Linear function or direct proportionality

Given the direct proportion function $y = -3x$:

• State the value of its gradient.

• Is the function increasing or decreasing?

• Calculate the value of the output for $x = 2$ and for $x = -1$.

State, with reasons, which of the following three graphs corresponds to a direct proportion function.

Three friends compare their cell phone data plans to see how many "gigs" they consume when watching high-definition videos. It is known that consumption is proportional to the viewing time.

• Plan A: $y = 0.8x$

• Plan B: $y = 1.2x$

• Plan C: $y = 0.5x$

Where $x$ is the time in hours and $y$ is the gigs consumed.

1. a) Which of the three plans consumes data fastest? Justify your answer using the concept of slope.
2. b) If a user has a daily limit of $6$ GB, how many hours of video could they watch with Plan B before running out of data?

Observe the following table of values relating the time an LED light bulb is turned on ($x$, in hours) to energy consumption ($y$, in watt-hours, Wh):

State whether it is a direct proportion function, calculate its constant of proportionality and write its equation.

State, with reasons, which of the following tables correspond to a direct proportion function. For those that do, calculate their constant of proportionality ($m$) and write their equation.

Table 1:

Table 2:

Table 3:

Find the gradient of the following direct proportion function represented on the graph and write its equation in the form $y = mx$.

Look at the following graph, which represents two direct proportionality functions, $r$ and $s$. They show the points obtained by two players in a video game over time (time is in minutes).

1. a) Find the equation of line $r$, given that it passes through the point $P(2, 10)$.

1. b) Find the equation of line $s$, given that it passes through the point $Q(4, 8)$.

1. c) What does the slope represent in this context?

The following graph shows the relationship between the volume of water ($V$) in a cylindrical container and the height ($h$) reached by the water level. Due to the shape of the container, this relationship is directly proportional.

• From the points shown on the graph, determine the constant of proportionality.

• Write the equation to calculate the volume as a function of height.

• What height will the water reach if we pour $7$ liters into the container?

Complete the following table knowing that the relationship between $x$ and $y$ is directly proportional with a constant of $m = 4$. Then, plot the points on a Cartesian plane.

You can do it on paper and send a photo with your answer.

Find the equation of the direct proportion function $y = mx$ whose graph passes through the point $A(-3, 12)$. Represent the function graphically, indicating at least two other points through which the line passes.

You can do it on paper and send a photo with your answer.

A delivery drone flies at a constant speed of $12$ meters per second ($m/s$).

1. a) Complete the following table relating flight time ($x$) to distance covered ($y$):

1. b) Represent the function graphically.
2. c) State the slope and explain its physical meaning in this problem.

A pizza delivery person on a bicycle travels at a constant speed of $20 \text{ km/h}$.

• Write the function relating the distance covered ($y$, in km) to the elapsed time ($x$, in hours).

• How many kilometers will they have traveled after $3$ hours of work?

In a cooking recipe, it states that for every $100 \text{ g}$ of flour you must use $2$ eggs.

• Calculate the constant of proportionality relating the number of eggs ($y$) to the amount of flour ($x$, measured in hundreds of grams).

• Write the equation of the function.

• If we have a bag of $500 \text{ g}$ of flour, how many eggs will we need according to this function?

In a recycling campaign, an NGO has determined that for every kilogram of PET plastic containers that is recycled, $1.5$ liters of oil are saved in the manufacture of new products.

1. a) Write the function that relates the kilograms of plastic recycled ($x$) to the liters of oil saved ($y$).

1. b) Create a table of values for $0, 5, 10$ and $20$ kg of plastic.

1. c) How many liters of oil will have been saved if the school manages to recycle $120$ kg of containers?

In an app store, downloading a pack of "virtual coins" for a game follows a linear function. We know that $500$ coins cost $270$ pesos.

1. a) Calculate the constant of proportionality (price per coin).
2. b) Write the algebraic expression of the function relating the number of coins ($x$) to the cost in pesos ($y$).
3. c) If a teenager has saved $900$ pesos, how many coins can they buy at most?

A 3D printing company charges schools only for the material used. It is known that $100$ grams of PLA filament cost $210$ pesos.

1. a) Determine the function that calculates the cost ($y$) according to the grams of material ($x$).
2. b) Is it an increasing or decreasing function? Why?
3. c) If the part you want to print for your Technology project weighs $45$ grams, how much will the printing cost you?

On a video game streaming platform, the streaming quality consumes data in proportion to the time of use. A user has checked that after playing for $15$ minutes, their router shows a consumption of $1.2$ gigabytes (GB).

• Determine the constant of proportionality $m$ that relates the data consumption ($y$, in GB) to the time ($x$, in minutes).

• Write the equation of the linear function that models this situation.

• If the user has a daily limit of $5$ GB before their speed is reduced, how many exact minutes and seconds can they play while maintaining that quality?

Two friends, Lucía and Marcos, are saving up for a trip. Lucía saves at a constant rate given by the function $f(x) = 750x$, where $x$ is the number of weeks and $y$ is the money saved in pesos. Marcos does not know his function, but he knows that in week $8$ he had saved exactly $5\,520$ pesos and that his savings are also proportional to time.

• Find the equation of Marcos's function, $g(x)$.

• Plot both functions on the same set of axes (sketch).

• Which of the two has a steeper "slope"? Interpret this result in terms of who saves faster.

• If the trip costs $18\,000$ pesos, how many weeks of difference will there be between the time Lucía gets the money and the time Marcos gets it?

Hooke's law states that the extension of a spring is directly proportional to the force applied to it. A spring measures $12$ cm at rest. When hanging a weight of $2$ kg, the spring measures $15$ cm.

• Note: The function of proportionality relates the force (or mass, in this case) to the extension (the increase in length), not to the total length. Calculate the extension produced by the $2$ kg.

• Determine the constant of proportionality $m$ ($cm/kg$).

• Write the function $y = mx$ relating extension $y$ to mass $x$.

• What will be the total length of the spring if we hang a weight of $5.4$ kg?