Algebraic Expression of a Function

Match each of the following sentences with its corresponding algebraic expression, where $x$ represents any number and $y$ represents the result of the function:

• a) The result is triple a number.

• b) The result is twice a number plus five.

• c) The result is half a number.

• d) The result is a number subtracted from ten.

• $y = \frac{x}{2}$

• $y = 10 - x$

• $y = 3x$

• $y = 2x + 5$

A computer repair technician charges $1\,200$ pesos for a home call-out and $900$ pesos for each hour of work.

• a) Write down the algebraic expression of the function relating the total bill cost ($y$) to the number of hours worked ($x$).

• b) How much will a repair that took $3$ hours cost?

A music streaming platform offers a monthly plan that costs $\text{RD\$}\ 12$ as a fixed fee and, in addition, charges $\text{RD\$}\ 0.50$ for each song the user decides to download to listen to offline. In the figure you can see the graph of this situation.

a) Write the algebraic expression of the function that relates the number of songs downloaded, $x$, to the total monthly cost, $y$.

b) State what type of function it is (linear, affine or inverse proportionality).

c) If in one month a user pays a total of $\text{RD\$}\ 24.50$, how many songs have they downloaded?

In a mobile video game, each "diamond chest" costs 1,50. pesos If we want to buy $x$ chests, the function that gives us the total price to pay is $y = 1.5x$. The figure shows its graph.

Calculate how much money we would spend if we decided to buy:

1. a) $2$ chests.

1. b) $5$ chests.

1. c) $10$ chests.

Find the value of the dependent variable $y$ for the values of the independent variable $x$ indicated in the following function:

$y = 4x - 3$

• a) For $x = 0$

• b) For $x = 1$

• c) For $x = 2$

• d) For $x = 5$

Given the function $f(x) = x^2 + 1$:

• a) Calculate the image of $x = 3$. That is, calculate $f(3)$.

• b) Calculate the image of $x = -2$. That is, calculate $f(-2)$.

• c) Calculate the image of $x = 0$. That is, calculate $f(0)$.

Copy and complete the following table of values in your exercise book for the function whose algebraic expression is $y = -2x + 1$:

Look at the following table of values relating the number of concert tickets to the total price. Find the algebraic expression that represents this function:

Look at the following table of values relating two quantities, $x$ and $y$:

a) Check whether this is an inverse proportion function.

b) Write down its algebraic expression.

c) Calculate the image value for $x = 10$.

In a physics laboratory, the variation of the temperature of a liquid as it is heated is being observed. At the start of the experiment, the liquid is at $15^\circ C$ and its temperature increases at a constant rate of $3^\circ C$ for every minute that passes.

a) Determine the algebraic expression to calculate the temperature $T$ as a function of time $t$ (in minutes).

b) What temperature will the liquid reach after a quarter of an hour of heating?

c) How much time must pass for the liquid to reach $60^\circ C$?

Given the function whose algebraic expression is $f(x) = \frac{2x - 6}{3}$:

a) Calculate the values of $f(0)$, $f(3)$ and $f(-3)$.

b) Determine the point where the function crosses the x-axis (the $X$ axis).

c) Plot the points you obtained on a coordinate system like the one in the figure (each tick mark is one unit) and join them to draw the function.

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

Find the algebraic expression of the linear function whose graph passes through the points $A(1, 4)$ and $B(3, 10)$. Explain step by step how you calculate the gradient ($m$) and the y-intercept ($n$).

A courier company charges a fixed collection fee plus a cost per kilometer distance for delivering parcels. We know that a delivery to a city located at $120\ km$ costs $\text{RD\$}\ 924$, and a delivery to another city at $200\ km$ costs $\text{RD\$}\ 1\,308$.

1. a) Find the slope of the function and state what it means in this context.
2. b) Find the algebraic expression of the function $y = f(x)$.
3. c) How much would the company charge for a parcel delivered to the same collection point ($0\ km$)? What is this value called in the function?

A surveillance drone takes off from the roof of a building located at a certain height above sea level. After $4\ segundos$ of constant vertical flight, the drone is at a height of $52\ metros$. At $10\ segundos$ of flight, it reaches $88\ metros$.

1. a) Determine the algebraic expression of the function relating the flight time ($x$, in seconds) to the drone's height ($y$, in meters).
2. b) How high is the roof of the building?
3. c) If the drone has a maximum flight ceiling of $150\ metros$, use the algebraic expression to calculate how many seconds it can ascend before reaching that limit.

Identify which of the following expressions corresponds to each of the graphs described below:

1. $y = 3x$

2. $y = -2x + 4$

3. $y = 5$

4. $y = \frac{4}{x}$

a) A horizontal line passing through the point $(0, 5)$.

b) A direct proportion function passing through the point $(2, 6)$.

c) A decreasing linear function that intersects the $Y$ axis at the point $(0, 4)$.

d) A hyperbola passing through the point $(1, 4)$.

