Linear functions f(x) = ax + b

Identify the gradient $m$ and the y-intercept $n$ of the following linear functions:

1. a) $y = -4x + 7$
2. b) $y = \frac{2}{5}x$
3. c) $y = -3$

Write the equation of the line that has a gradient of $5$ and whose y-intercept is $-2$.

Classify the following functions into direct proportion (strictly linear) function, linear function (affine function), or constant function:

1. a) $f(x) = 3x - 1$
2. b) $g(x) = \frac{x}{4}$
3. c) $h(x) = 8$

A first-degree polynomial function $f(x) = mx + n$ is sought that simultaneously satisfies three conditions:\n1. It is a decreasing affine function.\n2. It intersects the $y$-axis at a point located $4$ units below the origin of coordinates.\n3. The preimage of $2$ is equal to $-3$.\na) Determine with justification the exact values of the slope $m$ and the $y$-intercept $n$, and write the algebraic expression of $f(x)$.\nb) Justify from the sign of the slope why the function is decreasing.\nc) Check algebraically whether the point $Q(-6, 8)$ belongs to the graph of this function.

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?

Cian has the affine function $f(x) = 3x + 4$ and needs to calculate the image of $x = 10$. Since he knows that $f(5) = 3 \times 5 + 4 = 19$, he reasons as follows: "Since $10$ is twice $5$, the value of $f(10)$ must be exactly twice $19$, that is, $f(10) = 38$".\na) Identify Cian's conceptual error and explain why it is not valid to apply a direct rule of three to this type of functions.\nb) What value should the $y$-intercept $n$ have for Cian's reasoning to have been completely correct?\nc) Calculate the actual value of $f(10)$ using its formula and check algebraically how many units the value of $y$ actually increases by when $x$ increases by $5$ units.

The length $L$ of an elastic spring, measured in centimetres, is given by the function $L(p) = 15 + 0.2p$, where $p$ is the suspended weight in grams. Find the expression for the inverse function $p(L)$ and use it to calculate what weight was suspended if the spring measures exactly $22$ cm.

From the following graphical representation, determine the gradient, the y-intercept and the equation of the line.

Calculate the points of intersection with the coordinate axes (x-axis and y-axis) of the line $y = 3x - 12$.

Check algebraically whether the points $A(2, 5)$ and $B(-1, -3)$ lie on the graph of the function $y = 2x + 1$.

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 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 that is parallel to $y = 3x - 2$ and passes through the point $(-1, 4)$.

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 tank to the time elapsed and calculate in which minute the tank will be completely empty.

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 $2000$ €?