Transcript graphical transformations

FACTORING
• We have been looking at factoring polynomials
BUT we have only factored quadratic functions.
• We need to be able to factor anything not just
unFOIL.
• The ACT question required us to take a step
toward that.
• The first thing we need to do is factor out any
single value that is a part of each term.
• y = 3x4 – 6x3 – 3x2 + 6x
• y = 3x (x3 – 2x2 – x + 2)
• Graph the remaining function and guess at what
some of the factors could be. (Look at where it
crosses the x-axis)
• How do we check to see if something is a
factor?
• How do we check with normal numbers?
• Is 14656 divisible by 32?
• Can you still do long division?
8 14656
POLYNOMIAL DIVISION
• We can do the same thing with functions.
x  1 x 3 - 2x 2 - x  2
FACTORS
• Now we know that (x + 1) is a factor of the
original polynomial.
• x3 – 2x2 – x + 2 = (x + 1)(x2 – 3x + 2)
• What are all the factors of this polynomial?
• Factor the second factor:
• (x – 2)(x – 1) = x2 – 3x + 2
• So,
• x3 – 2x2 – x + 2 = (x + 1)(x – 2)(x – 1)
• What are the zeroes of this function?
• x = -1, 1, and 2
ZEROES OF POLYNOMIALS
• We already know that the degree of the
polynomial tells us how many times the function
can change direction.
• It also tells us how many zeroes it can have.
• A linear function is degree 1 and has one zero
(x-intercept)
• A quadratic function is degree 2 and can cross
the x-axis twice.
• The degree is also the number of zeroes the
function can have.
CROSSING X-AXIS
• Just because a function has a zero does not
mean it crosses the x-axis at that point.
• The function may just touch the x-axis at that
point: it is still a zero but does not cross.
• Look at y = x2 + 6x + 9
• It factors (x + 3)(x + 3) and just touches at x = -3
• You can have more than one factor the same
• Factor: y = x4 – 4x3 – 2x2 + 12x + 9
• y = (x + 1)(x + 1)(x – 3)(x – 3)
• The function touches at x = -1 and x = 3
MULTIPLICITY OF FACTORS
• If there are an odd number of a particular factor,
the function will cross the x-axis.
• If there are an even number of a particular
factor, the function will just touch the x-axis at
that point.
• The factors determine the shape of the graph.
POLYNOMIAL GRAPHS
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Look at the function we have already factored
y = x4 – 4x3 – 2x2 + 12x + 9
y = (x + 1)(x + 1)(x – 3)(x – 3)
What does the function do outside x = -1, x = 3
and in between -1 and 3?
x < -1: (-)(-)(-)(-) = + the function starts positive
x > 3: (+)(+)(+)(+) = + function ends positive
-1 < x < 3: (+)(+)(-)(-) = + function is positive too
The function is positive everywhere except the
zeroes and changes directions three times.
INTERMEDIATE VALUE THEOREM
• This theorem states the obvious
• If the function is continuous between two
values of x: a < x < b
• the function must pass through every
value between f(a) and f(b) as x goes from
a to b.
• The most practical application of this
theorem is that if f(a)<0 and f(b)>0, then
there must be a zero between a and b.
LONG DIVISION REMAINDERS
• Luckily, the example we had did not have a
remainder.
• What do we do with the remainder when we are
dealing with polynomials?
• The same thing we did with long division with
just numbers. The remainder will be
represented as a fraction.