Fundamental Math

IT100 Session 4: Algebra Review and Functions
20 Sept 2017 presented by Dr Bob Batzinger

Agenda

  • Simplification of relationships
  • Basic Algebraic strategies
  • Functions vs Non-Functions
  • Transformation of functions

1. Simplification

Some useful relationships

\[ \huge\begin{array}{rcl} (x+a)(x+b) &=& x^2 + (a+b)x + ab\\ (x+a)(x+a) &=& x^2 + 2a + a^2\\ (x+a)(x-a) &=& x^2 - a^2\\ (ax+b)(cx+d)&=& ac x^2 + (ad+bc)x + bd\\ \end{array} \]

Simplifying using additive inverses

\[ \huge\begin{array}{rcl} x + 3 & = & 5 \\ x = x + 3 -(3) & = & 5 - (3) = 2 \\ & & \\ x + 7 & = & 2x + 3 \\ 7 = x + 7 - (x) & = & 2x + 3 - (x) = x + 3\\ 4 = 7 - (3) & = & x + 3 - (3) = x \\ \end{array} \]

Simplifying using reciprocal values

\[ \huge\begin{array}{rcl} 2 x & = & 16\\ {2x \over (2)} & = & {16 \over (2)}\\ x & = & 8 \\ \end{array} \]

\[ \huge\begin{array}{rcl} {3\over x} &= & 36\\ 1 = {3(x)\over (3)x} &= & {36(x) \over (3)} = 12x\\ {1\over 12} = {1\over (12)} &=& {12x\over (12)} = x\\ \end{array} \]

Case 1

\[ \huge\begin{array}{c} \left({10h^2 - 9h - 9\over 2h^2 - 19h + 24}\right)\left({h^2 - 16h + 64\over 5h^2 - 37h - 24}\right)\\ {(5h+3)(2h-3)(h-8)(h-8)\over (2h-3)(h-8)(5h+3)(h-8)}\\ {(5h+3)(2h-3)(h-8)(h-8)\over (5h+3)(2h-3)(h-8)(h-8)}\\ 1\\ \end{array} \]

Case 2

\[ \huge\begin{array}{c} \left({x^2 - x - 6\over x^2 + x - 6}\right) \left({x^2 + 8x + 15\over x^2 - 9}\right)\\ {(x-3)(x+2)(x+3)(x+5)\over (x+3)(x-2)(x+3)(x-3)}\\ {(x+2)(x+5)\over (x-2)(x+3)}= {x^2 +7x + 10\over x^2 + x - 6}\\ \end{array} \]

Challenge

Simplify the following:
\[ \huge\begin{array}{c} \left({y^2 + 10y + 25\over y^2 + 11y + 30}\right)\\ & & \\ \left({2x^2 + 7x - 4\over 4x^2 + 2x - 2}\right)\\ & & \\ \left({c^2 + 2c - 24\over c^2 + 12c + 36}\right)\left({c^2 - 10c + 24\over c^2 - 8c + 16}\right)\\ \end{array} \]

2. Algebraic Strategies

Algebra

  • from Arabic “al-jabr”
    meaning “reunion of broken parts”

  • a systematic application of the basic rules to simplify and solve equations

Strategy for simultaneous equations

  1. Remove common multiples and simplify the equations separately
  2. Transform the equations placing variables on one side of the equals and constants on the other
  3. Combine terms
  4. Solve for one variable
  5. Substitute the value of the known variable to find the values of the others

Case 1

\[ \huge\begin{array}{rcl} 3x - 2y = 57; & & x + y = 44\\ & & \\ 2x + 2y=(2)(x + y) &=& (2)\times 44 = 88\\ & & \\ 5x = (3x-2y)+ 2x +2y &= & 57 + 88 = 145\\ x=5x/5&=&145 /5 = 29\\ & & \\ 29 + y &=& 44\\ y =29 + y -(29) & = & 44- (29)= 15\\ \end{array} \]

Case 2

\[ \huge\begin{array}{rcl} 8x - 3y = 8; & & x + 2y = 20\\ & & \\ 8x +16y = 8(x + 2y) &=& 8\times 20=160\\ 19y = 8x +16y -(8x - 3y) &=& 160 - 8 = 152\\ y = 19y/19 & = & 155/19 = 8\\ & & \\ x + 16 =x +2 \times 8 &= & 20\\ x = x + 16 - (16) &=& 20 -(16) = 4\\ \end{array} \]

Challenge

\[ \huge\begin{array}{rcl} 3x + 7 & = & 5x - 3\\ & & \\ {-2 \over (3x+5)} & = & 6x - 4\\ & & \\ {2x + 7\over x - 2} &=& 27\\ & & \\ 2x+5& = & 6y -15\\ x+8& = & y + 6\\ \end{array} \]

