Studying Lines
An electric scooter in Santo Domingo charges $\text{RD}\$10$ to unlock it and $\text{RD}\$5$ for each minute. Lucía's piggy bank starts with $\text{RD}\$900$ and grows by $\text{RD}\$150$ each week. In both cases the graph is a line, and with just two numbers (how much it rises per unit and where it starts) you know everything that is going to happen.
Points and lines in the plane
To read a line, you first need to know how to read points. Each point is written as $(x, y)$: first how far you move right (or left), and then how far you go up (or down).
A graph is the graph of a function when each value of $x$ has only one corresponding value of $y$. Trick: imagine a vertical line sliding across the graph; if at any moment it crosses the graph at two or more points, it is not a function.
A group of friends is designing a new logo for their gaming club on grid paper. Write down the coordinates of the key points shown on the graph that form the vertices of…
A group of friends is designing a new logo for their gaming club on grid paper. Write down the coordinates of the key points shown on the graph that form the vertices of the design.
For a graph to represent a function, each value of the independent variable $x$ must correspond to a single value of the dependent variable $y$. Look at the following…
For a graph to represent a function, each value of the independent variable $x$ must correspond to a single value of the dependent variable $y$. Look at the following graphs and state, giving reasons, which of them is not a function. To justify your answer, imagine vertical lines sweeping across each graph and notice how many points they cross it at.
Slope and y-intercept
Move $m$ and $n$. Notice that $m$ tilts the line and $n$ moves it up or down without turning it:
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$m > 0$: the line is increasing. $m < 0$: decreasing. $m = 0$: 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 $(0, 0)$?
A new music streaming service offers a subscription costing $\text{RD\$}\ 180…
A new music streaming service offers a subscription costing $\text{RD\$}\ 180$ 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 ( pesos) ($y$) | $180$ |
- b) Write down the equation of the direct proportion function that represents this situation.
Renting an electric scooter costs $\text{RD\$}\ 60$ as an initial unlock fee and $\text{RD\$}\ 30$ for each minute of use. The function relating the cost ($y$) to the…
Renting an electric scooter costs $\text{RD\$}\ 60$ as an initial unlock fee and $\text{RD\$}\ 30$ for each minute of use. The function relating the cost ($y$) to the time in minutes ($x$) is $y = 30x + 60$.
Calculate how much a trip lasting $10$ minutes will cost.
Horizontal lines, vertical lines, and proportionality
Some lines deserve their own name:
| Equation | Line | Is it a function? |
|---|---|---|
| $y = mx$ | passes through the origin | yes, direct proportion |
| $y = mx + n$ | crosses $Y$ at $n$ | yes, linear |
| $y = k$ (e.g. $y = 3$) | horizontal, $m = 0$ | yes, constant |
| $x = k$ (e.g. $x = -2$) | vertical | no |
A parking lot that charges $\text{RD}\$120$ per day no matter how many hours has $y = 120$: a constant function. Its table always repeats the same value.
A walker at constant speed covers distances proportional to the time: the graph is a line that passes through the origin and its slope is the constant of proportionality (the speed). To check whether a graph shows direct proportion, see that it passes through $(0, 0)$ and that the quotient $y/x$ is always the same.
A bicycle parking facility in the city center charges a flat rate of $\text{RD\$}\ 120…
A bicycle parking facility in the city center charges a flat rate of $\text{RD\$}\ 120$ per day, regardless of how many hours the bicycle is parked.
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a) Represent this situation using a table of values (hours vs cost).
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b) What type of function is it: linear, affine, or constant?
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c) What would its equation be?
Look at the following graph showing the journey of a person walking at a constant speed. State whether the relationship between the distance covered and the time is…
Look at the following graph showing the journey of a person walking at a constant speed. State whether the relationship between the distance covered and the time is directly proportional, and calculate its constant of proportionality (the gradient).
Match each of the following equations of lines with its corresponding description: 1. $y = 3$ 2. $x = -2$ 3. $y = 3x$ 4. $y = -x + 3$ A) Line passing through the origin…
Match each of the following equations of lines with its corresponding description:
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$y = 3$
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$x = -2$
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$y = 3x$
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$y = -x + 3$
A) Line passing through the origin with a positive gradient.
B) Constant function representing a horizontal line.
C) Line with a negative gradient and a y-intercept equal to $3$.
D) Vertical line that does not represent a function.
Calculating the slope from two points
If you know two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the line:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$$This makes sense: the slope is how much $y$ rises for each unit that $x$ advances, and here we measure both changes between the two points.
Once you have $m$, you substitute one of the points into $y = mx + n$ and solve for $n$.
It can also be read from a graph: $n$ is where the line crosses the $Y$ axis, and $m$ comes from two points that fall exactly on grid crossings.
