REQUISITOS Y ELEMENTOS DE LA JAULA ANTIVUELCO

DISEÑO DEL CHASIS BASE O LA JAULA ANTIVUELCO

A.3 REQUISITOS Y ELEMENTOS DE LA JAULA ANTIVUELCO

See the Teacher Resources at SpringBoard Digital for a student page for

this mini-lesson.

My Notes

Lesson 4-3

Transforming the Absolute Value Parent Function

6. Graph each function.

a. f(x) = |x − 4| b. f(x) = |x + 5|

7. Use the coordinate grid at the right.

a. Graph the parent function f(x) = |x| and the function g(x) = |2x|.

b. Describe the graph of g(x) as a horizontal stretch or horizontal shrink of the graph of the parent function.

8. Express regularity in repeated reasoning. Use the results from Item 7 to predict how the graph of h(x) = ¹₂x is transformed from the graph of the parent function. Then graph h(x) to confirm or revise your prediction.

–5 5

–2 –4 4 2

x y

f(x) = |x – 4| –5

–2 –4 2 4

x y

f(x) = |x + 5|

–5 5

–2 2 4 6

x y g(x) = |2x|

f(x) = |x|

–5 5

–2 2 4 6

x y

f(x) = |x|

h(x) = 12 x

A horizontal stretch or shrink by a factor of k maps a point (x, y) on the graph of the original function to the point (kx, y) on the graph of the transformed function.

Similarly, a vertical stretch or shrink by a factor of k maps a point (x, y) on the graph of the original function to the point (x, ky) on the graph of the transformed function.

MATH TIP

a horizontal shrink by a factor of 12

Sample prediction: a horizontal stretch by a factor of 2

Activity 4 • Piecewise-Defined Functions 67 continued ACTIVITY 4

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Technology Tip

ACTIVITY 4

^Continued

6 Predict and Confirm, Create Representations, Chunking the Activity Before students graph the functions given in Item 6, have them write down their predictions for the transformation that will take place with the parent absolute value graph. Have them confirm their predictions by graphing. Before moving on to the next items, sketch a couple of absolute value horizontal transformations, and ask students to write the equations of the functions you graphed.

7–8 Discussion Groups, Predict and Confirm, Critique Reasoning, Think-Pair-Share After presenting Items 7 and 8, have students work in small groups or with a partner to try to predict how the value of k in f x( ) = kx affects the graph of f(x) = |x|. In other words, what if g x( ) = 3 and x h x( ) = 13 ? Have students collaborate x and share their findings.

To graph the parent absolute value function on a TI graphing calculator, follow these basic steps:

1. Press the y = key in the upper left corner.

2. Press the CLEAR key to clear any previous functions.

3. Press the MATH key.

4. Press the right arrow one time to highlight the NUM command.

5. Select option 1, “abs(” for absolute value, by pressing ENTER . 6. Press the X,T,Θ,n ^key.

7. Press the ) key to close parentheses.

8. Press the GRAPH key.

Note: Use this basic process with the transformations of the parent absolute value function.

For additional technology resources, visit SpringBoard Digital.

My Notes

Lesson 4-3 Transforming the Absolute Value Parent Function

9. In the absolute value function f(x) = |kx| with k > 1, describe how the graph of the function changes compared to the graph of the parent function. What if k < −1?

10. In the absolute value function f(x) = |kx| with 0 < k < 1, describe how the graph of the function changes compared to the graph of the parent function. What if −1 < k < 0?

11. Each graph shows a transformation g(x) of the parent function f(x) = |x|.

Describe the transformation and write the equation of g(x).

–5 5

–2 2 4 6

x y _g(x)

f(x) = |x|

–5 5

–4 –2 2 4

x y

g(x) f(x) = |x|

The graph of f(x) is a horizontal shrink of the graph of the parent function by a factor of 1k when k > 1 or when k < −1.

The graph of f(x) is a horizontal stretch of the graph of the parent function by a factor of 1k when 0 < k < 1 or when −1 < k < 0.

vertical stretch by a factor of 3 or horizontal shrink by a factor of 13 ; g(x) = 3|x| or g(x) = |3x|

vertical translation down 3 units; g(x) = |x| − 3

68 SpringBoard^® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions continued

ACTIVITY 4

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ACTIVITY 4

^Continued

9–10 Close Reading, Marking the Text Students probably will not have too much difficulty expressing regularity in repeated reasoning and coming up with a generalization for Item 9, based upon the previous items.

However, Item 10 is written in such a way that you may want to emphasize some points. The functions in Items 9 and 10 are written the same, but students should look closely at the values of k. In Item 10, k is really a fractional value. Because of the way this is written, the only difference between the answers for Items 9 and 10 is the words shrink and stretch. The reason why the factors are both 1

k is because 1 k, when k is a fraction, is the inverse of the fractional value.

11 Summarizing Point out that the phrases “vertical stretch” and

“horizontal shrink” can both be used to describe what is taking place in Item 11a. However, the difference between these descriptions is the values of their factors.

