Transcript section1

Section 1-1
Models and Solving Equations
Section 6-1 Topics
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numerical models
algebraic models
graphical models
solving factored equations
fundamental connection
grapher failure and hidden behavior
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)
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)
Graphical Models
• a visible representation of a numerical or
algebraic model
• examples of graphical models: scatter
plots, function graphing, charts, etc.
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.)
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)
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