MATH 30-1Function Transformations Module One

Module / Unit One - Assignment Booklet

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Math 30-1: Module 1 Lesson Assignment 2 | P a g e

Math 30-1: Module 1 Lesson Assignment 3 | P a g e

Lesson 1: Horizontal and Vertical Translations1. Sketch the graph of the following, given y f(x).

a. y f(x) 4 b. y 2 f(x 3)

2. Refer to the tile puzzle, Image Shuffle, from the beginning of the lesson on the moodle site. Describe what it means to move a tile located at (x, y) given the following:

a. (x, y)(x 2, y)

b. (x, y)(x 3, y 1)

3. Use the graph to answer the following questions.

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a. Give an equation for the function y g(x) in the form y k f(x h).

c. Give an equation of the function y f(x) in the form y k g(x h).

4. The world population is approximately seven billion and still rapidly growing. How would the following transformations change the meaning of the graph?

Population Changes

a. The graph is translated 300 units to the left.

b. The graph is translated one billion units up.

Math 30-1: Module 1 Lesson Assignment 5 | P a g e

LESSON 1 SUMMARY

Refer to the Key Ideas section on page 12 of your text Pre-Calculus 12.

A transformation is a change in the shape or position of a figure or relation. A translation is a type of transformation that causes a “slide” in the graph. The new graph is the same size, shape, and orientation as the original but in a different position.

In a graph of the form y − k = f(x − h), k is the vertical translation from y = f(x). If k > 0, the translation is upwards. If k < 0, the translation is downwards.

In a graph of the form y − k = f(x − h), h is the horizontal translation from y = f(x). If h > 0, the translation is to the right. If h < 0, the translation is to the left.

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Lesson 2: Reflections 1. Consider the completed face from the Focus section of the lesson.

a. What type of reflection was used to complete the face?

b. Sketch a diagram that could be completed using a different kind of reflection.

c. What part(s) of the face in the diagram would represent an invariant point(s) of the reflection?

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2. Use the graph of y f(x) to answer the following questions.

a. Determine the domain and range of f(x).

b. Sketch y f(x) and determine its domain and range.

c. Sketch y f(x) and determine its domain and range.

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3. Three functions are shown: f(x) (x 2)2 2, g(x), and h(x). Determine an equation for g(x) and h(x) in the same form as f(x).

LESSON 2 SUMMARY

Refer to the Key Ideas section on page 27 of your text Pre-Calculus 12.

A reflection is a transformation that creates a mirror image of the original. A reflection in the x-axis changes all y-coordinates to –y. A reflection in the y-axis changes all x-coordinates to –x. A point that does not change during a transformation is called an invariant.

Now is a good time to update your summary table for the reflections you learned in this lesson

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Lesson 3: Stretches1. Determine a mapping of the form (x, y) (__, __) from y f(x) for each of the following.

a. y 3f(2x)

b.

c.

2. A point (6, 3) lies on f(x) from question 1. Determine the new location of the point for the functions in parts a, b, and c of question 1.

a. y 3f(2x)

b.

c.

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3. Complete these tasks for each of the following two graphs.

GRAPH 1 GRAPH 2

a. Determine an equation for g(x) in terms of f(x).

GRAPH 1 GRAPH 2

b. List any invariant points.

GRAPH 1 GRAPH 2

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c. Determine the domain and range of the functions f(x) and g(x).

GRAPH 1 GRAPH 2

4. a. Explain how you could use the graph of y x2 to graph .

b. Sketch the graph of .

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LESSON 3 SUMMARY Refer to the Key Ideas section on page 38 of your text Pre-Calculus 12.

When you multiply all of the x-coordinates or y-coordinates of a function by a constant, the change in shape is called a stretch. When you're stretching about the x-axis or the y-axis, it is useful to think of a function in terms of the equation y = af(bx). When using an equation of this form, a corresponds to a vertical stretch about the x-axis by a factor

of a. And b corresponds to a horizontal stretch about the y-axis by a factor of .

