First-degree polynomial functions
A polynomial of degree $1$ in $x$ has the form $mx+n$. As a function, $f(x)=mx+n$ is the simplest of the non-constant functions: its graph is always a straight line. Before moving on to the equations of a line in general form, it is worth establishing precisely what distinguishes a linear function from an affine one and from a constant one, and getting comfortable with calculating images.
The first-degree polynomial as a function
Identify the gradient $m$ and the y-intercept $n$ of the following linear functions: a) $y = -4x + 7$ b) $y = \frac{2}{5}x$ c) $y = -3$
Identify the gradient $m$ and the y-intercept $n$ of the following linear functions:
- a) $y = -4x + 7$
- b) $y = \frac{2}{5}x$
- c) $y = -3$
Write the equation of the line that has a gradient of $5$ and whose y-intercept is $-2$.
Write the equation of the line that has a gradient of $5$ and whose y-intercept is $-2$.
Calculating images and preimages
Calculating the image of $a$ means evaluating $f(a)$: substituting $a$ for $x$ in the expression. Calculating the preimage of $b$ (the values of $x$ such that $f(x)=b$) requires solving the first-degree equation $mx+n=b$.
Classify the following functions into direct proportion (strictly linear) function, linear function (affine function), or constant function: a) $f(x) = 3x - 1$ b)…
Classify the following functions into direct proportion (strictly linear) function, linear function (affine function), or constant function:
- a) $f(x) = 3x - 1$
- b) $g(x) = \frac{x}{4}$
- c) $h(x) = 8$
Find the equation of the line that has gradient $m = -2$ and passes through the point $(1, 4)$.
Find the equation of the line that has gradient $m = -2$ and passes through the point $(1, 4)$.
Find the equation of the line passing through the points $A(-2, 5)$ and $B(4, -7)$.
Find the equation of the line passing through the points $A(-2, 5)$ and $B(4, -7)$.
A point lying on the graph
A point $P(a,b)$ lies on the graph of $f$ if and only if $f(a)=b$. Checking this means substituting and comparing; it is different from determining the equation of a line, which in general requires two independent conditions (two points, or one point and the gradient).
A taxi fare is given by the function $C(x) = 1,50x + 3$, where $C(x)$ is the cost in euros and $x$ is the distance travelled in kilometres. How much will a journey of…
A taxi fare is given by the function $C(x) = 1,50x + 3$, where $C(x)$ is the cost in euros and $x$ is the distance travelled in kilometres. How much will a journey of $8$ kilometres cost?
Find the equation of the line that is parallel to $y = 3x - 2$ and passes through the point $(-1, 4)$.
Find the equation of the line that is parallel to $y = 3x - 2$ and passes through the point $(-1, 4)$.
From the table of values to the algebraic expression
A plumber charges $30$ € for a call-out fee, plus $15$ € for each hour of work. Write the algebraic expression of the function relating the cost to the time worked and…
A plumber charges $30$ € for a call-out fee, plus $15$ € for each hour of work. Write the algebraic expression of the function relating the cost to the time worked and calculate how much he will charge for a repair taking $3,5$ hours.
A water tank initially contains $800$ litres and is being emptied at a constant rate of $20$ litres per minute. Write the function relating the volume of water in the…
A water tank initially contains $800$ litres and is being emptied at a constant rate of $20$ litres per minute. Write the function relating the volume of water in the tank to the time elapsed and calculate in which minute the tank will be completely empty.
Find the equation of the line that passes through the point $(0, 5)$ and forms a right-angled triangle with the coordinate axes of area $10$ square units in the first…
Find the equation of the line that passes through the point $(0, 5)$ and forms a right-angled triangle with the coordinate axes of area $10$ square units in the first quadrant.
Modelling with first-degree functions
When translating a problem statement, you first identify whether there is a constant term independent of $x$ ($n\neq 0$, affine) or not ($n=0$, linear); then the rate of change $m$ (how much $y$ changes for each unit of $x$).
Find algebraically the point of intersection of the lines $y = 4x - 5$ and $y = -2x + 7$. Check that your result matches the point indicated on the following graph.
Find algebraically the point of intersection of the lines $y = 4x - 5$ and $y = -2x + 7$. Check that your result matches the point indicated on the following graph.
A trainer factory has monthly fixed costs of $5000$ €. In addition, producing each pair of trainers costs $15$ €. If the company sells each pair for $40$ €, write the…
A trainer factory has monthly fixed costs of $5000$ €. In addition, producing each pair of trainers costs $15$ €. If the company sells each pair for $40$ €, write the cost $C(x)$ and revenue $R(x)$ functions. How many pairs of trainers must they sell per month to make a profit of at least $2000$ €?
The line $r$ passes through the points $(1, 2)$ and $(3, 8)$. The line $s$ is parallel to $r$ and its y-intercept is half the y-intercept of $r$. Determine the equation…
The line $r$ passes through the points $(1, 2)$ and $(3, 8)$. The line $s$ is parallel to $r$ and its y-intercept is half the y-intercept of $r$. Determine the equation of the line $s$.