1129

Identify points for quadratic and linear function.  Construct the product.

The result looks consistent with a degree-3 polynomial.

(m x + k) ( a x^2 + b x + c) is degree 3.

Degree 3 factors into (x - x1) * ( irreducible quadratic) or into (x - z1) ( x - z3) ( x - z3).

The former is the case here.

We could get the equations of the linear and quadratic if we knew the x scale.  We can impose any x scale.

Exponential * linear ...

Dividing by small numbers:  stacks of thin f(x) blocks.

Sketch graphs of f(x) = (x - 3) and g(x) = (x+ 2).

• Identify the points on the x axis at which either f(x) = 0 or g(x) = 0.
• Identify the points on the x axis at which either | f(x) | = 1 or | g(x) | = 1.
• Identify the interval(s) of the x axis over which | f(x) | < 0.
• Identify the interval(s) of the x axis over which | g(x) | < 0.
• Identify the interval(s) of the x axis over which | f(x) | < 1.
• Identify the interval(s) of the x axis over which | g(x) | < 1.
• Identify the point(s) of the x axis for which f(x) = - g(x).

For the functions f(x) and g(x) of the preceding problem

• If f(x) = 0 then what is the value of the function f(x) + g(x)?
• If g(x) = 0 then what is the value of the function f(x) + g(x)?
• If f(x) = - g(x) then what is the value of the function f(x) + g(x)?
• If f(x) < 0 then what is the value of the function f(x) + g(x)?
• If f(x) > 0 then what is the value of the function f(x) + g(x)?
• Can you use this information to construct a graph of f(x) + g(x)?

For the functions f(x) and g(x) of the preceding problem

• If f(x) = 0 then what is the value of the function f(x) * g(x)?
• If g(x) = 0 then what is the value of the function f(x) * g(x)?
• If f(x) = 1 then what is the value of the function f(x) * g(x)?
• If g(x) = 1 then what is the value of the function f(x) * g(x)?
• If f(x) < 0 then what can we say about the value of the function f(x) * g(x)?
• If g(x) < 0 then what is the value of the function f(x) * g(x)?
• If f(x) < 0 and g(x) < 0 then what can we say about the value of g(x)?
• If f(x) < 0 and g(x) > 0 then what can we say about the value of g(x)?
• If f(x) > 0 and g(x) < 0 then what can we say about the value of g(x)?
• If f(x) > 0 and g(x) > 0 then what can we say about the value of g(x)?
• If | f(x)| = 1 then what is the value of the function f(x) * g(x)?
• If | g(x)| = 1 then what is the value of the function f(x) * g(x)?