Linear Function or Direct Proportionality

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

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 $x = -1$.

Three friends compare their cell phone data plans to see how many gigabytes 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 gigabytes 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?

In a sweet shop, each bag of sweets costs $\$30$. We want to study the relationship between the number of bags we buy ($x$) and the total money we pay ($y$).

Identify:

• The independent variable.

• The dependent variable.

• The constant of proportionality ($m$).

• The equation of the function relating both quantities.

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.

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

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 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.

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.

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.

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?

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 recycled, $1.5$ liters of oil are saved in the manufacture of new products.

1. a) Write the function relating the kilograms of recycled plastic ($x$) to the liters of oil saved ($y$).
2. b) Create a table of values for $0, 5, 10$ and $20$ kg of plastic.
3. 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 $90$ 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 $300$ 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 $70$ 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?

A robotics laboratory is testing a prototype moving in a straight line at a constant speed. It is known that the function relating the distance covered ($d$) to the time ($t$) is a direct proportion function. At $t = 4.5$ seconds, the robot has covered exactly $11.25$ meters.

• Calculate the slope of the function and state what physical quantity it represents in this context.

• Write the algebraic expression of the function.

• If the robot must complete a $0.05$-kilometer track, how long will it take? Express the result in seconds.

In a design workshop, they use a laser cutting machine that consumes energy in proportion to the area of the cut material. It has been observed that to cut $4$ square pieces of $15$ cm side each, the machine consumes $180$ Joules (J).

• Calculate the energy consumption per square centimeter cut ($cm^2$).

• Define the linear function that relates total energy consumption ($y$) to the surface area cut ($x$).

• How much energy would the machine consume to cut a single rectangular panel measuring $0.4$ meters long by $25$ centimeters wide?

Two friends, Lucía and Marcos, are saving up for a trip. Lucía saves at a constant rate given by the function $f(x) = 250x$, 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 $1\,840$ 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 $6\,000$ pesos, how many weeks of difference will there be between the time Lucía gets the money and the time Marcos gets it?

A scaled map shows a direct proportion relationship between the distance measured on paper ($x$, in millimeters) and the real distance on the ground ($y$, in kilometers). We know that a real distance of $150$ km is represented on the map by a segment of $60$ mm.

• Determine the equation of the linear function that converts mm on the map to real km.

• What real distance separates two cities if they are $14.4$ centimeters apart on the map? (Be careful with the units).

• If we want to represent a hiking route of $12,500$ meters, how many millimeters will it measure on the map?

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?

A plastic recycling company estimates that the weight of polyethylene obtained ($y$) is directly proportional to the number of empty bottles processed ($x$). When processing a container with $120$ bottles, $3.3$ kg of recycled material is obtained.

• Find the constant of proportionality and explain its meaning.

• If the company receives an order to manufacture parts that require $55$ kg of plastic, how many bottles will they need to recycle?

• If for each recycled bottle the company receives a subsidy of $0.30$ pesos, write the linear function relating the money received ($D$) to the weight of plastic obtained ($y$). (Note: you will need to combine the bottle-weight relationship with the bottle-money relationship).