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 $y = mx + n$.
Given the following graph representing an affine function, complete the following steps: a) Visually identify the y-intercept ($n$). b) Calculate the slope ($m$) using…
Given the following graph representing an affine function, complete the following steps:
- a) Visually identify the y-intercept ($n$).
- b) Calculate the slope ($m$) using two points on the line that it passes through exactly (for example, the intercepts with the axes).
- c) Write the explicit equation of the line 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.
Lines in real-life situations
When something changes at a constant rate, the equation is $y = mx + n$: $n$ is the initial value and $m$ is the change per unit. If the value decreases, $m$ is negative.
Notice the real domain and range: here $x$ only goes from $0$ to $10$ and $y$ from $0$ to $40$, because neither time nor height can be negative.
If the slope isn't given, you get it from two data points. For example, a battery that goes from 90% to 45% in 2.5 hours has $m = \dfrac{45 - 90}{2.5} = -18$ (it drops 18% each hour).
A drone is initially at a height of $40$ meters. Upon pressing the "return to base" button, the drone begins to descend at a constant speed of $4$ meters per second. a)…
A drone is initially at a height of $40$ meters. Upon pressing the "return to base" button, the drone begins to descend at a constant speed of $4$ meters per second.
- a) Define the independent and dependent variables in this problem.
- b) Write down the equation of the function that relates the height of the drone ($y$) to the time elapsed ($x$) since the button was pressed.
- c) What is the height of the drone after $6$ seconds? How long does it take to land completely (height $0$)?
A water tank with a capacity of $1000$ liters has a leak. Every hour that passes, the volume of water decreases by a constant amount. $3$ hours after detecting the leak,…
A water tank with a capacity of $1000$ liters has a leak. Every hour that passes, the volume of water decreases by a constant amount. $3$ hours after detecting the leak, $850$ liters remain.
a) Find the equation of the linear function relating the elapsed time ($x$) to the volume of water in the tank ($y$).
b) If the leak is not repaired, how much water will remain after $10$ hours?
c) Plot the function and state the domain and range of the function, taking into account the physical context of the problem (time cannot be negative and volume cannot be less than zero).
A laptop battery discharges linearly. At 10:00 in the morning, the battery was at $90\%$. At 12:30 in the morning, the level had dropped to $45\%$. a) If we let $x$ be…
A laptop battery discharges linearly. At 10:00 in the morning, the battery was at $90\%$. At 12:30 in the morning, the level had dropped to $45\%$.
a) If we let $x$ be the time elapsed in hours since 10:00, find the equation of the function giving the percentage of remaining battery.
b) What is the gradient of the function? Explain its meaning in this real-life context.
c) At what exact time will the battery run out completely if consumption remains the same?
A digital content creator analysed the growth of their followers. During the first month, their growth followed a linear function passing through points $A(2, 500)$ and…
A digital content creator analysed the growth of their followers. During the first month, their growth followed a linear function passing through points $A(2, 500)$ and $B(5, 1250)$, where $x$ is the day of the month and $y$ is the number of followers. However, from day 15 onwards, due to a viral video, their growth became another linear function parallel to the previous one, but which on day $x = 16$ had already reached $5000$ followers.
a) Find the equation of the linear function for the first section (days 1 to 15).
b) Find the equation of the new function from day 15 onwards.
c) How many followers did they have on day 20 according to this second function?
Parallel lines
If the point is not on the $Y$ axis, you substitute its coordinates to find $n$. For example, parallel to $y = \frac{3}{2}x + 2$ through $P(-4, -1)$: $m = \frac{3}{2}$ and $-1 = \frac{3}{2}(-4) + n \Rightarrow -1 = -6 + n \Rightarrow n = 5$, that is, $y = \frac{3}{2}x + 5$.
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.
In a drone competition, "Drone A" follows a straight-line trajectory represented by the function $y = \frac{3}{2}x + 2$. A "Drone B" is to be launched whose trajectory…
In a drone competition, "Drone A" follows a straight-line trajectory represented by the function $y = \frac{3}{2}x + 2$. A "Drone B" is to be launched whose trajectory is a straight line parallel to that of "Drone A", but must pass through the point of a beacon located at the coordinates $P(-4, -1)$.
a) Determine the equation of the linear function describing the trajectory of "Drone B".
b) At what point will the trajectory of "Drone B" cross the y-axis (axis $Y$)?
c) Plot both trajectories on a graph and state whether either of them is a direct proportion function.
You can do it on paper and send a photo with your answer.
