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**1-3 Transforming Linear functions**

Chapter 1 1-3 Transforming Linear functions

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Warm up Instructions: Name the parent function of the following problems: 1.y= 3 π₯ 2 +15 2. π¦= 1 π₯ + 1 4 3. y= 3π₯+ 1 2 4. π¦=2 3π₯

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Warm up answer Instructions: Name the parent function of the following problems: 1.y= 3 π₯ Answer: Quadratic Function 2. π¦= 1 π₯ Answer: Rational Function 3. y= 3π₯+ 1 2 Answer: Linear function 4. π¦=2 3π₯ Answer; Square root function

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**Objectives The student will be able to: Transform linear functions**

Solve problems involving linear transformations

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**Transforming linear Functions**

What is a transformation? Answer: A transformation is a change in the position, size, or shape of a figure or graph. What is a Linear function? Answer: is a function, meaning we have an input and an output, that can be written in the form π π₯ =ππ₯+π. Its graph is a line. If we transforming linear functions , we can say we are changing the linear function either the way it looks in the graph or the equation.

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**Transforming Linear Functions**

There are four ways we can transform the linear function by : Just remember the x changes

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**Transforming Linear functions**

Just remember y changes

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**Transforming linear functions**

Just remember y is the mirror so the one that changes is the x

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**Transforming Linear functions**

Just remember x is the mirror so the one that changes is the y

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Example 1 Let g(x) be the indicated transformation of f(x).Write the rule for g(x). π π₯ =3π₯+2; g(x) is a horizontal shift 3 units to the right. Solution: π π₯ =π π₯β3 subtract 3 from the input π π₯ =3 π₯β evaluate f at x-3 π π₯ =3π₯β9+2 Simplify π π₯ =3π₯β7

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Example 2 Let g(x) be the indicated transformation of f(x).Write the rule for g(x). π π₯ =π₯+2; g(x) is reflected about the y-axis. Solution: π π₯ =π(βπ₯) change the input of f π π₯ = βπ₯ +2 Simplify π π₯ =βπ₯+2

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**Student practice Example 3**

Let g(x) be the indicated transformation of f(x).Write the rule for g(x). π π₯ =6π₯+2; g(x) is a vertical shift (vertical translation) 3 units down.

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**Student Practice Example 4**

Let g(x) be the indicated transformation of f(x).Write the rule for g(x). π π₯ =6π₯+2; g(x) is a reflection across the x-axis.

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**Lets combine transformations Example 5**

Let g(x) be the indicated transformation of f(x).Write the rule for g(x). π π₯ =2π₯β6; g(x) is a vertical shift (vertical translation) 3 units down followed by a reflection across the x-axis .Solution: First lets take care of the vertical translation π π₯ =π π₯ β3 π π₯ = 2π₯β6 β3 substitute π π₯ =2π₯β9 simplify

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Example 5 continue Then we continue with the reflection across the x-axis π π₯ =βπ π₯ π π₯ =β 2π₯β9 π π₯ =β2π₯+9

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**Stretches and compression**

Stretches and compressions change the slope of a linear function. If the line becomes steeper, the function has been stretched vertically or compressed horizontally. If the line becomes flatter, the function has been compressed vertically or stretched horizontally.

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**Stretches and compressions**

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Example 6 Let g(x) be a vertical compression of f(x) = 3x + 2 by a factor of Write the rule for g(x) and graph the function. Solution: Vertically compressing f(x) by a factor of replaces each f(x) with a Β· f(x) where a = 4 . π π₯ =πβπ π₯ =4βπ π₯ π π₯ =4β(3π₯+2) substitute π π₯ =12π₯+8 simplify

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**Student Practice Example 7**

Let g(x) be a horizontal compression of f(x) = 5x - 2 by a factor of 1/3 . Write the rule for g(x) and graph the function.

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**Now lets put everything together**

Example 8: Let g(x) be a horizontal compression of f(x) = 6x - 5by a factor of 1/3 followed by a vertical translation 4 units up . Lets h(x) be the horizontal compression and g(x) the vertical translation. Write the rule for g(x) and graph the function. π π₯ =β 1 π π₯ π π₯ =β π₯ =β 3π₯ β π₯ =6 3π₯ β5=18π₯β5

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**Example 8 continue Now lets take care of the translation π π₯ =β π₯ +4**

π π₯ = 18π₯β5 +4 substitute π π₯ =18π₯β1 simplify

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Student practice Do all worksheet

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Homework Page 28 from book problems 2 to 6 and 12 to14.

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Closure Today we talked about transforming linear functions through translating and reflecting . Tomorrow we are going to see scatter plots and the best fit line.

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Have a great day!!!

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Section 1.6 Transformation of Functions

Section 1.6 Transformation of Functions

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