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

− − <