Skill 4Graphs of Functions

Skill 4a: Increasing / Decreasing and Constant FunctionsSkill 4b: Even / Odd Symmetry of GraphsSkill 4c: Recognize Functions from the Library of Parent Functions

Identity, Squaring (Quadratic), Cubic, Square Root, Cube Root, Absolute Value, (Exponential, Logarithmic), Reciprocal, Greatest Integer, Piecewise

Skill 4d: Function TransformationsRigid Transformations (Left/Right; Up/Down; Reflections)Non-Rigid Transformations (Vertical and Horizontal Stretch / Shrink)Combinations of Transformations

Skill 4e: Writing Variation FunctionsDirect / Inverse / Joint

Skill 4a: Increasing / Decreasing and Constant Functions

Identify the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant

1. 2.

3. 4.

A function is increasing on an interval if for any values x=a and x=b on the interval with a<b, f (a)< f (b).

A function is decreasing on an interval if for any values x=a and x=b on the interval with a<b, f (a )>f (b).

A function is constant on an interval if for any values x=a and x=b on the interval, f (a )= f (b).

Skill 4b: Even and Odd Functions

8 thru 10. Identify each function as even, odd, or neither.8. 9. 10.

4. 5. 6.

A function is odd if for each x in the domain of f (x),f ( x )=−f (−x).

A function is even if for each x in the domain of f (x), f ( x )=f (−x ).

Given an equation to determine if the function is even, odd, or neither compare the results of f (a )∧ f (−a). If the results are the same, the function is even. If the results are opposites, the function is odd. If there is no relation, the function is neither.

7. f ( x )=3 x2−2 8. f ( x )=x3−5x

9. f ( x )=2x2−4 x+2 10. f ( x )=4 x3−x+2

Skill 4c: Recognize Functions from the Library of Parent Functions

Library of Parent Functions

Constant Linear (identity) Square Cubic

f (x)=C f (x)=x f (x)=x2 f ( x )=x3

Absolute Value Square Root Cube Root

f (x)=¿x∨¿ f (x)=√x f (x)=3√x

Greatest Integer Piecewise Reciprocal

f ( x )= ⟦ x ⟧ f ( x )={ 6x2−5

−x−1

x←3−3≤x<1x>1

f ( x )=1x

Logarithmic Exponential Logistic Sinusoidal

f (x)=log (x ) f (x)=2x f (x)= 11+e− x

f (x)=sin (x )

Name the parent function for each graph below:

1. 2. 3.

Skill 4d: Function TransformationsRigid Transformations (Vertical Shifts; Horizontal Shifts; Reflections)Complete the following chart.

f ( x )=x2 ;g (x )= x2−5; h ( x )=x2+3

Sketch the graph of f ( x ) , g ( x ) and h(x )

How are these graphs the same?

How are they different?

x f ( x ) x g ( x ) x h(x )-2 -2 -2-1 -1 -10 0 01 1 12 2 2

1. The graph of f (x) is shown. A) Sketch a graph of g(x ), if

g ( x )=f ( x )+6

B) Sketch a graph of h(x ), if

h ( x )=f (x )−3

2. g ( x )=f ( x )−4. If f (3 )=6 , f (−1 )=−4 ,and f (7 )=3.

A) What is g (7 )?

B) What is g (3 )?

x f (x) x g(x )

Complete the following charts. f ( x )=x2 ;g (x )=(x−3)2 ;h (x )=(x+2)2

x f (x) x g(x ) x h(x )-3-2-10123

Sketch the graph of f ( x ) , g ( x ) and h(x )

How are these graphs the same?

How are they different?

3. The graph of f (x) is shown.

A) Sketch a graph of g(x ), if g ( x )=f ( x+4 )

B) Sketch a graph of h(x ), if h ( x )=f (x−2 )

4. g ( x )=f ( x−4 ). If f (3 )=6 , f (−1 )=−4 ,and f (7 )=3. A) What is g (7 )? B) What is g (3 )?

x f (x) x g(x )

Complete the following chart. f ( x )=√x ;g ( x )=f (−x )=√−x ; h ( x )=−f ( x )=−√ x

x f (x) x g(x ) x h(x )0149

Sketch the graph of f ( x ) , g ( x ) and h(x )

How are these graphs related?