Given the functions $f(x) = ax + 3$ and $g(x) = -2x + b$: a) If both lines are parallel and line $g(x)$ passes through the point $P(1, 4)$, calculate the values of $a$…
Given the functions $f(x) = ax + 3$ and $g(x) = -2x + b$:
a) If both lines are parallel and line $g(x)$ passes through the point $P(1, 4)$, calculate the values of $a$ and $b$.
b) Once $a$ and $b$ have been found, calculate the point where line $f(x)$ crosses the x-axis (axis $X$).
c) Is either of these functions a constant function? Justify your answer.
Intercepts with the axes and area of the triangle
A line crosses the axes at two points that are easy to calculate:
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$Y$ axis: set $x = 0$. You get $(0, n)$.
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$X$ axis: set $y = 0$ and solve for $x$. From $0 = mx + n$ you get $x = -\dfrac{n}{m}$ (if $m \neq 0$).
If the line does not pass through the origin, these two intercepts and the origin form a right triangle with the axes. Its legs are $|n|$ and $\left|\dfrac{n}{m}\right|$, so:
$$\text{Area} = \frac{\text{base} \times \text{height}}{2} = \frac{|n| \cdot \left|\frac{n}{m}\right|}{2}$$When the problem gives the area but not $n$, use the fact that the line passes through a known point: substitute the point to write $m$ in terms of $n$, and set up the area equation with that single unknown.
It is known that a linear function is represented graphically by a line passing through the point $A(3, -2)$. Furthermore, we know that this line forms a triangle with…
It is known that a linear function is represented graphically by a line passing through the point $A(3, -2)$. Furthermore, we know that this line forms a triangle with the coordinate axes whose area is $4$ square units, and its y-intercept is a positive integer.
a) Determine the value of the y-intercept ($n$).
b) Calculate the gradient ($m$) of the line.
c) Write the equation of the line and check that the point $A$ belongs to it.
Intersection of lines and comparing rates
To find the point where two lines intersect, set their equations equal: it is the value of $x$ where both give the same $y$. In problems, $n$ is usually the fixed cost and $m$ the cost per unit.
Another example: the line $y = 2x - 5$ and the line $y = 3$ intersect where $2x - 5 = 3$, that is $x = 4$, at the point $(4, 3)$.
Two electric scooter rental companies have different rates. EcoRolling charges $\text{RD\$}\ 12…
Two electric scooter rental companies have different rates. EcoRolling charges $\text{RD\$}\ 12$ per minute of use with no fixed fee. FastGlide charges a fixed unlocking fee of $\text{RD\$}\ 120$ and then $\text{RD\$}\ 6$ per minute.
- a) Write the function relating total cost ($y$) to usage time in minutes ($x$) for each company.
- b) State which of the two is a direct proportion (linear) function and which is an affine function.
- c) Above how many minutes of travel does FastGlide start to be the cheaper option?
Lucía is saving money in a piggy bank. Initially, she has $\text{RD\$}\ 900$, and each week she decides to put in $\text{RD\$}\ 150$ from her allowance. a) Complete the…
Lucía is saving money in a piggy bank. Initially, she has $\text{RD\$}\ 900$, and each week she decides to put in $\text{RD\$}\ 150$ from her allowance.
- a) Complete the following table of values:
| Weeks ($x$) | $0$ | $1$ | $2$ | $4$ | $10$ |
|---|---|---|---|---|---|
| Money saved ($y$) | $900$ |
- b) Write down the equation of the function.
- c) If Lucía wants to buy a video game costing $\text{RD\$}\ 3\,300$, how many weeks will she have to wait according to this function?
Two fiber optic companies offer the following monthly tariffs: NetRápido: a fixed $1\,200$ pesos per month plus $30$ pesos for each extra GB downloaded. FibraTotal: no…
Two fiber optic companies offer the following monthly tariffs:
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NetRápido: a fixed $1\,200$ pesos per month plus $30$ pesos for each extra GB downloaded.
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FibraTotal: no fixed fee, but charges $78$ pesos 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?
Find the equation of the line $r$ that satisfies the following conditions simultaneously: 1. It is decreasing. 2. Its slope is twice (in absolute value) the slope of the…
Find the equation of the line $r$ that satisfies the following conditions simultaneously:
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It is decreasing.
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Its slope is twice (in absolute value) the slope of the line that passes through the points $C(0, 1)$ and $D(2, 2)$.
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It passes through the intersection point of the lines $y = 2x - 5$ and $y = 3$.
a) First calculate the slope of the line that passes through $C$ and $D$.
b) Determine the intersection point mentioned in condition 3.
c) Write the final equation of the line $r$.
Back to the beginning: the equation for Lucía's piggy bank is $y = 150x + 900$: it starts at 900 pesos ($n = 900$) and increases 150 pesos per week ($m = 150$). The scooter charges $y = 5x + 10$.