 Determine the value of k for which the expression can be factored using a special product pattern: x 3 + 6x 2 + kx + 8  The “x” = x, and the “y” = 2.  The pattern is: y = x 3 + 3x 2 y + 3xy 2 + y 3 = (x + y) 3  Substituting gives: y = x 3 + 3x 2 *2 + 3x = (x + 2) 3  Cleaning house gives: y = x 3 + 6x x + 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?

 Parent functions are the most basic form of the function. Examples include:  y = x  y = x 2  y = x 3  Let’s look at variations on the parent function x 2 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 = ax 2 + 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 = x 2 :  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 = x 2 (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 = x 2 – 4. What are the zeros of the graph?  (2, 0) and (-2, 0)  Look at the graph y = x 2. What are the zeros of the graph?  (0, 0) with duplicity of two  Look at the graph y = x 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) = x 2  g(x) = 2x 2  h(x) = x  i(x) = -(x 2 – 9)

 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.

 If you need something else to memorize: Even DegreeOdd Degree (Including 1 st ) a > 0Rise to the left Rise to the right Fall to the left Rise to the right a < 0Fall to the left Fall to the right Rise to the left Fall to the right

 Multiplying the whole equation reflects the graph across the x-axis.  Make a graphic organizer w/ equations & graphs Changing the ConstantChanging the leading Coefficient constant > 0Move up, same size and shape |a| > 0Stretch Vertically constant < 0Move down, same size and shape 0 < |a| < 1Compress Vertically

 Pg 128 # 1 – 9 all

 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.

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

 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) = x is an even function since: f(-x) = (-x) = x = f(x)  Again, look at the Excel graph of cubic and quadratic

 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) = x 3 is an odd function since: f(-x) = (-x) 3 = -x 3 = -f(x)  A function f can be neither even or odd.  Example: f(x) = x is neither since: f(-x) = (-x) = -x  -f(x) or f(x)

 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)

 Even Function: 1. Reflects across the y-axis 2. (x, y)  (-x, y) 3. f(-x) = f(x)  Odd Function: 1. Reflects across the origin 2. (x, y)  (-x, -y) 3. f(-x) = -f(x)

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

 Make a table, plot the functions and describe the transformation X f(x) g(x) h(x) i(x)

 Make a table, plot the functions and describe the transformation X f(x) g(x) h(x) i(x)

 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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