Algebraic expression of a function
"Each chest of diamonds costs €1.50." That sentence hides a formula: $y = 1{,}5x$. Knowing how to translate a statement, a table or a graph into an algebraic expression is the most powerful tool you have, because with it you can calculate any value without having to measure or count.
From a statement to the formula
Look in the text for two pieces of information: a fixed amount (if there is one) and an amount per unit.
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…
Match each of the following sentences with its corresponding algebraic expression, where $x$ represents any number and $y$ represents the result of the function:
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a) The result is triple a number.
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b) The result is twice a number plus five.
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c) The result is half a number.
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d) The result is a number subtracted from ten.
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$y = \frac{x}{2}$
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$y = 10 - x$
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$y = 3x$
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$y = 2x + 5$
A computer repair technician charges $20$ € for a home call-out and $15$ € for each hour of work. a) Write down the algebraic expression of the function relating the…
A computer repair technician charges $20$ € for a home call-out and $15$ € for each hour of work.
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a) Write down the algebraic expression of the function relating the total bill cost ($y$) to the number of hours worked ($x$).
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b) How much will a repair that took $3$ hours cost?
A music streaming platform offers a monthly plan that costs $12\,€$ as a fixed fee and, in addition, charges $0,50\,€$ for each song the user decides to download to…
A music streaming platform offers a monthly plan that costs $12\,€$ as a fixed fee and, in addition, charges $0,50\,€$ 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 $24,50\,€$, how many songs have they downloaded?
From a table to the formula
If the table always increases by the same amount when $x$ goes up by 1 unit, it is a linear function (of the form $y = mx + n$): that constant amount is $m$.
| $x$ | $1$ | $2$ | $3$ | $4$ |
|---|---|---|---|---|
| $y$ | $8$ | $11$ | $14$ | $17$ |
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: | Number of…
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:
| Number of tickets ($x$) | $1$ | $2$ | $3$ | $4$ |
|---|---|---|---|---|
| Price in € ($y$) | $12$ | $24$ | $36$ | $48$ |
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…
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 ($n$).
Evaluating the function: calculating $f(a)$
You can also work backwards: if you know $y$ and want the $x$ that produces it, you solve for $x$.
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…
Find the value of the dependent variable $y$ for the values of the independent variable $x$ indicated in the following function:
$y = 4x - 3$
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a) For $x = 0$
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b) For $x = 1$
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c) For $x = 2$
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d) For $x = 5$
Copy and complete the following table of values in your exercise book for the function whose algebraic expression is $y = -2x + 1$: | $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$…
Copy and complete the following table of values in your exercise book for the function whose algebraic expression is $y = -2x + 1$:
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|---|
| $y$ |
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…
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.
When the formula is not linear
Not all the functions in a problem are $y = mx + n$. Pay attention to what stays constant.
We want to design a rectangular poster whose total area is always $36\ m^2$. a) Write down the algebraic expression of the function that gives the height ($y$) of the…
We want to design a rectangular poster whose total area is always $36\ m^2$.
- a) Write down the algebraic expression of the function that gives the height ($y$) of the poster as a function of its base length ($x$).
- 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?
- c) What type of function is it? Explain what happens to the height as the base becomes extremely large.
A group of Year 8 students wants to hire a coach for an end-of-year trip that costs $840\ €$. a) Write down the algebraic expression relating the price each student must…
A group of Year 8 students wants to hire a coach for an end-of-year trip that costs $840\ €$.
- a) Write down the algebraic expression relating the price each student must pay ($y$) to the number of students who finally sign up ($x$).
- 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.
- c) If the students decide that they cannot pay more than $35\ €$ 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.