The idea of a linear function. Types
A plumber charges a fixed fee plus a price per hour. A tank empties at a constant rate. A flat-rate tariff never changes, no matter how much you use. All three situations are first-degree functions, but they are not all the same type. Learning to tell them apart is the first step towards working with them.
Linear, affine or constant
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The line $y = 0,7x$ passes through the origin: it is linear.
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The line $y = 0,5x + 40$ crosses the $Y$-axis at $40$, not at $0$: it is affine.
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The line $y = 70$ is horizontal: it is constant, it always equals $70$ whatever $x$ is.
Identify the gradient ($m$) and y-intercept ($n$) of the following linear functions: a) $y = 5x - 2$ b) $y = -\frac{3}{4}x + 1$ c) $y = 8$ d) $y = \frac{x}{2}$
Identify the gradient ($m$) and y-intercept ($n$) of the following linear functions:
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a) $y = 5x - 2$
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b) $y = -\frac{3}{4}x + 1$
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c) $y = 8$
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d) $y = \frac{x}{2}$
Classify the following functions into direct proportion functions (linear functions), linear functions (affine functions), or constant functions: a) $y = -3x + 5$ b)…
Classify the following functions into direct proportion functions (linear functions), linear functions (affine functions), or constant functions:
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a) $y = -3x + 5$
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b) $y = \frac{4}{5}x$
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c) $y = -7$
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d) $y = x$
Gradient and y-intercept
In $y = mx + n$, $m$ is the gradient (how much the line rises or falls for each unit that $x$ increases) and $n$ is the $y$-intercept (the value of $y$ when $x = 0$, the point where the line crosses the $Y$-axis).
Given the linear function $f(x) = -2x + 9$, calculate the images of $x = 3$, $x = -1$ and $x = \frac{1}{2}$.
Given the linear function $f(x) = -2x + 9$, calculate the images of $x = 3$, $x = -1$ and $x = \frac{1}{2}$.
Does the point lie on the line?
To check whether a point $P(a, b)$ lies on the graph of $f$, substitute $x = a$ into the formula: if the result is $b$, the point lies on the graph; if not, it does not.
Determine mathematically whether the points $A(2, 5)$ and $B(-1, -3)$ lie on the graph of the function $y = 3x - 1$.
Determine mathematically whether the points $A(2, 5)$ and $B(-1, -3)$ lie on the graph of the function $y = 3x - 1$.
Calculate the gradient of the line passing through the points $A(0, 4)$ and $B(2, 10)$.
Calculate the gradient of the line passing through the points $A(0, 4)$ and $B(2, 10)$.
From the table to the equation
If a table of values corresponds to a direct proportion function, the quotient $\frac{y}{x}$ is always the same: that constant is $m$.
Write the equation of the direct proportion function whose graph passes through the point $P(4, -12)$.
Write the equation of the direct proportion function whose graph passes through the point $P(4, -12)$.
A direct proportion function is given by the following table of values. Find its algebraic expression $y = f(x)$. $x$: $2, 4, 6$ $y$: $7, 14, 21$
A direct proportion function is given by the following table of values. Find its algebraic expression $y = f(x)$.
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$x$: $2, 4, 6$
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$y$: $7, 14, 21$
Observe the following graph showing two lines, $A$ and $B$. Determine the algebraic expression for each of them by analysing the points they pass through.
Observe the following graph showing two lines, $A$ and $B$. Determine the algebraic expression for each of them by analysing the points they pass through.
Modelling real-life situations
When translating a problem into a function, first identify whether there is a fixed fee (affine, $n \neq 0$) or not (linear, $n = 0$); then work out how much $y$ changes for each unit of $x$ (that is $m$).
A plumber charges a fixed call-out fee of $40$ €, plus $25$ € per hour of work. Write the function relating the total cost of the service ($y$) to the hours worked…
A plumber charges a fixed call-out fee of $40$ €, plus $25$ € per hour of work. Write the function relating the total cost of the service ($y$) to the hours worked ($x$). How much will a repair that took $3,5$ hours cost?
A spring measures $15$ cm at rest. When weights are attached to it, it is found that for each additional kilogram the spring stretches by $3$ cm (according to Hooke's…
A spring measures $15$ cm at rest. When weights are attached to it, it is found that for each additional kilogram the spring stretches by $3$ cm (according to Hooke's law). Write the linear function relating the total length of the spring ($L$) to the attached mass in kilograms ($m$). What length will it have if a mass of $4,5$ kg is attached?
Two phone companies offer different plans. In the graph of the monthly cost in euros ($C$) as a function of the minutes used ($t$), the line for Plan A passes through…
Two phone companies offer different plans. In the graph of the monthly cost in euros ($C$) as a function of the minutes used ($t$), the line for Plan A passes through the points $(0, 10)$ and $(200, 20)$, while that of Plan B passes through the origin $(0, 0)$ and $(200, 20)$. From the graph, determine the algebraic expression of each plan and explain, by solving the corresponding system, from how many minutes it is better to choose Plan A over Plan B.
Back to the beginning: The plumber's cost is affine (there is a fixed call-out fee), the emptying of the tank is linear (the water drained is proportional to time) and the flat-rate tariff is constant (the amount does not vary with use).