Studying straight lines
An electric scooter charges €1 to unlock and €0,50 per minute. A piggy bank starts with €15 and grows by €2,50 every week. In both cases the graph is a straight line, and with just two numbers (how much it rises per unit and where it starts) you know everything that is going to happen.
Gradient and y-intercept
Move $m$ and $n$. Notice that $m$ tilts the line, while $n$ raises or lowers it without turning it:
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$m > 0$: the line is increasing. $\quad m < 0$: decreasing. $\quad m = 0$: a horizontal line.
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The greater $|m|$ is, the steeper the line.
Given the linear function $y = -2x$, state with reasons: a) What is the value of its gradient ($m$)? b) Is the function increasing or decreasing? c) Does the graph pass…
Given the linear function $y = -2x$, state with reasons:
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a) What is the value of its gradient ($m$)?
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b) Is the function increasing or decreasing?
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c) Does the graph pass through the origin $(0, 0)$?
Renting an electric scooter costs $1 \text{ €}$ as an initial unlock fee and $0,5 \text{ €}$ for each minute of use. The function relating the cost ($y$) to the time in…
Renting an electric scooter costs $1 \text{ €}$ as an initial unlock fee and $0,5 \text{ €}$ for each minute of use. The function relating the cost ($y$) to the time in minutes ($x$) is $y = 0,5x + 1$.
Calculate how much a trip lasting $10$ minutes will cost.
A new music streaming service offers a subscription costing $3 \text{ €}$ for each month purchased. a) Complete the following table of values relating the time in months…
A new music streaming service offers a subscription costing $3 \text{ €}$ for each month purchased.
- a) Complete the following table of values relating the time in months ($x$) to the total cost ($y$):
| Months ($x$) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Cost (€) ($y$) | $3$ |
- b) Write down the equation of the direct proportion function that represents this situation.
Calculating the gradient from two points
If you know two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the line:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$$Find the gradient ($m$) and the y-intercept ($n$) of the line shown in the following graph. Then, write its equation in the form $y = mx + n$.
Find the gradient ($m$) and the y-intercept ($n$) of the line shown in the following graph. Then, write its equation in the form $y = mx + n$.
Find the equation of the line passing through points $A(-2, 5)$ and $B(4, -7)$. To do this: 1. First, calculate the gradient $m$ using the formula…
Find the equation of the line passing through points $A(-2, 5)$ and $B(4, -7)$. To do this:
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First, calculate the gradient $m$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
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Substitute the gradient and one of the points into the equation $y = mx + n$ to find the value of $n$.
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Write down the final equation and state whether the function is increasing or decreasing, justifying your answer based on the value of the gradient.
Kenia is saving money in a piggy bank. Initially, she has $15\ €$, and each week she decides to put in $2,50\ €$ from her pocket money. a) Complete the following table…
Kenia is saving money in a piggy bank. Initially, she has $15\ €$, and each week she decides to put in $2,50\ €$ from her pocket money.
- a) Complete the following table of values:
| Weeks ($x$) | $0$ | $1$ | $2$ | $4$ | $10$ |
|---|---|---|---|---|---|
| Money saved ($y$) | $15$ |
- b) Write down the equation of the function.
- c) If Kenia wants to buy a video game costing $55\ €$, how many weeks will she have to wait according to this function?
Parallel lines
Determine the equation of the line that satisfies the following two conditions: It is parallel to the line $y = -3x + 5$. It passes through the point $P(0, 4)$. Explain…
Determine the equation of the line that satisfies the following two conditions:
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It is parallel to the line $y = -3x + 5$.
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It passes through the point $P(0, 4)$.
Explain what property two lines must share to be parallel and how you have used the point $P$ to find the y-intercept.
Draw on the same coordinate axes the graphs of the functions $y = 2x - 1$ and $y = 2x + 3$. a) What geometric relationship do you observe between the two lines? b)…
Draw on the same coordinate axes the graphs of the functions $y = 2x - 1$ and $y = 2x + 3$.
- a) What geometric relationship do you observe between the two lines?
- b) Justify your previous answer by comparing their equations.
- c) Will these two lines intersect at any point? Explain your reasoning.
You can do it on paper and send a photo with your answer.
Comparing tariffs with lines
In problems, $n$ is usually the fixed charge (the subscription, the unlocking fee) and $m$ the cost per unit (per minute, per GB). To find out when one tariff or the other is better value, you set the two functions equal.
Two electric scooter rental companies have different rates. The company EcoRolling charges $0,20\ €$ for each minute of use with no fixed fee. The company FastGlide…
Two electric scooter rental companies have different rates. The company EcoRolling charges $0,20\ €$ for each minute of use with no fixed fee. The company FastGlide charges a fixed fee of $2\ €$ for unlocking the scooter and then $0,10\ €$ for each minute. The graph shows both rates.
- a) Write the function that relates the total cost ($y$) to the usage time in minutes ($x$) for each of the companies.
- b) State which of the two is a direct proportionality (linear) function and which is an affine function.
- c) After how many minutes of travel does the FastGlide option start to be cheaper?
Two fibre optic companies offer the following monthly tariffs: NetRápido: a fixed $20$ € per month plus $0,50$ € for each extra GB downloaded. FibraTotal: no fixed fee,…
Two fibre optic companies offer the following monthly tariffs:
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NetRápido: a fixed $20$ € per month plus $0,50$ € for each extra GB downloaded.
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FibraTotal: no fixed fee, but charges $1,30$ € for each GB downloaded.
a) Write the algebraic expression of the function relating monthly cost ($y$) to extra GB ($x$) for each company.
b) Above how many extra downloaded GB does NetRápido start to be better value? (Find the point of intersection of both lines).
c) If in one month you use $30$ extra GB, what is the price difference between the two?
Back to the start: The piggy bank's equation is $y = 2{,}50x + 15$, where $x$ represents the weeks elapsed and $y$ the money saved: it starts from an initial amount of €$15$ ($n = 15$) and increases at a rate of €$2{,}50$ per week ($m = 2{,}50$).