Objectives:
1.
To evaluate and graph piecewise defined functions
2.
To internalize the graphs of various parent functions
3.
To perform translations on the graphs of functions
Assignment:
• P. 49: 35, 37
• P. 71-2: 43, 44, 47
• P. 73: 71, 72
• P. 81: 20, 21, 29, 30,
37, 38, 43, 45, 48
• Shifting Worksheet
• Read: P. 76-78

Piecewise Functions Scaling
Reflecting Translating

You will be able to graph and evaluate piecewise functions

Determine whether the graph shown represents a function.

A piecewise defined function is defined by more than one equation. Each equation corresponds to a different part of the domain of the function.
f ( x )






3 x

1 ,
2 x

1 ,
3 , if if

2
 if x x
 
2 x

1

1

Evaluate g ( x ) at the values below.
g ( x )

2 x

1 ,
3 x

1 , if x

1 if x

1
1.
g (1)
2.
g (5)

Graph g ( x ).
g ( x )

2 x

1 ,
3 x

1 , if x

1 if x

1

Method 1:
1.
Rather than starting at the y -intercept, start at the domain’s breaking point.
Use the slope to graph the partial line in the correct direction.
2.
Repeat for each piece of your function.

Method 2:
1.
Graph one of the equations in the piecewise function as you normally would.
2.
Erase the part of the graph that you don’t need according to the domain of the piece.
3.
Repeat for each piece of your function.

The absolute value parent function is defined by f ( x ) = | x |.
Write this function as a piecewise function.
-5 f x = x
-2
-4
4
2
5

Graph 𝑓 𝑥 =
− 1
2 𝑥 − 6, 𝑥 ≤ −4 𝑥 + 5, 𝑥 > −4
.

Write a piecewise defined function for the graph shown.
This is called a step function . I wonder why.

The function 𝑓 𝑥 = 𝑥 is called the greatest integer function . This function outputs the greatest integer less than or equal 𝑥 .
Evaluate each of the following:
1
1.
2
2.
−
1
2
3.
3
4.
1.1
5.
−1.1

The function 𝑓 𝑥 = 𝑥 is called the greatest integer function . This function outputs the greatest integer less than or equal 𝑥 .
Graph 𝑓 𝑥 = 𝑥 over the domain −3, 3

You will be able to graph parent functions and detail their important characteristics

Who is the simplest member of your family?
Well, in math, the simplest member of a family of functions is called the parent function .
Linear Parent Function Family of Linear Functions 𝑦 = −𝑥 + 5 𝑦 = 3𝑥– 7 𝑦 =
1
2 𝑥 − 9 𝑦 = 𝑥

Who is the simplest member of your family?
Well, in math, the simplest member of a family of functions is called the parent function .
Family of Quadratic Functions Quadratic Parent Function 𝑦 = −𝑥 2 + 5 𝑦 = 3𝑥 2 – 7 𝑦 =
1
2 𝑥 2 – 9 𝑦 = 𝑥 2

Who is the simplest member of your family?
Well, in math, the simplest member of a family of functions is called the parent function .
Family of Functions Parent Function
A group of functions that share common characteristics
Simplest member of the family

In your groups, graph each parent function by hand using a few key points. Then list the domain and range in both interval and set notation. Finally, indicate each graph’s symmetry.

You will be able to perform vertical and horizontal shifts on parent functions

One of the handiest ways to graph a function is by transformations on parent functions. This includes scaling , reflecting , and translating ; what we’ll call SRT .

Each of these transformations ( scaling , reflecting , and translating ) come in a vertical, y , and a horizontal, x , variety.
y a f ( c
 x h

)
 k

If you are given the graph of the function y = f ( x ), then the graph of y
  k is translated vertically k units.
• Upward if k > 0
• Downward if k < 0

By hand, graph each of the following on the
1.
same coordinate plane: y
 x
2.
y x 3
3.
y x 3

If you are given the graph of the function y = f ( x ), then the graph of y

(
 h ) is translated horizontally h units.
• Left if h > 0
• Right if h < 0

It might seem a bit weird that when k is positive, the graph moves up—in the positive y direction—while when h is positive, the graph moves left—in the negative x -direction. This is not all that weird if you consider that k is really on the wrong side of the equation: y

(
 h )
 k y k f ( x
 h )

It might seem a bit weird that when k is positive, the graph moves up—in the positive y direction—while when h is positive, the graph moves left—in the negative x -direction. This is not all that weird if you consider that k is really on the wrong side of the equation: y x 3 y x 5
This is still a vertical translation of 3 units up.

By hand, graph each of the following on the
1.
same coordinate plane: y
 x
2.
y
 x

3
3.
y
 x

3

By hand, graph each of the following on separate coordinate planes.
1.
𝑦 = 𝑥 + 1
2.
𝑦 = 𝑥 + 1

Objectives:
1.
To evaluate and graph piecewise defined functions
2.
To internalize the graphs of various parent functions
3.
To perform translations on the graphs of functions
Assignment:
• P. 49: 35, 37
• P. 71-2: 43, 44, 47
• P. 73: 71, 72
• P. 81: 20, 21, 29, 30,
37, 38, 43, 45, 48
• Shifting Worksheet
• Read: P. 76-78