View
47
Download
0
Category
Preview:
DESCRIPTION
Given in class on Feb 6th
Citation preview
Math 31 – Sign Diagrams Class Workbook
Instructions: Use algebraic / reasoning methods where possible (or – your graphing calc!) to sketch each indicated graph.
Then, follow the instructions give. The first is already done.
1. (a) Sketch the line 6 3( )f x x= − , label the x and y intercepts.
2. (a) Sketch the curve 26( )f x x x= − − , label the x and y intercepts.
3. (a) Sketch the curve 2 3( ) ( )( )f x x x x= − + − , label the x and y intercepts.
(b) On the number line below, label the x-intercept with a solid
dot (�). This is a called a critical value of the function.
(c) Your number line above has been split into two intervals by
the critical value. Indicate the sign, pos (+) or neg (-) for
each of the two intervals.
(b) On the number line below, label the x-intercepts with a
solid dot (�).
(c) Your number line above has been split into four intervals by
the critical value. Indicate the sign, pos (+) or neg (-) for
each of the two intervals.
(b) On the number line below, label each x-intercept with a
solid dot (�).
(c) Your number line above has been split into three intervals
by the critical value. Indicate the sign, pos (+) or neg (-) for
each of the two intervals.
Graph is below -axis Graph is above -axis
4. (a) Sketch the curve 2( )f x x= , label the x and y intercepts.
5. (a) Sketch the curve 2
3( )
xf x
x=
−, label the x and y intercepts.
6. (a) Sketch the curve , label the x and y intercepts.
(b) On the number line below, label the x-intercept with a solid
dot (�).
(c) Your number line above has been split into two intervals by
the critical value. Indicate the sign, pos (+) or neg (-) for
each of the two intervals.
(b) On the number line below, label the x-intercept with a solid
dot (�) and the undefined value with a hollow dot (o).
(c) Your number line above has been split into intervals by the
critical value. Indicate the sign, pos (+) or neg (-) for each of
the two intervals.
(b) On the number line below, label the x-intercept with a solid
dot (�) and the undefined value with a hollow dot (o).
(c) Your number line above has been split into intervals by the
critical value. Indicate the sign, pos (+) or neg (-) for each of
the two intervals.
The SIGN DIAGRAM of a function is used to indicate:
• The x-values for which the functions is equal to 0 (�) , or is undefined (o). These are called the critical values.
• The intervals where the function is positive (+) or negative (-).
7. Construct a sign diagram for each function:
STEPS to constructing a SIGN DIAGRAM � Find the critical values of the function. (Values of x that make the function zero or undefined) This may
involve factoring.
� Plot the critical values on a number line, solid dot (�) for zeros, and hollow dots (o) for restricted values.
� Determine the sign (+) or (-) for the furthest left interval, that is, for “large x”.
� Proceed leftward alternating signs at all* critical values.
*Alternate signs UNLESS the factor that provides the critical value has an even multiplicity. In that case, the curve
“bounces” on the x-axis / do NOT change sign.
8. Construct a sign diagram for each function:
(a) 37 6y x x= − −
(b) 22 15
1( )
x xf x
x
+ −=−
(a) (b)
When x is large (more than 3)
first factor, (x-3) is positive Second factor (x+1)
is also positive… As is final factor, (x+2)
SO, pos times a pos times
a pos is…POSITIVE!
All three factors are positive
when x is larger than 3
(c) 22 15
3( )
x xf x
x
+ −=−
(d) 4
3
1 2
5
( )( )( )
( )
x xf x
x
− +=−
(e) 2
2
2
25
( )( )
xf x
x
−=−
(f) 2 6 53 1 3( ) ( ) ( )y x x x= − + +
9. Given the graph of each function, construct a sign diagram:
(c)
(e)
(a) (b)
(d) (f)
Using Sign Diagrams to Solve Inequalities Step � Ensure that the inequality expression is set to 0.
Step � Construct a Sign Diagram for the expression.
Sept � Use the signs to determine the solution interval. Express in both interval notation and on a number line.
1.
2. or
3. 2 6 0x x− − ≥
4. 2( 3)( 1) 0x x x− + <
5. 3 22 10 0x x x+ − ≤
6. 3 2(2 )( 3) ( 1) 0x x x− − + <
7. 2
20
( 3)
x
x
+ ≤−
8. 2 10
03
x
x
− >−
9. 2 12
0x x
x
− − ≥
10. 2 12
0x x
x
− − <
Recommended