# Cubic Functions

Document Sample

```					•   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
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
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
matches.

• 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
the graph.
Example B
• Find an equation for the graph
shown.
• 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
• The shape is correct, but you need
to reflect it across the x-axis. You
also need to vertically stretch the
graph.
• 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
• You can also use what you
cubic equations from general
form to factored form. Look
at the graph of y=x3-3x+2 at
right.
• It has x-intercepts x=2 and
x=1.
Most cubic equations you’ve
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
y=(x+2)(x-1)2.
Example C
• Find the exact x-
intercepts of y=x3+2x2-
7x-2.
• 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

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 94 posted: 4/19/2011 language: English pages: 15
How are you planning on using Docstoc?