Mathematics

Elementary Functions

Study of lines, parabolas, rational, exponential, logarithmic, and trigonometric functions.

The line is a 1st-degree polynomial function. The slope-intercept form of a line is:
y=mx+ny = mx + n
Where:
  • nn: yy-intercept (the point where it crosses the Y-axis).
  • mm: slope, which is calculated given two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) as:
m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
To graph it, we find two points (for example, the axis intercepts, by setting x=0x = 0 and y=0y = 0).
Table of values to graph a line
Special cases
  • Horizontal line: Its equation is of the form y=ny = n.
  • Vertical line: Its equation is of the form x=kx = k.
  • Line passing through the origin: Its equation is of the form y=mxy = mx.
Graphs of horizontal, vertical lines and lines passing through the origin
Equation of a Line and Linear Interpolation
Given two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the point-slope equation of the line passing through them is:
y−y1=y2−y1x2−x1⋅(x−x1)y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} \cdot (x - x_1)
Linear interpolation consists of finding an unknown third point between two known points, assuming the relationship between the variables is linear.

Example: Knowing that a 5 km taxi ride costs 16,anda12kmridecosts16, and a 12 km ride costs 30, calculate the cost of a 10 km ride.

The function we are looking for, yy, is the price of the trip as a function of the distance in kilometers, xx. We know two points of this function: (5,16)(5, 16) and (12,30)(12, 30). Applying the formula:
y−16=30−1612−5⋅(x−5)y - 16 = \frac{30 - 16}{12 - 5} \cdot (x - 5)
Simplifying, we get:
y=2x+6y = 2x + 6
To find the requested value (how much a 1010 km trip costs), we substitute x=10x = 10 into the obtained expression:
y=2⋅10+6y = 2 \cdot 10 + 6
The price of a 1010 km trip will be y=$26y = \$26.
It is a 2nd-degree polynomial function of the form:
y=ax2+bx+cy = ax^2 + bx + c
  • Domain: Dom=RDom = \mathbb{R}
  • If a>0a > 0: the parabola is concave up ∪\cup (opens upwards), and it has a minimum at the vertex.
  • If a<0a < 0: the parabola is concave down ∩\cap (opens downwards), and it has a maximum at the vertex.
  • The xx-coordinate of the vertex is given by:
xv=−b2ax_v = -\frac{b}{2a}
  • The constant term cc is the yy-intercept.
To graph it, we find the vertex and the intercepts with the axes.
Table of values of the parabola with vertex and intercepts
Graph of the parabola with vertex and intercepts
Quadratic interpolation
Given three points (x1,y1),(x2,y2),(x3,y3)(x_1, y_1), (x_2, y_2), (x_3, y_3), the equation of the parabola passing through them is determined by solving the system:
{ax12+bx1+c=y1ax22+bx2+c=y2ax32+bx3+c=y3\begin{cases} ax_1^2 + bx_1 + c = y_1 \\ ax_2^2 + bx_2 + c = y_2 \\ ax_3^2 + bx_3 + c = y_3 \end{cases}
Where the unknowns to be determined are the parameters of the parabola: a,b,ca, b, c.
Rational functions are of the form:
f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}
Where P(x)P(x) and Q(x)Q(x) are polynomials.
  • Domain: Dom=R−{Q(x)=0}Dom = \mathbb{R} - \{Q(x) = 0\}
  • It has asymptotes:
    • Vertical: At the roots of the denominator that are not roots of the numerator.
    • Horizontal: If degree of P(x)≤P(x) \le degree of Q(x)Q(x).
    • Slant (Oblique): If degree of P(x)=P(x) = degree of Q(x)+1Q(x) + 1.
Inverse Proportionality Function
f(x)=kxwith k≠0f(x) = \frac{k}{x} \quad \text{with } k \neq 0
It has the following characteristics:
  • Domain Dom=R−{0}Dom = \mathbb{R} - \{0\}.
  • It has no axis intercepts.
  • Vertical asymptote at x=0x = 0.
  • Horizontal asymptote at y=0y = 0.
Graph of an inverse proportionality function
They are of the form:
y=f(x)ny = \sqrt[n]{f(x)}
  • If nn is even: The domain will be the interval where f(x)≥0f(x) \ge 0.
  • If nn is odd: Dom=RDom = \mathbb{R}.
Graph of the square root function (even index)
Graph of the cube root function (odd index)
The Sine Function
f(x)=sin⁡x \displaystyle f(x) = \sin x
It has the following characteristics:
  • Domain Dom=RDom = \mathbb{R}
  • Periodic with period P=2πP = 2\pi
  • yy-intercept at (0,0)(0, 0)
  • xx-intercepts: x=kπ(k∈Z)x = k\pi \quad (k \in \mathbb{Z})
  • Relative maximums: x=π/2+2kπ(k∈Z)x = \pi/2 + 2k\pi \quad (k \in \mathbb{Z})
  • Relative minimums: x=3π/2+2kπ(k∈Z)x = 3\pi/2 + 2k\pi \quad (k \in \mathbb{Z})
The Cosine Function
f(x)=cos⁡x \displaystyle f(x) = \cos x
It has the following characteristics:
  • Domain Dom=RDom = \mathbb{R}
  • Periodic with period P=2πP = 2\pi
  • yy-intercept at (0,1)(0, 1)
  • xx-intercepts: x=π/2+kπ(k∈Z)x = \pi/2 + k\pi \quad (k \in \mathbb{Z})
  • Relative maximums: x=2kπ(k∈Z)x = 2k\pi \quad (k \in \mathbb{Z})
  • Relative minimums: x=π+2kπ(k∈Z)x = \pi + 2k\pi \quad (k \in \mathbb{Z})
Overlapping graphs of the sine and cosine functions
The Tangent Function
f(x)=tan⁡x \displaystyle f(x) = \tan x
It has the following characteristics:
  • Dom=R−{π/2+kπ}(k∈Z)Dom = \mathbb{R} - \{\pi/2 + k\pi\} \quad (k \in \mathbb{Z})
  • Period: P=πP = \pi
  • It has no maximums or minimums.
  • Vertical asymptotes:
    x=π/2+kπ(k∈Z)x = \pi/2 + k\pi \quad (k \in \mathbb{Z})
Graph of the tangent function showing its vertical asymptotes
The natural logarithmic function:
f(x)=ln⁡x \displaystyle f(x) = \ln x
or in general, any logarithmic function:
f(x)=log⁡bx \displaystyle f(x) = \log_b x
They have the following characteristics:
  • Dom=(0,∞)Dom = (0, \infty)
  • xx-intercept at (1,0)(1, 0)
  • Vertical asymptote at x=0x = 0
Graph of the logarithmic function
The natural exponential function:
f(x)=ex \displaystyle f(x) = e^x
or in general, any exponential function (inverse of the logarithmic function):
f(x)=ax \displaystyle f(x) = a^x
They have the following characteristics:
  • Domain Dom=RDom = \mathbb{R}
  • yy-intercept at (0,1)(0, 1)
  • Horizontal asymptote at y=0y = 0
Graph of exponential functions with base greater and less than 1
A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain.

