Linear function or direct proportion

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

Three friends compare their mobile data plans to see how many "Gigas" 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 Gigas consumed.

1. a) Which of the three plans consumes data fastest? Justify your answer using the concept of gradient.
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 $1,50 \text{ €}$. 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 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 ($\text{m/s}$).

1. a) Complete the following table that relates flight time ($x$) to distance traveled ($y$):

1. b) Graph the function on Cartesian axes (time on the horizontal axis and distance on the vertical axis).

1. c) State what the slope is and explain what physical meaning it has in this problem.

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

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 kilometres will they have travelled after $3$ hours of work?

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?

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?