Find the algebraic expression of a linear function whose graph is parallel to the line passing through the points $P(1, 5)$ and $Q(3, 1)$, and which also intersects the y-axis ($Y$) at the same point as the function $g(x) = 4x - 7$.

1. a) First calculate the gradient of the line passing through $P$ and $Q$.
2. b) Identify the y-intercept of $g(x)$.
3. c) Write down the final expression for the required function.

We want to surround several square posters with an elastic string for a school fair. The relationship between the side length of the square ($x$) and the total perimetre of string required ($y$) is given by a function.

• a) Write down the algebraic expression of this function.

• b) If the side of the square measures $12$ cm, how much string will we need?

We want to build a rectangular frame for a poster using a wooden strip with a total length of $120 cm$. If we call $x$ the length of the base of the rectangle:

a) Write down the algebraic expression relating the height of the rectangle ($y$) to its base ($x$). Hint: remember that the perimetre is the sum of all the sides.

b) Write down the algebraic expression that gives the area ($A$) of the rectangle as a function of its base ($x$).

c) If the base is $25 cm$ long, what will the area of the frame be?

We want to design a rectangular poster whose total area is always $36\ m^2$.

1. a) Write down the algebraic expression of the function that gives the height ($y$) of the poster as a function of its base length ($x$).
2. b) If by regulations the poster cannot have a base smaller than $2\ metros$ or larger than $12\ metros$, what is the domain of this function in this real-life context?
3. c) What type of function is it? Explain what happens to the height as the base becomes extremely large.

In a physics laboratory, the variation of the pressure of a gas as the volume of its container changes is being studied, keeping the temperature constant (Boyle's Law). It has been observed that when the volume is $2\ litros$, the pressure is $6\ atmósferas$.

1. a) Knowing that the relationship is one of inverse proportion, write down the algebraic expression relating pressure ($y$) to volume ($x$).
2. b) Using the algebraic expression, calculate what volume the container should have so that the pressure is reduced to $1.5\ atmósferas$.
3. c) Plot the graph of the function for volume values between $1$ and $12\ litros$.

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

A group of 8th grade students wants to rent a bus for an end-of-year trip that costs $\text{RD\$}\ 50\,400$.

1. a) Write down the algebraic expression relating the price each student must pay ($y$) to the number of students who finally sign up ($x$).
2. b) Initially $30$ students sign up, but a week before, $5$ drop out. Use the algebraic expression to calculate how much the price increases for the students who do go on the trip.
3. c) If the students decide that they cannot pay more than $\text{RD\$}\ 2\,100$ each, what is the minimum number of people who must attend for the trip to go ahead? Explain how you obtained this from the algebraic expression.

A gardener charges a fixed call-out fee of $\text{RD\$}\ 1\,200$ plus $\text{RD\$}\ 900$ per hour of work. His competitor does not charge a call-out fee, but his rate is $\text{RD\$}\ 1\,320$ per hour.

a) Write down the algebraic expression for the cost of each gardener's service ($y_1$ and $y_2$) as a function of the hours worked ($x$).

b) If we need a job that will take exactly $4$ hours, which of the two gardeners is cheaper for us?

c) Is there a number of working hours for which both charge the same?

A video game streaming platform offers two monthly plans for content creators. With the Pro Plan, the creator receives a fixed amount of $\text{RD\$}\ 15$ per month and, in addition, $\text{RD\$}\ 0.20$ for each new subscriber. With the Partner Plan there is no fixed amount; on the contrary, the platform deducts $\text{RD\$}\ 5$ per month for management costs and the creator receives $\text{RD\$}\ 0.35$ for each new subscriber.

1. a) Write the algebraic expression of the function that represents the net monthly income for each plan, where $x$ is the number of subscribers and $y$ is the total money received (in pesos).

1. b) If a streamer estimates that they will get between $60$ and $70$ subscribers this month, which plan is more beneficial for them? Justify your answer by calculating the values of $y$ for each plan for $x=60$ and for $x=70$ and comparing them.

Two identical water tanks, with a capacity of $500$ liters, are in different situations. Tank A is full and is being emptied at a constant rate of $8$ liters per minute. Tank B contains $120$ liters and is being filled at a constant rate of $12$ liters per minute.

Let $t$ be the time in minutes, counted from the moment they start emptying and filling, and $y$ the volume of water in liters.

1. a) Write the algebraic expression of the volume-time function for each tank ($y_A$ and $y_B$).

1. b) Using the expressions above, calculate the exact moment at which both tanks will have the same amount of water.

1. c) What volume of water will each tank contain at that moment?