Transformations and Operations y - k = af[b(x - h)] LESSON...

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Example 1

b) Stretches and Reflections. Given the transformation y = -2f(-x), Brian applies the vertical stretch first, then reflects about the x-axis, and finally reflects about the y-axis. Rebecca reflects about thex-axis, reflects about the y-axis, and finally applies the vertical stretch. Who drew the correct graph?

a) Stretches. Given the transformation y = 2f( x), Brian applies the vertical stretch first,

Brian’s Graph Rebecca’s Graph


Brian and Rebecca are investigating transformation combinations.

followed by the horizontal stretch. Rebecca applies the horizontal stretch first, followed by the vertical stretch. Who drew the correct graph?

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12

Transformations and OperationsLESSON TWO - Combined Transformations

Lesson Notesy - k = af[b(x - h)]

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c) Stretches and Translations. Given the transformation y - 3 = 2f(x), Brian applies the translation first, then the stretch. Rebecca applies the stretch first, followed by the translation. Who drew the correct graph?


d) Reflections. Given the transformation y = -f(-x), Brian applies the reflection about the x-axis first,followed by the reflection in the y-axis. Rebecca applies the reflections in the opposite order, startingwith the reflection in the y-axis, followed by the reflection in the x-axis? Who drew the correct graph?




e) Reflections and Translations. Given the transformation y = -f(x) + 2, Brian applies the reflectionabout the x-axis first, then the vertical translation. Rebecca applies the vertical translation first, followed by the reflection about the x-axis. Who drew the correct graph?


f) Translations. Given the transformation y - 7 = f(x - 3), Brian applies the horizontal translation first, followed by the vertical translation. Rebecca applies the vertical translation first, followed by the horizontal translation. Who drew the correct graph?



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Example 2 Draw the transformation of each graph.Include a mapping calculation for at least one of the points on the graph.

b) y = f(-x)13



a) y = 2f( x)13


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c) y = -f(2x)

d) y = - f(-x)12


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Example 3

a) y + 2 = -f(x)

b) y = f(- x) + 114

Draw the transformation of each graph.Include a mapping calculation for at least one of the points on the graph.


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c) y = - f(2x) - 1

d) y - 4 = 3f(x - 2)

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Example 4

a) y = f[ (x - 1)]

b) y = f(2x + 6)

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Draw the transformation of each graph.Include a mapping calculation for at least one of the points on the graph.

c) y + 2 = f(3x - 6)

d) y = f(-x - 4)13


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Example 5

a)

b)

The graph of y = f(x) is vertically stretchedby a factor of 3, reflected about the x-axis,and translated 2 units to the right.

The graph of y = f(x) is horizontally stretchedby a factor of 2, reflected about the y-axis,and translated 3 units up.

For each transformation description, write the transformation equation.Use mappings to draw the transformed graph.


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c)

d)

The graph of y = f(x) is horizontally stretched13

The graph of y = f(x) is vertically stretched

by a factor of , reflected about the x-axis,

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by a factor of , reflected about the x-axis,

and translated 2 units left.

reflected about the y-axis, and translated 5 units up.


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Example 6 Write a sentence describing each transformation, then write the transformation equation.

a)

b)

Original graph:

Transformed graph:

Think of the dashed line as a placeholder for where the graph was in the past, and the solid line is where the graph is now.


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c)

d)


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Original graph: f(x) = 3x2 + 6

Transformation: y + 5 = f(x)13

Example 7 Derive the equation of the transformed graph. Draw the original and transformed graphs.

a)

b)

Transformation: y = 2f(x - 3)Original graph: f(x) = |x + 1|


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Original graph: f(x) = 2|x - 3|

Transformation: y = f(- x)12

c)

d)

Transformation: y - 4 = -f(x + 1)Original graph: f(x) = (x - 4)2 - 1


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Example 8 The point (-2, 6) exists on the graph of y = f(x). For each transformationequation, determine the new location of the point.

a) y = -2f(x - 3)

b) y = f(-2x)12

c)

Method 1: Step-by-step transformation. Method 2: Use a Mapping




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d) y = f(4x - 8)

f)

e)





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Example 9 Answer the following questions.

a) The point (m, n) exists on the graph of y = f(x). If the transformation y - 5 = 2f(x) is applied tothe graph, the image point is (4, 7). Find the values of m and n.

b) The point (m, n) exists on the graph of y = f(x). If the transformation y = f(-x + 6) is applied tothe graph, the image point is (2, 5). Find the values of m and n.

Method 1: Step-by-step reverse transformation. Method 2: Use a Mapping



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c) The point (m, n) exists on the graph of y = f(x). If the transformation is applied to the graph, the image point is (-15, 1). Find the values of m and n.

d) The point (m, n) exists on the graph of y = f(x). If the transformation is applied to the graph, the image point is (-21, -9). Find the values of m and n.




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Example 10

The goal of the video game Space Rocks is to pilot a spaceship through an asteroid field without colliding with any of the asteroids.

a) If the spaceship avoids the asteroid by navigating to the position shown, describe the transformation.

b) Describe a transformation that will let the spaceship pass through the asteroids.

Original position of ship

Final position of ship


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c) The spaceship acquires a power-up that gives it greater speed, but at the same time doublesits width. What transformation is shown inthe graph?

d) The spaceship acquires two power-ups. The first power-up halves the original width of the spaceship, making it easier to dodge asteroids. The second power-up is a left wing cannon. What transformation describes the spaceship’s new size and position?

e) The transformations in parts (a - d) may not be written using y - k = af[b(x- h)].Give two reasons why.

Transformations and Operations y - k = af[b(x - h)] LESSON...

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