = x \times y$.
A group of cyclists has to cover $240\ \text{km}$ between two stages. The average speed $v$ at which they cycle will determine the time $t$ it takes them. a) Write the…
A group of cyclists has to cover $240\ \text{km}$ between two stages. The average speed $v$ at which they cycle will determine the time $t$ it takes them.
- a) Write the function that gives the time $t$ in terms of the speed $v$.
- b) Complete the table in your notebook for $v = 10$, $15$, $20$, $30$ and $40\ \text{km/h}$.
- c) Is it an inverse proportion function? What is the value of $k$?
Some painters have to paint a fence. A single painter would take $12$ days working at the same rate. a) Write the function connecting the number of painters $x$ with the…
Some painters have to paint a fence. A single painter would take $12$ days working at the same rate.
- a) Write the function connecting the number of painters $x$ with the number of days $y$ they take.
- b) How long do $4$ painters take? And $6$?
- c) Does it make sense to give decimal values to $x$ in this context? What about negative values? Explain what values $x$ can take.
An inverse proportion function passes through the point $P(4, 6)$. a) Calculate its constant $k$ and write its equation. b) Complete a table of values with $x = 1$, $2$,…
An inverse proportion function passes through the point $P(4, 6)$.
- a) Calculate its constant $k$ and write its equation.
- b) Complete a table of values with $x = 1$, $2$, $3$, $6$, $8$ and $12$.
- c) Plot the hyperbola in your notebook for positive values of $x$.
- d) Does it also pass through the point $(-3, -8)$? Check this using the equation.