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Table of Contents

Polynomial

$P(x) = a_0+a_1x+a_2x^2+a_3x^3+...+ a_nx^n$

  • examples of polynomials:
    \[\begin{align*} f(x) &= 2x^3 + 3x^2 + 7x - 3 \\ g(x) &= x^4 + 3x + 4 \\ t(x) &= x \end{align*}\]

Constant

$f(x) = a \quad \textrm{where} \quad y = a$

  • Dependent variable, $y$, has always the same value for all values of $x$, independent variable.

  • It’s a horizontal line

  • example: $f(x) = -3$

    x y
    0 -3
    1 -3
    3 -3
    4 -3

Affine

$f(x) = mx + n$ where $m$ is the slope and $n$ the origin of ordinates.

  • It is a polynomial function.

  • Grade 1

  • $P(0,0) \not\in f(x)$

  • example: $f(x) = -3x + 5$

Lineal

$f(x) = mx$

  • It is a polynomial function.

  • Grade 1.

  • $P(0,0) \in f(x)$

  • example: $f(x) = -3x$

Identity

$f(x) = x$

  • image of $x$ is always $x$

  • It is a polynomial function.

  • Grade 1.

  • $P(0,0) \in f(x)$

  • the slope, $m$, is always 1.

  • example: $f(x) = x$

Quadratic

$f(x) = a^2 + bx + c$ $\neq$ $a != 0$

  • They have always a vertical parabole shape.

    • a > 0 ,, concave \/
    • a < 0 ,, convex /\

  • Find parabole vertex

    • $V(V_x, V_y)$
    • $Vx = -b/2a$
    • $Vy = f(Vx)$

  • example: $f(x) = x^2 - 6x + 3$

    • Calculate vertex of $f(x) = x^2 - 6x + 3$:

      • $f(x) = x^2 - 6x + 3$

      • $Vx = -(-6) / (2*1) = 6/2 = 3$

      • $V_y = f(3) = 3^2 - 6*3 + 3 = 9 - 18 + 3 = -6$

      • $V(3, -6)$

Cubic

$f(x) = ax^3 + bx^2 + cx + d$ $\mid$ $a \neq 0$

  • example: $f(x) = x^3$

Rational

\[f(x)= \dfrac{P(x)}{Q(x)}\]

  • P grade and Q grade can be different

  • when P grade is equal to Q grade => horizontal asymptote

  • when Q grade is greater than P grade => horizontal asymptote

  • when y, P grade is greater in one unit than Q grade => oblique asymptote

  • example: $f(x) = \frac{1}{x}$

  • example: $f(x) = \frac{1}{x^2}$

Exponential

$f(x) = a^x$

  • if a > 0 funtion goes up

  • if a < 0 function goes down

  • if x > 0 fuction goes right

  • if c < 0 fuction goes left

  • example, $f(x) = 2^x$

    • example, $f(x)=\left(\dfrac{1}{3}\right)^x$
      • $f(x)=\left(\dfrac{1}{3}\right)^x = (3^{-1})^x = 3^{-x}$
      • $f(x) = 3^{-x}$
      • x < 0, then to left
      • 3 > 0, then up

Logarithm

$f(x)=log_a(x)$ $g(x)=log_a(x-p)$

  • there is a vertical asymptote and it is where (x) = 0. in $h(x)=log_2(x-2)$ , the asymptote is in $x-2=0$ then $x=2$.

  • $f(x)=nlog_a(x)$

    • x > 0, to the right
    • n > 0, up

  • example: $f(x) = log_2(x)$

Trigonometrical

  • basic trigonometrical functions are sin(x), cos(x) and tan(x).

  • $f(x) = sin(x)$

    x y
    π/2 1
    π 0
    3π/2 -1

  • $f(x) = cos(x)$

    x y
    0 1
    π/2 0
    π -1
    3π/2 0

  • $f(x) = tan(x)$

    x y
    0 0
    1 1.55
    -1 -1.55

Discontinuous

$f(x) = \left\{ \begin{array}{lr} g(x) \\ t(x) \\ ...\end{array}\right\}$

  • example: \[ f(x) = \left\{ \begin{array}{lr} x+3, &amp; x \le 1 \\ x-2, &amp; x&gt;1 \end{array}\right\} \]

    x+3
    x y
    1 4
    0 3
    -1 2
    x-2
    x y
    1 -1
    2 0

  • example: \[ f(x) = \left\{ \begin{array}{lr} 2, &amp; x \le 0 \\ x^2+2, &amp; x&gt;0 \end{array}\right\} \]

  • example: \[ f(x) = \left\{ \begin{array}{lr} x, &amp; x &lt; 1 \\ 2, &amp; 1 \le x \le 4 \\ 3-x, &amp; 4 &lt; x \end{array}\right\} \]