Cubic Functions

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Cubic Functions Powered By Docstoc
					•   To explore general patterns and characteristics
    of cubic functions
•   To learn formulas that model the areas of
    squares and the volumes of cubes
•   To explore the graphs of cubic functions and
    transformations of these graphs
•   To write the equation of a cubic function from
    its graph
• In this lesson you’ll learn about
  cubic equations, which are
  often used to model volume.
  You’ll see that some of the
  techniques you’ve used with
  quadratic equations can be applied
  to cubic equations, too.
•   The volume of the cube
    at right is 64 cubic
    centimeters cm3, so you
    can fill the cube using
    64 smaller cubes, each
    measuring 1 cm by 1 cm
    by 1 cm.
•   The edges of a cube
    have equal length, so
    you can write its volume
    formula as
    volume=(edge length)3.
•   The cubing function,
    f(x)=x3, models volume.
    Each edge length, or
    input, gives exactly one
    volume, or output.
•   The edge length of a
    cube with a volume of
    64 is 4. So you can
    write 43=64.
•   You call 4 the cube
    root of 64 and the
    number 64 a perfect
    cube because its cube
    root is an integer.
•   Then you can express
    the equation as 4  3 64
•   You can evaluate cubes
    and cube roots with
    your calculator.
Graphs of cubic functions have different and
interesting shapes. In the window -5≤x≤5 and -
4≤y≤4, the parent function y=x3 looks like the
graph shown.
Example A
• Write an equation for each graph.
Rooting for Factors
• In this investigation you’ll discover a relationship
  between the factored form of a cubic equation
  and its graph.
• List the x-intercepts for each of these graphs.
• Each equation below matches exactly
  one graph in the previous step. Use
  graphs and tables to find the

• Describe how the x-intercepts you
  found in Step 1 relate to the factored
  forms of the equations in Step 2.
Now you’ll write an
  equation from a graph.
• Use what you
  discovered in Steps 1–
  3 to write an equation
  with the same x-
  intercepts as the graph
  shown. Graph your
  equation; then adjust
  your equation to match
  the graph.
Example B
• Find an equation for the graph
• There are three x-intercepts on the
  graph. They are x=0, 1, and 2.
• Each intercept helps you find a
  factor in the equation. These
  factors are x, x-1, and x-2.
• Graph the equation
  y=x(x-1)(x-2) on your calculator.
• The shape is correct, but you need
  to reflect it across the x-axis. You
  also need to vertically stretch the
• Check the y-value of your graph at
  x=1. The y-value is 2. You need it
  to be 4, so multiply by 2. The
  correct equation is y=2x(x-1)(x-2).
  Check this equation by graphing it
  on your calculator.
• You can also use what you
  know about roots to convert
  cubic equations from general
  form to factored form. Look
  at the graph of y=x3-3x+2 at
• It has x-intercepts x=2 and
  Most cubic equations you’ve
  explored have had three
  roots. Where is the third one?
  Notice that the graph just
  touches the axis at x=1. It
  doesn’t actually pass through
  the axis. The root at x=1 is
  called a double root, and the
  factor x-1 is squared.
• Graph the factored form
Example C
 • Find the exact x-
   intercepts of y=x3+2x2-
 • The graph shows that 2
   is an x-intercept of the
   function. This means
   that (x-2) is a factor.
   You can approximate the
   other two roots by
   tracing, but to find exact
   algebraic solutions you
   need to factor.
 • You can do this using a
   rectangle diagram.
                                         
x 3  2x 2  7x  2   x  2 x 2  4x  1
                                          
 x 3  2x 2  7x  2   x  2 x 2  4x  1

Use the quadratic formula to find the
zeros of x2  4x  1

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