 # Unit 3 Functions (Linear and Exponentials)

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Unit 3 Functions (Linear and Exponentials)
Parent Functions and Transformations

Transformation of Functions
Recognize graphs of common functions Use shifts to graph functions Use reflections to graph functions Graph functions w/ sequence of transformations

The following basic graphs will be used extensively in this section
The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

The identity function f(x) = x

The exponential function

The square root function

The absolute value function

The cubic function

The rational function

Transformations happen in 3 forms: (1) translations (2) reflections (3) stretching.

The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function. Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.

Vertical Translation OUTSIDE IS TRUE! Vertical Translation
the graph of y = f(x) + k is the graph of y = f(x) shifted up k units; the graph of y = f(x)  k is the graph of y = f(x) shifted down k units.

Horizontal Translation
INSIDE LIES! Horizontal Translation the graph of y = f(x  h) is the graph of y = f(x) shifted right h units; the graph of y = f(x + h) is the graph of y = f(x) shifted left h units.

Recognizing the shift from the equation, examples of shifting the function f(x) =
Vertical shift of 3 units up Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

Example Explain the difference in the graphs Vertical Shift Up 3 Units
Horizontal Shift Left 3 Units Vertical Shift Up 3 Units

Use the basic graph to sketch the following:

Combining a vertical & horizontal shift
Example of function that is shifted down 4 units and right 6 units from the original function.

Use the basic graph to sketch the following:

The graph of y = - f (x) is the graph of y = f (x) reflected about the x-axis. Reflection about the y-Axis The graph of y = f (-x) is the graph of y = f (x) reflected about the y-axis.

Stretching and Shrinking Graphs
Let f be a function and c a positive real number. If a > 1, the graph of y = a f (x) is y = f (x) vertically stretched by multiplying each of its y-coordinates by a. If 0 < a < 1, the graph of y = a f (x) is y = f (x) vertically shrunk by multiplying each of its y-coordinates by a. g(x) = 2x2 f (x) = x2 10 9 8 h(x) =1/2x2 7 6 5 4 3 2 1 -4 -3 -2 -1 1 2 3 4

The big picture…

Sequence of Transformations
A function involving more than one transformation can be graphed by performing transformations in the following order. 1. Horizontal shifting (Parentheses) 2. Vertical stretching or shrinking (Multiply) 3. Reflecting (Multiply) 4. Vertical shifting (Add/Subtract)

Example Use the graph of f(x) = x3 to graph g(x) = -2(x+3)3 - 4

A combination If the parent function is Describe the graph of
The parent would be horizontally shifted right 3 units and vertically shifted up 6 units

If the parent function is
What do we know about The graph would be vertically shifted down 5 units and vertically stretched two times as much.

It would be a cubic function reflected across the x-axis and horizontally compressed by a factor of ½.

Transformations of Exponential Functions

Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions. State the domain, range and the horizontal asymptote for each.

Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions. State the domain, range and the horizontal asymptote for each.

Transformations of Graphs of Exponential Functions
Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions. State the domain, range and the horizontal asymptote for each.

Graph using transformations and determine the domain, range and horizontal asymptote.
B) A) D) C)