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**Unit 3 Functions (Linear and Exponentials)**

Parent Functions and Transformations

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**Transformation of Functions**

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

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**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.

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**The identity function f(x) = x**

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**The exponential function**

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**The quadratic function**

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**The square root function**

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**The absolute value function**

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The cubic function

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The rational function

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

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**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.

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**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.

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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.

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**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.)

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**Use the basic graph to sketch the following:**

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**Combining a vertical & horizontal shift**

Example of function that is shifted down 4 units and right 6 units from the original function.

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**Use the basic graph to sketch the following:**

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**Reflection about the x-Axis**

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.

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**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 the graph of 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 the graph of 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

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The big picture…

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**Sequence of Transformations**

A function involving more than one transformation can be graphed by performing transformations in the following order. 1. Horizontal shifting 2. Vertical stretching or shrinking 3. Reflecting 4. Vertical shifting

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

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**Example Explain the difference in the graphs Vertical Shift Up 3 Units**

Horizontal Shift Left 3 Units Vertical Shift Up 3 Units

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**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

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**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.

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**What can we tell about this graph?**

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

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**Transformations of Exponential Functions**

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**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.

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**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.

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**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.

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**Graph using transformations and determine the domain, range and horizontal asymptote.**

B) A) D) C)

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