Graphing Functions Piecewise Functions

1
Graphing Functions - Piecewise Functions

• Note graphing piecewise functions on a
  calculator is not usually required by
  instructors. The following directions on
  graphing piecewise functions are mainly for the
  curious.

2
Graphing Functions - Piecewise Functions

• Graphing piecewise functions is one of the
  biggest challenges facing students of
  mathematics. These can be done on the
  calculator, but it does take some time to key in
  the expression. An example will be done now,
  with an explanation of the process later.

3
Graphing Functions - Piecewise Functions

• Enter function f into Y1 using the TEST menu
  to enter the inequality symbols.
• Y1 (x2)(x ? 0) (x 1)(0 ? x)(x ? 3) (6)(x
  ? 3)

Slide 2
4
Graphing Functions - Piecewise Functions

• Use a ZOOMZStandard window to graph the
  function.

• To get a better graph, press WINDOW and make
  the following changes. Then press GRAPH.

Slide 3
5
Graphing Functions - Piecewise Functions

• The short vertical line at x 3 does not
  belong to the graph. To correct this, press
  MODE, ...

move the cursor to Dot in the fifth row and press
ENTER.
Press GRAPH and the result is the completed
graph.
Slide 4
6
Graphing Functions - Piecewise Functions

• The graph does not show which endpoints are
  open and which are closed. Press TRACE and
  enter 3 (if not already at x 3).

• Notice that the cursor is on the line y 6, as
  it should be.

• Try x 0. It appears the other closed point
  is on the line y x 1.

Slide 5
7
Graphing Functions - Piecewise Functions

• The rest of this module is to explain the
  reasoning behind the function that was typed into
  Y1 in the previous example.

• Enter the function into Y1 ...

and use a ZOOMZStandard window for the graph.
Slide 6
8
Graphing Functions - Piecewise Functions

• Now go back to Y1 and make the following
  changes

(1) Enter Y1 according to the following screen
...
(2) and then select ? from the TEST menu.
(3) Complete Y1 as follows and press GRAPH.
Slide 7
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Graphing Functions - Piecewise Functions

• Only the right branch of the parabola is
  graphed. To see why this is so, consider (x ? 0)
  in the equation line.

• A test inequality, such as (x ? 0) , evaluates
  as either 0 or 1. It is 0 when the statement is
  false, and 1 when the statement is true.

• When x ? 0, (x ? 0) 0. This yields (x2)(0)
  0 for the function value, and the point (x,0) is
  plotted on the graph.

Slide 8
10
Graphing Functions - Piecewise Functions

• Note that (x,0) is a point on the x axis and
  does not show up on the graph.

• When x ? 0, (x ? 0) 1. This yields (x2)(1)
  x2 for the function value, which results in a
  point (x, x2) that is plotted on the graph.

?
?
Slide 9
11
Graphing Functions - Piecewise Functions

• Another way to see this is in a table. Go to
  TBLSET and make the following changes.

• Now go to TABLE.

• Compare the graph of function f with the table.

Slide 10
12
Graphing Functions - Piecewise Functions

• When entering the function into Y1, the
  following format was used. Y1 (x2)(x ?
  0) (x 1)(0 ? x)(x ? 3) (6)(x ? 3)

Slide 11
13
Graphing Functions - Piecewise Functions

• If x -2, then Y1 takes the following form

Y1 (x2)(x ? 0) (x 1)(0 ? x)(x ? 3) (6)(x
? 3) (-2)2 (1) (-2 1) (0)
(1) (6) (0) (-2) 2
0 0 4
In similar fashion, for all x ? 0, only the
first part of the piecewise function y x2 is
nonzero. On the domain x ? 0, y x2 is the
only part of the function that is graphed.
Slide 12
14
Graphing Functions - Piecewise Functions

• If x 1, then y x 1 is the only nonzero
  term Y1 (x2)(x ? 0) (x 1)(0 ? x)(x
  ? 3) (6)(x ? 3) (1)2 (0) (1
  1) (1) (1) (6) (0)
  0 2
  0 2

Thus, for all x in the domain 0 ? x ? 3, only
the second part of the piecewise function y x
1 is nonzero, and is the only part of the
function that is graphed.
Slide 13
15
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