5. The graph of f (x) is shown. A) Sketch a graph of g(x ), if g ( x )=f (−x ).

B) Sketch a graph of h(x ), if h ( x )=−f ( x )

6. g ( x )=f (−x )and h ( x )=− f (x). If f (3 )=6 , f (−4 )=2 ,and f (−3 )=−4. A) What is g (3 )? B) What is h (3 )?

x f (x) x g(x )

x f (x) x h(x )

Non-Rigid Transformations (Vertical and Horizontal Stretch / Shrink)Complete the following chart.

f ( x )=√x ;g ( x )=3 f ( x)=3√x ;h ( x )=12f ( x )=1

2 √ x

x f ( x ) x g(x ) x h(x )0149

Sketch the graph of f ( x ) , g ( x ) and h(x )

7. The graph of f (x) is shown. A) Sketch a graph of g(x ), if g ( x )=4 f ( x )

B) Sketch a graph of h(x ), if

h ( x )=12f ( x )

8. g ( x )=6 f ( x )and h ( x )=15f (x ). If f (2 )=−5 , f (0 )=3 ,and f (−2 )=7.

A) What is g (2 )? B) What is h (2 )?

x f (x) x g(x )

x f (x) x h(x )

Non-Rigid Transformations (Vertical and Horizontal Stretch / Shrink)Complete the following chart.

f ( x )=x2 ;g (x )=f (2x )=(2 x)2 ;h (x )=f ( 12 x )=(12x)2

x f ( x ) x g ( x ) x h ( x )

Sketch the graph of f ( x ) , g ( x ) and h(x )

9. The graph of f (x) is shown. A) Sketch a graph of g(x ), if g ( x )=f (3 x )

B) Sketch a graph of h(x ),

if h ( x )=f ( 12 x)

10. g ( x )=f (2 x )and h ( x )=f ( 12 x). If f (1 )=−3 , f (2 )=2 ,and f (4 )=5.

A) What is g (2 )? B) What is h (2 )?

x f ( x ) x g ( x )

x f ( x ) x h(x )

Combinations of Transformations

Order of Transformations1) Horizontal Shifts g ( x )=f (x+C1 )2) Horizontal Compressions (Including Reflections across y-axis) g ( x )=f (C2 x )3) Vertical Compressions (including reflections across x-axis) g ( x )=C3 f ( x )4) Vertical Shifts g ( x )=f ( x )+C4

g ( x )=C3 f (C2 x+C1 )+C4

Suppose the graph of f ( x ) is given. Describe how the graph of g(x ) can be obtained.

1. g ( x )=f ( x−3 )+7 2. g ( x )=6−f ( x)

To obtain the graph of f , do the following transformations;

To obtain the graph of f , do the following transformations;

3. g ( x )=4 f ( x−2 )+1 4. g ( x )=12f (4−x )−3

To obtain the graph of f , do the following transformations;

To obtain the graph of f , do the following transformations;

f ( x )=√x, is shown as a dotted line. Using transformations of f (x), write the equation of the graph shown as a solid line. 5. 6. 7.

8. The graph of f(x) is shown as a dotted line. Match each graph with the correct equation.

____ 3 f ( x )+3

____ −f ( x+3 )−2

____ f ( x )−8

____ f (−x−7 )

____ f ( x−3 )+5

____ f ( x+6 )+5

Skill 4e: Writing Variation FunctionsDirect Variation

If y is directly proportional to x, the quantities x and y are related by the equationy=kx

Inverse VariationIf y is inversely proportional to x the quantities x and y are related by the equation

y= kx

Joint VariationIf z is jointly proportional to x and y the quantities x , y and z are related by the equation

z=kxy

Write an equation that represents each statement:1. R varies directly with x 2. W is inversely proportional to m

3. A is directly proportional to the square of r 4. S is jointly proportional to x and y

Express the statement below as an equation. Use the given information to find the constant of proportionality.5. z varies directly as a. If a = 6, then z = 4

6. T varies directly with y and inversely with x. If x = 2 and y = 6, then T = 12

7. The cost C of printing a newspaper is jointly proportional to the number of pages, p, in the newspaper, and n, the number of newspapers printed. Write an equation for the cost of printing newspapers if it costs $10,000 to print 8000 copies of a newspaper containing 20 pages.