Transcript Math I, Sections 2.5 – 2.9 Graphing Polynomials

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Determine the value of k for which the expression can
be factored using a special product pattern:
x3 + 6x2 + kx + 8
The “x” = x, and the “y” = 2.
The pattern is:
y = x3 + 3x2y + 3xy2 + y3 = (x + y)3
Substituting gives:
y = x3 + 3x2*2 + 3x22 + 23 = (x + 2)3
Cleaning house gives:
y = x3 + 6x2 + 12x + 8 = (x + 2)3
So k = 12
Standards: MM1A1c Graph transformations of basic
functions including vertical shifts, stretches, and
shrinks, as well as reflection across the x- and y-axis.
MM1A1d Investigate and explain characteristics of a
function: domain, range, zeros, intercepts, intervals of
increase and decrease, maximum and minimum
values, and end behavior
MM1A1h. Determine graphically and algebraically
whether a function has symmetry and whether it is
even, odd or neither.
Today’s question: What do the graphs of different
polynomial functions look like and how do they move?
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Parent functions are the most basic form of the
function. Examples include:
y=x
y = x2
y = x3
Let’s look at variations on the parent function
x2 using the Excel file and see what we can
discover.
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A quadratic function is a nonlinear function
that can be written in standard form
y = ax2 + bx + c, where a ≠ 0
Every quadratic function has a U-shaped graph
called a parabola.
The lowest or highest point on a parabola is the
vertex.
The line that passes through the vertex and
divides the parabola into two symmetric parts
is called the axis of symmetry.
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Compared to y = x2:
What happens when c is > 0?
The graph moves up vertically the amount of c,
but keeps same size and shape
What happens when c is < 0?
The graph moves down vertically the amount
of c, but keeps same size and shape
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Compared to y = x2 (a = 1)
What happens when a is increased?
The graph is stretched vertically
What happens when a is decreased?
The graph is compressed vertically if 0 < a < 1
What happens if we multiply the function by -1?
It is reflected across the x-axis.
Where do the ends of the graph go if a > 0?
Ends “raise” to the left and right
Where do the ends of the graph go if a < 0?
Ends “fall” to the left and right
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Look at the graph y = x2 – 4. What are the zeros
of the graph?
(2, 0) and (-2, 0)
Look at the graph y = x2. What are the zeros of
the graph?
(0, 0) with duplicity of two
Look at the graph y = x2 + 4. What are the
zeros of the graph?
There are no real zeros or roots.
Zeros, roots, intercepts, solutions are all the
same – they are where the graph crosses the xaxis.
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Describe and compare the movement and end
conditions of the following graphs relative to
f(x) = x2
g(x) = 2x2
h(x) = x2 - 9
i(x) = -(x2 – 9)
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The end behavior of a function’s graph is the
behavior of the graph as x approaches positive
(+ ) or negative infinity (- ).
Look at the Excel graph of cubic and quadratic
What determines the end conditions?
The end conditions are established by the
highest degree term.
End conditions for all even degree functions
are the same as the quadratic
End condition for all odd degree functions are
the same as the cubic.
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If you need something else to memorize:
Even Degree
a>0
a<0
Odd Degree
(Including 1st)
Rise to Rise to
Fall to
Rise to
the left the right the left the right
Fall to
Fall to
Rise to
Fall to
the left the right the left the right
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Multiplying the whole equation reflects the graph
across the x-axis.
Changing the Constant
constant > 0
Move up,
same size
and shape
constant < 0 Move down,
same size
and shape
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Changing the leading
Coefficient
|a| > 0
Stretch
Vertically
0 < |a| < 1
Compress
Vertically
Make a graphic organizer w/ equations & graphs
 Pg
128 # 1 – 9 all
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Make a table and graph the following
functions:
f(x) = |x|
g(x) = 2 *|x|
h(x) = 2 *|x| - 3 and
q(x) = -(2 *|x|-3)
{NOTE: q(x) is the same as -1 * (2 *|x|-3)}
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Explain each transformation.
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Use the Excel file for the functions and Geo
Sketch for the points to help explain.
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A function f is an even function if f(-x) = f(x).
The graph of even functions are symmetric
about the y-axis.
Example: f(x) = x2 + 4 is an even function since:
f(-x) = (-x)2 + 4 = x2 + 4 = f(x)
Again, look at the Excel graph of cubic and
quadratic
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A function f is an odd function if f(-x) = -f(x).
The graph of odd functions are symmetric
about the origin.
Example: f(x) = x3 is an odd function since:
f(-x) = (-x)3 = -x3 = -f(x)
A function f can be neither even or odd.
Example: f(x) = x3 + 4 is neither since:
f(-x) = (-x)3 + 4 = -x3 + 4  -f(x) or f(x)
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Shapes are really moved and reflected a data
point at a time.
What is change in the data point (x, y) to reflect
it across the x-axis?
(x, y)  (x, -y)
What is the change in the data point (x, y) to
reflect it across the y-axis?
(x, y)  (-x, y)
What is the change in the data point (x, y) to
reflect it across the origin?
(x, y)  (-x, -y)
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3.
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3.
Even Function:
Reflects across the y-axis
(x, y)  (-x, y)
f(-x) = f(x)
Odd Function:
Reflects across the origin
(x, y)  (-x, -y)
f(-x) = -f(x)
Pg 128 # 10 – 15 all
 Pg 129 # 8, 12 and 13
 This is a total of 9 problems
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Make a table, plot the functions and describe
the transformation
f ( x)  x
X
f(x)
g(x)
h(x)
i(x)
g ( x)  x  3 h( x)  ( x  3) i( x)  3 x
0
1
2
3
4
5
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Make a table, plot the functions and describe
the transformation
f ( x)  x
g ( x)  x  3 i( x)  3 x h( x)  ( x  3)
X
0
1
2
3
4
5
f(x)
0
1
1.4
1.7
2
2.2
g(x)
3
4
4.4
5.7
6
6.2
h(x)
-3
-4
-4.4
-5.7
-6
-6.2
i(x)
0
3
4.2
5.2
6
6.7
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What is the domain of the parent function?
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The domain is greater than or equal to zero
What is the range of the parent function?
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The range is greater than or equal to zero
What happens when a > 1?
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Vertical stretch
What happens when 0 < a < 1?
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Vertical shrink
What happens when the right side of the function
is multiplied by a -1?
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The function is reflected across the x-axis
What happens when the constant > 0?
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Shifts the curve up by the constant.
What happens when the constant < 0?
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Shifts the function down by the constant.
Is this an even or odd function? Why?
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Neither since it is not symmetrical around the y-axis
or the origin
How would we reflect the curve across the
origin?
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Change (x, y) to (-x, -y)
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Page 138 - 139, # 1, 2, 3, 5 & 8