Example:
f(x)={x2+4x+3x<0x+3x≥0f(x) = \begin{cases} x^2 + 4x + 3 & x < 0 \\ x + 3 & x \ge 0 \end{cases}
  • In the interval (−∞,0)(-\infty, 0), the function f(x)f(x) is the parabola y=x2+4x+3y = x^2 + 4x + 3.
  • In the interval [0,∞)[0, \infty), the function f(x)f(x) is the line y=x+3y = x + 3.
Each branch (sub-function) is graphed by analyzing its respective characteristics:
Table of values for each branch of the piecewise function
Graph of the piecewise function combining a parabola and a line
Absolute value functions: They can be rewritten as piecewise functions. We find the roots of the expression inside the absolute value and construct a sign chart. For the intervals where the expression is negative, we multiply it by −1-1.

Example:
f(x)=3x+∣x2−4∣  ⟹  x2−4=0  ⟹  x=−2 ; x=2f(x) = 3x + |x^2 - 4| \implies x^2 - 4 = 0 \implies x = -2 \text{ ; } x = 2
Sign chart for the polynomial x^2 - 4
f(x)={3x+(x2−4)if x≤−23x−(x2−4)if −2<x<23x+(x2−4)if x≥2f(x) = \begin{cases} 3x + (x^2 - 4) & \text{if } x \le -2 \\ 3x - (x^2 - 4) & \text{if } -2 < x < 2 \\ 3x + (x^2 - 4) & \text{if } x \ge 2 \end{cases}