Inverse Proportionality 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.

State whether each statement is true or false and justify your answer with an example or a counterexample:

1. a) 'Every decreasing function is an inverse proportion function.'

1. b) 'In the function $y = \frac{k}{x}$, if the value of $x$ doubles, the value of $y$ is halved.'

1. c) 'The hyperbola $y = \frac{10}{x}$ intersects the line $y = 10$ at a single point.'

1. d) 'The functions $y = \frac{2}{x}$ and $y = \frac{-2}{x}$ are symmetrical with respect to the $X$-axis.'

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.

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 liters 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 $V = 0$?

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.

In a regional cake competition, the prize of $\text{RD\$}\ 600$ is shared equally among all the winners, as shown in the figure.

1. a) Write the function that gives the prize per person $y$ in terms of the number of winners $x$.

1. b) How much does each person receive if there are $3$ winners? And if there are $8$?

1. c) The graph of this function is not a complete hyperbola, but only some points on it. Explain why.

An external battery of $20,000\ \text{mAh}$ discharges depending on the power consumption of the device it powers. The duration $t$ in hours depends on the average consumption $c$ in $\text{mA}$: the higher the consumption, the shorter it lasts.

1. a) Knowing that $t = \frac{20,000}{c}$, calculate how many hours it lasts with a consumption of $800\ \text{mA}$.

1. b) What is the consumption if the battery has lasted $40$ hours?

1. c) If the consumption doubles, what happens to the duration? Justify your answer using the equation.

Decision task. A school's PTA hires a $54$-seat bus for a trip to the science museum. The bus costs $\text{RD\$}\ 378$ and the cost is shared equally among the students who sign up, as shown in the figure.

1. a) Write the function that gives the price per student $y$ in terms of the number of attendees $x$. What values can $x$ take?

1. b) How many students must sign up so that the price per person does not exceed $\text{RD\$}\ 14$?

1. c) On the first day, $21$ students sign up. How much would each one pay? If $6$ more sign up later, how much would be refunded to each person?

1. d) Why does the graph of this function get closer and closer to the $X$-axis without touching it? What does this mean in this context?

To lift a bucket of water with a lever, the principle of moments (law of the lever) is applied: $F \times d = 600$, where $F$ is the force in newtons and $d$ is the distance from the pivot to the end being pushed, in centimeters.

1. a) Write the function that gives the force in terms of distance.

1. b) What force must be applied when pushing at $50\ \text{cm}$? And at $150\ \text{cm}$?

1. c) A child can apply a maximum force of $80\ \text{N}$. What is the minimum distance from the pivot at which they must push?

1. d) Interpret the vertical asymptote of the graph: what would happen when pushing very close to the pivot?

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.