Increasing the value of a will make the graph seem taller; however, increasing the value of b will make the graph seem narrower, not wider.

Now is a good time to update your summary table for the stretches you learned in this lesson.

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Lesson 4: Combining Transformations

1. The mapping can be used with the transformation of y f(x) to

Explain how this mapping could be used to graph the function

from the graph of y f(x).

2. Determine an equation of the function f(x) in terms of g(x) and g(x) in terms of f(x).

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3. a. Given y f(x), sketch

b. Determine the domain and range of y f(x) and y g(x).

y f(x) y g(x)

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LESSON 4 SUMMARY

Refer to the Examples in section 1.2 between pages 21 to 24 of your text Pre-Calculus 12.

It is possible to perform multiple transformations on a function. The order in which you apply the transformations is important in determining the resulting function. The form of the equation of the function described in the diagram is commonly used when transforming a function, as the different parameters are easily interpreted. Using this form implies that the function is stretched and reflected before it is translated.

Multiple Transformations reviews how some specific functions of the form

behave when you change the different parameters.

Make sure you spend time using the geobra applet that allows you to adjust all parameters for quadratic equations. This will really help you understand how changing parameters causes transformations .

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Lesson 5: Inverse of a Relation1. The inverse function f 1(x) is shown in the diagram.

a. Explain how you could sketch the graph of y f(x).

b. Sketch the graph of y f(x) on the grid provided.

c. Determine the equation for f(x). Label the graph.

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2. Use the graph to answer the following questions.

a. Will the inverse of f(x) be a function? How can you tell?

b. Graph the inverse of the relation on the grid provided.

c. Determine the domain and range of f(x).

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d. Determine the domain and range of the inverse of f(x). How do these values compare to the domain and range of f(x)?

3. Determine a mapping of the form (x, y) (__,__) for the inverse of a relation. Explain how you determined this mapping.

4 a. Determine the inverse of f(x) 2x2 4.

b. Restrict the domain of f(x) so its inverse is a function.

c. Sketch both f(x) and its inverse on the same grid using the restricted domain from part b. Use different colors for each graph. Label the graph.

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LESSON 5 SUMMARY

Refer to the Key Ideas section on page 51 of your text Pre-Calculus 12.

If relations f and g undo each other, they are called inverses. If f and g are both functions, g is often written as f −1. You have seen that the domain and range of the function become the range and domain of the inverse, respectively. This will always be true of inverses.

To determine an inverse graphically, switch the x- and y-coordinates of each point of the original graph to determine points of the inverse. The inverse of a function may or may not be a function. Just as you can check if a relation is a function by using the vertical line test, a horizontal line test can be used on the original graph to determine if the inverse of that relation will be a function. If it is possible for a horizontal line to cross the graph of a relation at more than one point, the inverse will not be a function.

This diagram shows y = f(x) and its inverse. The inverse is not labelled y = f −1(x) because the inverse is not a function.

You have also found that it is possible to determine the equation of an inverse given the equation of a function. One method used is to replace f(x) with y, switch x and y, and then solve for y. Then f(x) is replaced with y to avoid equations that contain the variable f(x), which can be confusing. Take note of when it is appropriate to use function notation to represent an inverse and when it is not appropriate.

Although this is the end of the transformations module, many of these ideas will recur throughout the course. In Module 2 you will focus on radical functions.

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MODULE 1 - SUMMATIVE ASSIGNMENT

Solve the following questions found in your text book. Show full solutions and attach to this module. Hand in to your instructor. Module one is now complete. Once you have received your corrected work, review your instructor comments and prepare for your module one test.

Text: Pre-Calculus 12

Section1.1 Pages 12 to 15, # 11, 12, 18c, 18d

Section 1.2 Pages 28 to 31, #6, 7, 14, 15

Section 1.3 Pages 38 to 43, #9, 10

Section 1.4 Pages 51 to 55, #7a, 9a, 9e,13a, 15a,17