3. Functions vs Non-Functions

Function Definition

A function is a relation in which each possible input value leads to exactly one output value.

\[ \huge\begin{eqnarray} Input & \rightarrow & \hbox{Function} & \rightarrow & Output\\ x\qquad & & f(x) & & {}\qquad y\\ \hbox{domain} & & y = {x \over 5} & & \hbox{range}\\ \large [0, 1, 2 ... n] & & & & \large [0, 0.2, 0.4 ... n/5] \\ \end{eqnarray} \]

Common non-function relationships

Circle
\( \huge 25= x^2 + y^2 \)
circle

Ellipse
\( \huge 25 = 2x^2+ 5y^2 \)
circle

Hyperbola

\( \huge 25= x^2 y^2 \)
circle

\( \huge 25=\left(x+1\right) \left(x-5\right) \left(y-1\right) \left(y+4\right) \)
circle

Egg and Heart Shapes

\( \huge 25 = \frac{x^2}{0.5} + \frac{5y^2}{1-0.1x} \)
egg

\( \huge\left(x^2+y^2-2ax\right)^2 = 4a^2\left(x^2+y^2\right) \) flower

Flowers

\( \huge\left(x^2+y^2\right)^3 = 4x^2y^2 \)
flower

Challenge

  • Determine why these are considered non-functions in this form?

  • Is there a domain or mathematical transform that would make any of these a function?

  • Answer the questions on pg 20: Numbers 52-69

  • Give answers for Applied Exercises on pg 21: Numbers 88-90

4. Inverse Functions

An inverse function will convert output back to the corresponding input value.

\[ \huge\begin{eqnarray} Input & \rightarrow & \hbox{Function} & \rightarrow & Output\\ x\qquad & & f(x) & & {}\qquad y\\ & & & & \\ Output & \rightarrow & \hbox{Inverse Function} & \rightarrow & Input\\ y\qquad & & f\prime(y) & & {}\qquad x\\ \end{eqnarray} \]

Challenge

Input \( \huge(x) \) Function Output \( \huge (y) \) Inverse Function Reverted \( \huge (x_2) \) Does \( \huge x = x_2 \) ?
\( \Large [-5,-2,0,2,5] \) \( \huge y = 3x \) \( \Large [-15,-6,0,6,15] \) \( \huge x_2 = y / 3 \) \( \Large [-5,-2,0,2,5] \) True
\( \Large [-5,-2,0,2,5] \) \( \huge y = x^2 \) \( \Large [25,4,0,4,25] \) \( \huge x_2 = \sqrt{y} \)
\( \Large [-5,-2,0,2,5] \) \( \huge y = 3x + 4 \) \( \Large [-11,-2,4,10,19] \)
\( \Large [-5,-2,0,2,5] \) \( \huge y = 1/x \)
\( \Large [-5,-2,0,2,5] \) \( \huge y = |x|/x \)

type: section

5. Function Transforms

Function transforms

Transform Formula Parabola Line Normal Curve
(N) None \( \huge y = f(x) \) \( \huge y = x^2 \) \( \huge y = 2x \) \( \Huge y = {1\over \sigma\sqrt{2\pi}} e^{-{(x-\mu)^2\over 2 \sigma^2)}} \)
(A) Horizontal \( \huge y = f(x+A) \) \( \huge y = (x+A)^2 \) \( \huge y = 2(x + A) \) \( \Huge y = {1\over \sigma\sqrt{2\pi}} e^{-{(x+A-\mu)^2\over 2 \sigma^2)}} \)
(B) Vertical \( \huge y = f(x) + B \) \( \huge y = x^2 + B \) \( \huge y = 2x + B \) \( \Huge y = B+ {1\over \sigma\sqrt{2\pi}} e^{-{(x-\mu)^2\over 2 \sigma^2)}} \)
© Amplitude \( \huge y = C\times f(x) \) \( \huge y = C x^2 \) \( \huge y = C(2x) \) \( \Huge y = {C\over \sigma\sqrt{2\pi}} e^{-{(x-\mu)^2\over 2 \sigma^2)}} \)

Graph of Transformations

transforms

transforms

Normal Graph

transforms

Challenge

  • Compare the graph of \( \huge f(x+A) \) to that of \( \huge f(x) \)

  • Compare the graph of \( \huge f(x) + B \) to that of \( \huge f(x) \)

  • Compare the graph of \( \huge f(x)\times C \) to that of \( \huge f(x) \)

6. Topic for Session 5

Properties of Linear functions

Read Precalculus Chapter 2