Exercises: Types of linear functions. First-degree polynomial.

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) $h(x) = 8$

Find the equation of the line that has gradient $m = -2$ and passes through the point $(1, 4)$.

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?

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 $y = 2x + 1$.

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

Marcos has the affine function $f(x) = 3x + 4$ and needs to calculate the image of $x = 10$. Since he knows that $f(5) = 3 \cdot 5 + 4 = 19$, 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 Marcos'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 Marcos'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.

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.

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.

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.

Determine the value of the parameter $k$ such that the line with equation $y = kx + 5$ passes through the point $(3, -4)$.

The relationship between temperature in degrees Celsius ($^\circ C$) and degrees Fahrenheit ($^\circ F$) is linear. Water freezes at $0^\circ C$, which is equivalent to $32^\circ F$, and boils at $100^\circ C$, which is equivalent to $212^\circ F$. Find the linear function to convert degrees Celsius ($x$) to degrees Fahrenheit ($y$).

Calculate the area of the triangle formed by the line $y = -\frac{3}{4}x + 3$ intersecting the coordinate axes in the first quadrant.

The fare of an urban taxi service in a Spanish capital city consists of a fixed "flag drop" of $1,80$ € right at the start of the journey and a cost of $1,15$ € for each completed kilometre.\na) Write the analytical expression of the function $C(x)$ that gives the total cost of the trip in euros as a function of the kilometres $x$ travelled. Justify whether it is a linear, affine, or constant function, and interpret what the slope and the $y$-intercept represent in the context of the problem.\nb) A customer claims: "If I take an $18$ km trip, I will pay exactly three times as much as if I take a $6$ km trip". Justify mathematically whether this statement is true or false by calculating both amounts.\nc) If a person has a maximum budget of $20$ €, set up and solve an equation or inequality to determine how many full kilometres (whole number) they can travel at most.

According to meteorological measurements by AEMET in the troposphere, the temperature decreases uniformly by approximately $0,65$ °C for every $100$ m increase in altitude. In Cercedilla, a town located $1200$ m above sea level, the thermometer reads $17$ °C at midday.\na) Before formulating any equation, mentally estimate what approximate temperature there will be at the Navacerrada pass ($1860$ m in altitude): will it be closer to $12$ °C or $15$ °C? Explain your reasoning by rounding the difference in altitude to whole hundreds.\nb) Model the temperature using a first-degree function $T(h) = mh + n$, where $h$ is the altitude in metres. Determine with justification the value of the slope $m$ (in °C/m) and the value of the $y$-intercept $n$.\nc) Calculate with the mathematical function the exact value of the temperature at the Navacerrada pass and compare it with your initial estimate to assess whether it was accurate.

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.

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 of the line $s$.

A car park charges $2,50$ € for the first hour (or fraction thereof). After the first hour, it charges $0,05$ € for each additional minute. Write the piecewise function $C(t)$ representing the total cost in euros as a function of time $t$ in minutes that a vehicle is parked.

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.

The following graph shows the distance of a cyclist from home as a function of time. Determine the piecewise algebraic expression of the function $d(t)$ and calculate the constant speed of the cyclist during the return journey.

Given the lines $y = (2m - 1)x + 5$ and $y = 3x - n$, determine the values of the parameters $m$ and $n$ such that both equations represent exactly the same line (coincident lines).

Find the values of $a$ and $b$ such that the line $y = ax + b$ simultaneously satisfies two conditions: being parallel to the line $y = \frac{1}{2}x$ and passing through the point of intersection of the lines $y = 2x - 4$ and $y = -x + 5$.

Consider the following table of experimental values corresponding to an unknown function $f(x)$:\n$x$: $-3$, $1$, $3$, $7$\n$y$: $14$, $6$, $2$, $-6$\n\nTo find out whether the table can be described by a first-degree function and to find its formula, two students propose different strategies:\n- Lucía's method: She sets up a system of two equations with unknowns $m$ and $n$ using the first two rows. Then she substitutes the $x$-values of the other two rows to verify if the $y$-values match.\n- David's method: He calculates the ratio $\dfrac{\Delta y}{\Delta x}$ between each row and the next; if they all yield the same result, he asserts that the function is a first-degree polynomial, takes that value as the slope $m$, and solves for $n$ using any point.\n\na) Compare both methods and justify which of them allows one to rule out sooner that the function is of the first degree in the event that one of the rows contains a measurement error.\nb) Use the method you consider most efficient to prove whether the data correspond to a straight line. If so, classify it with justification (linear or affine) and find its formula $f(x) = mx + n$.\nc) Without explicitly calculating the value of $x$, reason a priori whether the preimage of $-100$ will be a positive or negative number based on the properties of the obtained function. Then, calculate it exactly.

Consider the family of functions $f_k(x) = (2k - 4)x + (3k + 6)$, where $k$ is a real parameter that is fixed in each case.\na) Determine the value of $k$ for which the function is constant. Write its equation and describe the geometric position of its graph with respect to the coordinate axes.\nb) Find the value of $k$ that turns $f_k$ into a linear function (of direct proportionality). What is the value of its slope in this case?\nc) Determine analytically whether there exists any value of $k$ such that the line passes through the point $A(2, 4)$ and, at the same time, the resulting function is strictly increasing. Strictly justify your conclusion.\nd) For $k = 1$, find the intersection points of the line with the coordinate axes and calculate the area of the triangle that this line forms together with the Cartesian axes.