Determine the value of k for which the expression can be factored using a special product pattern: x...

Warm-Up

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

Graphing Polynomials1A1c (2.10 - 2.12)

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 behaviorMM1A1h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd or neither.

Graphing Polynomials1A1c (2.10 - 2.12)

Today’s question: What do the graphs of different polynomial functions look like and how do they move?

Transformations 1A1c (2.10)Graphing y = ax2 + c

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.


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.


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


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


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 x-axis.


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)

End Conditions 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.

End Conditions areControlled by the Degree and Leading

Coefficient

If you need something else to memorize:

Even Degree Odd Degree(Including 1st)

a > 0 Rise to the left

Rise to the

right

Fall to the left

Rise to the

right

a < 0 Fall to the left

Fall to the

right

Rise to the left

Fall to the

right

Transformations from Parent Function

Multiplying the whole equation reflects the graph across the x-axis.

Make a graphic organizer w/ equations & graphs

Changing the Constant

Changing the leading Coefficient

constant > 0

Move up, same size and shape

|a| > 0 Stretch Vertically

constant < 0

Move down,

same size and shape

0 < |a| < 1 Compress Vertically

Practice

Pg 128 # 1 – 9 all

Warm-Up 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)}

Explain each transformation.

Even – Odd FunctionsTeacher Notes

Use the Excel file for the functions and Geo Sketch for the points to help explain.

Even – Odd Functions

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

Even – Odd Functions (3.1)

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)

Reflecting Across Axis and Origin

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)

Summary: Even Function:1. Reflects across the y-axis2. (x, y) (-x, y)3. f(-x) = f(x)

Odd Function:1. Reflects across the origin2. (x, y) (-x, -y)3. f(-x) = -f(x)

Practice

Pg 128 # 10 – 15 all Pg 129 # 8, 12 and 13 This is a total of 9 problems

Graphing Square Root Functions

Make a table, plot the functions and describe the transformation

xxf )(

X 0 1 2 3 4 5

f(x)

g(x)

h(x)

i(x)

3)( xxg )3()( xxh xxi 3)(


Make a table, plot the functions and describe the transformation

xxf )(

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

3)( xxg )3()( xxhxxi 3)(


What is the domain of the parent function? The domain is greater than or equal to zero

What is the range of the parent function? The range is greater than or equal to zero

What happens when a > 1? Vertical stretch

What happens when 0 < a < 1? Vertical shrink

What happens when the right side of the function is multiplied by a -1?

The function is reflected across the x-axis


What happens when the constant > 0? Shifts the curve up by the constant.

What happens when the constant < 0? Shifts the function down by the constant.

Is this an even or odd function? Why? Neither since it is not symmetrical around the

y-axis or the origin How would we reflect the curve across

the origin? Change (x, y) to (-x, -y)


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Determine the value of k for which the expression can be factored using a special product pattern: x...

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