Inverse proportion function

Consider the function $y = \frac{12}{x}$.

1. a) What is the value of the constant of inverse proportion $k$?

1. b) Copy and complete the table in your notebook for $x = 1$, $2$, $3$, $4$, $6$ and $12$.

1. c) What happens to the values of $y$ as $x$ gets larger?

State which of the following relationships are of inverse proportion and, in those cases, write their equation in the form $y = \frac{k}{x}$:

1. a) $y = \frac{3}{x}$

1. b) $y = \frac{x}{3}$

1. c) $x \times y = 10$

1. d) $y = -5x$

1. e) $y = \frac{1}{2x}$

Without plotting them, state which quadrants the branches of each of these hyperbolas lie in and reason why:

1. a) $y = \frac{7}{x}$

1. b) $y = \frac{-4}{x}$

1. 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.

1. a) Match each curve with its equation.

1. b) What effect does the value of $k$ have on the position of the hyperbola relative to the origin?

1. c) Where would you draw $y = \frac{6}{x}$? Justify your answer.

Consider the function $y = \frac{8}{x}$.

1. a) Does its graph intersect the $X$-axis? And the $Y$-axis? Justify your answers using the equation.

1. 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?

1. c) Explain in your own words what it means for the coordinate axes to be asymptotes of the hyperbola.

In a physics experiment, a piston compresses a gas at constant temperature and it is found that the pressure $p$ and the volume $V$ satisfy $p \times V = 6$, with the volume in litres and the pressure in atmospheres, as shown in the figure.

1. a) Write the function that gives the pressure in terms of the volume.

1. b) What pressure does the gas exert when it occupies $1.5\ \text{L}$?

1. c) What volume does it occupy when the pressure is $0.5\ \text{atm}$?

1. d) What happens to the pressure if the volume becomes very small? Could it ever be $V = 0$?

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.

1. a) Write the function that gives the time $t$ in terms of the speed $v$.

1. b) Complete the table in your notebook for $v = 10$, $15$, $20$, $30$ and $40\ \text{km/h}$.

1. c) Is it an inverse proportion function? What is the value of $k$?

The graph in the figure represents an inverse proportion function, as shown below.

1. a) What is the value of $y$ when $x = 2$? And when $x = 6$?

1. b) Using the values above, calculate the constant $k$ and write the equation of the function.

1. c) What is the value of $y$ when $x = 12$?

Some painters have to paint a fence. A single painter would take $12$ days working at the same rate.

1. a) Write the function connecting the number of painters $x$ with the number of days $y$ they take.

1. b) How long do $4$ painters take? And $6$?

1. 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 $P(4, 6)$.

1. a) Calculate its constant $k$ and write its equation.

1. b) Complete a table of values with $x = 1$, $2$, $3$, $6$, $8$ and $12$.

1. c) Plot the hyperbola in your notebook for positive values of $x$.

1. d) Does it also pass through the point $(-3, -8)$? Check this using the equation.

Investigate whether the line $y = x + 1$ intersects the hyperbola $y = \frac{6}{x}$:

1. a) Set up the equation $\frac{6}{x} = x + 1$ and transform it into a quadratic equation.

1. b) Solve it and write the coordinates of the points of intersection.

1. c) Check the result graphically by plotting both functions in your notebook.

1. d) Would the line $y = x + 8$ intersect the same hyperbola? Justify your answer without solving the equation.