Table of a Function

For a relation to be a function, each value of the independent variable ($x$) must correspond to a single value of the dependent variable ($y$).

State which of the following tables does NOT represent a function and explain why:

Table A:

Table B:

A cyclist travels along a route at a constant speed of $20$ km/h. The following table shows the distance covered depending on the elapsed time:

From the table, state:

• a) Which is the independent variable and which is the dependent variable?

• b) Is it a continuous or discrete function? Reason your answer.

Given the function defined by the algebraic expression $y = 3x - 2$, build a table of values by calculating the image ($y$) for the following values of the independent variable:

$x = \{-2, -1, 0, 1, 2\}$.

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

An e-scooter rental company charges a fixed unlocking fee of $40$ pesos and then $10$ pesos per minute of use.

Complete the table relating the time of use to the total rental cost:

Given the function $y = -2x + 3$, complete the following table of values including negative numbers and decimals:

In light of the results, state whether the function is increasing or decreasing.

We want to study how the perimeter of a square changes depending on the length of its side. Remember that a square has four equal sides and that the perimeter is the sum of all of them.

Complete the table of values and answer: what is the constant of proportionality between the perimeter and the side?

Observe the relationship between the side length of a square and its area. We know that if the side length is $x$, the area is given by the function $y = x^2$.

Copy and complete the table in your notebook:

Is this a direct proportion function? Justify your answer based on the values in the table.

Observe the following table of values relating two quantities:

• a) Check that the product $x \times y$ is constant for all pairs of values. What is this value?

• b) What type of function is it (linear, affine, or inverse proportion)?

A group of friends wants to rent a vacation home that costs $\$4\,800$ for the weekend. The price each person must pay ($y$) depends on the number of people going ($x$).

The table relating both quantities is as follows:

1. a) Complete the missing values.
2. b) Write down the formula for the function.
3. c) What type of function is it (linear, affine or inverse proportion)?

In a physics laboratory, the relationship between the pressure and volume of a gas at constant temperature (Boyle's Law) is being studied. The data obtained are collected in the following table of values:

a) Check whether it is an inverse proportion function by calculating the constant of proportionality $k = V \times P$ for each pair of values.

b) What volume will the gas occupy if the pressure increases to $20 \text{ atm}$?

c) If the volume is reduced to a quarter of the initial volume ($2 \text{ liters}$), what happens to the pressure according to the constant found?

Find the algebraic expression of the function corresponding to the following table of values:

The temperature of a freezer has been measured since it was switched on. The data obtained are shown in the following table:

1. a) What is the initial temperature of the freezer?

1. b) By how many degrees does the temperature drop every $5$ minutes? Is it a uniform decrease?

1. c) If the trend continues, what will the temperature be after $25$ minutes?

Given the following table of values for an affine function, find the missing values and determine the algebraic expression that defines it:

a) Calculate the increase in $y$ for each unit increase in $x$ to find the slope $m$.

b) Complete the gaps in the table based on the pattern found.

c) Write the algebraic expression of the function in the form $y = mx + n$.

A movie streaming service has an offer: the first month costs only $40$ pesos and each additional month costs $160$ pesos.

Write the function that relates the number of months subscribed ($x$) to the total cost ($y$) and complete the following table of values:

A courier company charges $\$80$ for sending a package, plus a variable amount of $\$24$ for each kilogram of mass.

1. a) Identify the independent variable and the dependent variable.
2. b) Create a table of values relating the weight of the package (from $1 \text{ kg}$ to $5 \text{ kg}$) to the total delivery cost.
3. c) Write the algebraic expression of the function.

A drone takes off from the ground and ascends at a constant speed of $3$ meters per second.

1. a) Draw up a table of values relating the flight time (in seconds) to the height reached (in meters) for the times $t = 0, 2, 4, 6, 8, 10$.
2. b) What height will the drone reach after $12$ seconds?
3. c) How long will it take to reach a height of $45$ meters? Complete the table if necessary to answer.

A telecommunications company offers a data plan for "gamers" that consists of a fixed fee of $\$12$ per month. This fee includes the first $20 \text{ GB}$ of download. After that, each extra $\text{GB}$ costs $\$1.5$.

a) Complete the following table of values that relates the data usage in a month ($x$) to the total cost of the bill ($y$):

b) Is this relationship a direct proportionality function? Justify your answer by analyzing the quotients in the table.

c) Determine the algebraic expression of the function for the section where the usage is above $20 \text{ GB}$.

A team of cavers is descending into a deep chasm. They have observed that the temperature increases at a constant rate as they descend. At ground level ($0 \text{ m}$) the temperature is $18 \text{ °C}$. For every $150 \text{ m}$ of descent, the temperature increases by $2.25 \text{ °C}$.

a) Construct a table of values showing the temperature for depths of $0, 300, 600, 900$ and $1200$ meters below ground level.

b) Calculate the rate of increase (slope) of this function, expressing how much the temperature increases per meter of descent.

c) If at the bottom of the chasm the thermometer reads $45 \text{ °C}$, use the logic of the table to determine their depth.

Two streaming platforms are competing for customers. The platform "CineTotal" charges a fixed fee of $\$160$ and $\$10$ for each new release movie rented. The platform "EstrenosYA" has no fixed fee, but charges $\$50$ for each new release movie.

a) Draw up a comparative table of values for renting $0, 2, 4, 5$ and $6$ movies on both platforms.

b) Analyzing the table, from how many movies onwards does a subscription to "CineTotal" start to be more cost-effective?

c) What type of function does each platform represent? (Linear, affine, or constant).

A gardener uses a $500 \text{ litros}$ water tank for watering. The tank loses water at a constant rate of $12.5 \text{ litros}$ per minute when the watering is on. The graph shows how the volume of water in the tank varies over time.

a) Create a table of values relating the elapsed time ($0, 5, 10, 20, 30$ minutes) to the amount of water remaining in the tank.

b) Identify the value of the y-intercept (initial amount) and the slope (rate of loss) from the data in your table.

c) How long will it take for the tank to be exactly half full?

A professional cyclist carries out endurance training. The table shows the distance covered as a function of time, but over different intensity sections:

a) State whether the speed was constant throughout the training by analysing the increments in the table.

b) What distance will they have covered after $3 \text{ hours and } 15 \text{ minutes}$ if they maintain the same speed as in the last section?

c) Plot the points from the table on a coordinate system. Are all the points aligned on the same straight line passing through the origin?

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

An NGO has calculated that the cost of sending humanitarian aid packages depends on the number of trucks used. The fixed logistics costs are $\$50\,000$ and the operating cost of each truck is $\$24\,000$.

a) Complete the table of values relating the number of trucks ($1, 2, 5, 10$) to the total cost of the shipment.

b) If the NGO's maximum budget for a shipment is $\$400\,000$, add a column to the table calculating the remaining money in each case.

c) Is the "remaining money" function an increasing or decreasing function? Justify your answer by observing the table.