Unit 4Name______

Transformations of Functions

Day 1 / Graphing Piecewise Functions
Day 2 / Applications of Piecewise Functions
Day 3 / Quiz Review
Day 4 / QUIZ
Day 5 / Transformations: Up/Down, Left/Right
Day 6 / Transformations: Stretch/Shrink
Day 7 / Transformations: Reflect
Day 8 / Transformations Review
Day 9 / Unit 4 Review
Day 10-11 / Unit 4 TEST

Unit 4 Vocabulary:

Word / Meaning / Where to find more info
Explicit Notation
Function Notation
Parent Function
Piecewise Function
Step Function
Transformation / Explain the transformation that takes place / Where to find more info

Graphing Piecewise Functions

A piecewise function is a function that is defined on a sequence of intervals.

Graph the following piecewise functions:

1)

Find the following values:

a) / b) / c)

2)

Find the following values:

a) / b) / c)

3)

Applications of Piecewise Functions

1) Evaluate for the following values:

a) / b) / c) / d) / e)

2) Given the graph:

a) What is the rate of change of the graph on the

interval where x > –1?

b) What is the rate of change of the graph on the

interval when x < –1?

c) At what x-value(s) will the y-value be 3?

3) Lynn usually has her bangs cut every three months. The graph at the right shows the growth of her bangs over the past year.

a) Write a story to describe her bang lengths for the past year.

b) Does the graph indicate that Lynn’s bangs continue to grow at a consistent rate? Explain.

4) The equation to determine the weekly earnings of an employee at The Pizza Shop is given by , where x is the number of hours worked.

Determine the difference in salary, in dollars, for an employee who works 48 hours versus one who works 35 hours.

Determine the number of hours an employee must work in order to earn $500. Explain how you arrived at this answer.

_____ 5) According to the United States Postal Service (in 2014), first-class letters are weighed and priced as shown in the chart at the right.

Which of the following graphs correctly displays this information regarding the pricing of first-class letters?

Function Notation vs. Explicit Notation

“Up/Down” Transformations of Functions

Function Notation
For any real number x,
the following functions are given
in terms of the function f :


/ Explicit Notation (Explicit Formula)
If asked to write an explicit formula for and , you simply plug the expression for in for in that equation.
So if , then explicitly:
______and ______
Parent Functions
The parent function for any quadratic function is
The parent function for any absolute value function is
1) Graph:


/ 2) Graph:



3) How does adding 3 on the outside affect the graphs above?
4) How does subtracting 4 on the outside affect the graphs above?
Recap: These are external shifts because the number is being added or subtracted outside of the or the
When we add a constant on the outside, it translates the graph up that many units. / When we subtract a constant on the outside, it translates the graph down that many units.

does just what you would expect!

“Left/Right” Transformations of Functions

3) Graph:


/ 4) Graph:



3) How does adding 3 on the inside affect the graphs above?
4) How does subtracting 4 on the inside affect the graphs above?
Recap: These are internal shifts because the number is being added or subtracted inside of the or the
When we add a constant on the inside,
it translates the graph left that many units. / When we subtract a constant on the inside, it translates the graph right that many units.

does the opposite of what you would expect!

5) Let where x can be any real number. Write a formula for each function whose graph is the transformation of the graph of f given by the instructions below.

Function Name / Transformation from / Function Notation
(in terms of ) / Explicit Notation
/ Moves down 6 units
/ Translates up 10 units

6) Let where x can be any real number. Write a formula for each function whose graph is the transformation of the graph of f given by the instructions below.

Function Name / Transformation from / Function Notation
(in terms of ) / Explicit Notation
/ Moves left 2 units
/ Translates right 7 units

“Stretching/Shrinking”

Transformations of Functions

1) Graph:



How does the number being multiplied to on the outside change the graph of the parabola? / 2) Graph:



How does the number being multiplied to on the inside change the graph of an absolute value function?
Recap: If the number being multiplied is outside of the parenthesis or absolute value bars, it is considered a vertical stretch or shrink because it is “done to y”
If the number being multiplied is inside of the parenthesis or absolute value bars, it is considered a horizontal stretch or shrink because it is “done to x.”
When we multiply the function by a number greater than 1,
the graph vertically stretches
(or horizontally shrinks). / When we multiply the function by a number between 0 and 1 ,
the graph vertically shrinks
(or horizontally stretches).
Fractions Flatten!

3) Let where x can be any real number. Write a formula for each function whose graph is the transformation of the graph of f given by the instructions below.

Function Name / Transformation from / Function Notation
(in terms of ) / Explicit Notation
/ Horizontally stretched by a scale factor of 0.6
/ Vertically stretched by a scale factor of 4

4) Graph:

How does multiplying by 3 change the graph of the parabola?

______

How does multiplying by change the graph of the parabola?

______

5) Graph:

How does multiplying by 4 change the

graph of an absolute value function?

______

How does multiplying by change the

graph of an absolute value function?

______

“Reflecting” Transformation of Functions

1) Graph:

/ 2) Graph:


How does multiplying or by a –1 change the graphs above?
Recap: Multiplying or by a –1 will reflect the graph over the x-axis!

3) Let where x can be any real number. Write a formula for each function whose graph is the transformation of the graph of f given by the instructions below.

Function Name / Transformation from / Function Notation
(in terms of ) / Explicit Notation
/ Is reflected over the x-axis and moves left 2 units.
/ Is reflected over the x-axis and translates down 7 units

Without graphing, explain how the functions will differ:

4)

5)

6)

7)

Another function notation:

Sometimes you may see a function written as:
/ All it means is:

8) If , find:

a) / b) / c)