Models and Solving Equations

1
Section 1-1

• Models and Solving Equations

2
Section 6-1 Topics

• numerical models
• algebraic models
• graphical models
• solving factored equations
• fundamental connection
• grapher failure and hidden behavior

3
Numerical Models

• data is analyzed to gain insights into phenomena
• data is often in the form of a table or list
• sometimes the trends are easy to see but in some
  cases a more sophisticated tool is needed to do
  the analysis (computer spreadsheet)

4
Algebraic Models

• formulas and equations are created from numbers
  to make further predictions for unknown values
• some of the formulas we will use are ones we
  learned from other math courses or science
  courses (d rt, A lw, F ma)

5
Graphical Models

• a visible representation of a numerical or
  algebraic model
• examples of graphical models scatter plots,
  function graphing, charts, etc.

6
Solving Factored Equations

• zero-product property if ab 0, then a
  0 or b 0.
• to solve an equation, move everything to one
  side so the other side equals 0, factor the
  polynomial, then set each factor equal to 0 and
  solve it
• if the poly will not factor then other methods
  are needed (quad. formula, graphing, etc.)

7
Fundamental Connection

• If a is a real number that solves the equation
  f (x) 0, then these statements are
  equivalent
• a is a root (or solution) of the equation f(x)
  0
• a is a zero of y f (x)
• a is an x-intercept of the graph of y f (x)
• (x a) is a factor of f (x)

8
Grapher Failure and Hidden Behavior

• sometimes the graphing calculator will have
  problems correctly showing the graph you want to
  see for a function
• maybe the window is too large or too small
• the grapher has trouble with functions with a
  vertical asymptote
• maybe there are hidden zeros for a graph