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

Identify the slope $m$ 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 slope of $5$ and a y-intercept of $-2$.

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

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 a slope of $m = -2$ and passes through the point $(1, 4)$.

The fare of a taxi is given by the function $C(x) = 1.50x + 3$, where $C(x)$ is the cost in dollars and $x$ is the distance traveled in miles. How much will a trip of $8$ miles cost?

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

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

From the following graph, determine the slope, the y-intercept, and the equation of the line.

Find the equation of the line that passes 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 service call, plus $\$15$ for each hour of work. Write the algebraic expression of the function that relates the cost to the time worked and calculate how much they will charge for a repair that takes $3.5$ hours.

A water tank initially contains $800$ gallons and is emptying at a constant rate of $20$ gallons per minute. Write the function that relates the volume of water in the tank to the elapsed time and calculate in which minute the tank will empty completely.

Find analytically 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 $k$ so that the line with equation $y = kx + 5$ passes through the point $(3, -4)$.

The relationship between the temperature in degrees Celsius ($^\circ C$) and degrees Fahrenheit ($^\circ F$) is linear. We know that 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 that allows converting degrees Celsius ($x$) to degrees Fahrenheit ($y$).

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

Find the equation of the line that passes through the point $(0, 5)$ and forms a right triangle of area $10$ square units with the coordinate axes in the first quadrant.

A sneaker factory has fixed monthly costs of $\$5000$. In addition, producing each pair of sneakers costs $\$15$. If the company sells each pair for $\$40$, write the cost function $C(x)$ and the revenue function $R(x)$. How many pairs of sneakers 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 parking lot charges $\$2.50$ for the first hour (or fraction of an hour). After the first hour, it charges $\$0.05$ for each additional minute. Write the piecewise function $C(t)$ that represents the total cost in dollars as a function of the time $t$ in minutes that a vehicle is parked.

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

The following graph shows the distance a cyclist is from their home as a function of time. Determine the piecewise analytical expression of the function $d(t)$ and calculate what the cyclist's constant speed was during the return trip.

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

Find the values of $a$ and $b$ so 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 $y = -x + 5$.