Universal Access My Notes

Lesson 4-3

Transforming the Absolute Value Parent Function

Example A

Describe the transformations of g(x) = 2|x + 3| from the parent absolute value function and use them to graph g(x).

Step 1: Describe the transformations.

g(x) is a horizontal translation of f(x) = |x| by 3 units to the left, followed by a vertical stretch by a factor of 2.

Apply the horizontal translation first, and then apply the vertical stretch.

Step 2: Apply the horizontal translation.

Graph f(x) = |x|. Then shift each point on the graph of f(x) by 3 units to the left. To do so, subtract 3 from the x-coordinates and keep the y-coordinates the same.

Name the new function h(x). Its equation is h(x) = |x + 3|.

Step 3: Apply the vertical stretch.

Now stretch each point on the graph of h(x) vertically by a factor of 2. To do so, keep the x-coordinates the same and multiply the y-coordinates by 2.

Solution: The new function is g(x) = 2|x + 3|.

Try These A

For each absolute value function, describe the transformations represented in the rule and use them to graph the function.

a. h(x) = −|x − 1| + 2 b. k(x) = 4|x + 1| − 3

–6 –2 2

–2 2 4 6

x y

–4

–8 4

h(x) = |x + 3|

f(x) = |x|

–6 –2 2

–2 2 4 6

x y

–4

–8 4

g(x) = 2|x + 3| h(x) = |x + 3|

To graph an absolute value function of the form

g(x) = a|b(x − c)| + d, apply the transformations of f(x) = |x| in this order:

1. horizontal translation 2. reflection in the y-axis and/or

horizontal shrink or stretch 3. reflection in the x-axis and/or

vertical shrink or stretch 4. vertical translation MATH TIP

You can check that you have graphed g(x) correctly by graphing it on a graphing calculator.

TECHNOLOGY TIP

translate to the right 1 unit, refl ect over the x-axis, and translate up 2 units

translate 1 unit to the left, vertically stretch by a factor of 4, then translate 3 units down

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ACTIVITY 4

^Continued

Example A Activating Prior

Knowledge This concept of unraveling multiple transformations to a parent absolute value function is similar to following orders of operations with arithmetic operations.

Try These A Answers a.

–5 5

–2 –4 2 4

x y

h(x) = –|x – 1| + 2

–5 5

–2 –4

2 4

x y

k(x) = 4|x + 1| – 3

For students having difficulty with the concept of absolute value, explain that an absolute value represents a distance, or measure, of a number from the origin of a number line.

Furthermore, a measurement cannot be a negative value.

Keeping that in mind, when applying this concept to absolute value functions and their graphs, this is why they remain above the x-axis unless written with a negative sign preceding the absolute value bars.

My Notes

Lesson 4-3 Transforming the Absolute Value Parent Function

Check Your Understanding

12. Graph the function g(x) = |−x|. What is the relationship between g(x) and f(x) = |x|? Why does this relationship make sense?

13. Compare and contrast a vertical stretch by a factor of 4 with a horizontal stretch by a factor of 4.

14. Without graphing the function, determine the coordinates of the vertex of f(x) = |x + 2| − 5. Explain how you determined your answer.

LESSON 4-3 PRACTICE

15. The graph of g(x) is the graph of f(x) = |x| translated 6 units to the right.

Write the equation of g(x).

16. Describe the graph of h(x) = −5|x| as one or more transformations of the graph of f(x) = |x|.

17. What are the domain and range of f(x) = |x + 4| − 1? Explain.

18. Graph each transformation of f(x) = |x|.

a. g(x) = |x − 4| + 2 b. g(x) = |2x| − 3 c. g(x) = −|x + 4| + 3 d. g(x) = −3|x + 2| + 4

19. Attend to precision. Write the equation for each transformation of f(x) = |x| described below.

a. Translate left 9 units, stretch vertically by a factor of 5, and translate down 23 units.

b. Translate left 12 units, stretch horizontally by a factor of 4, and reflect over the x-axis.

5 –2

–4 2

x y

70 SpringBoard^® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions continued

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ACTIVITY 4

^Continued

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to transformations of functions.

Answers

12. g(x) and f(x) are the same function.

This relationship makes sense because |x| = |−x|.

–4 –2 2 4

2 4 6

x y

6 –6

13. Sample answer: Both transformations stretch points on the original graph away from an axis. A vertical stretch maps a point (x, y) on the original graph to point (x, 4y) on the transformed graph. A horizontal stretch maps a point (x, y) on the original graph to point (4x, y) on the transformed graph.

14. (−2, −5); Sample explanation: The graph of f(x) is a translation of the graph of the absolute value parent function by 2 units left and 5 units down. The vertex of the graph of the absolute value parent function is (0, 0), so the vertex of the graph of f(x) must be (−2, −5).

ASSESS

Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning.

See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity.

LESSON 4-3 PRACTICE 15. g(x) = |x − 6|

16. a reflection over the x-axis and a vertical stretch by a factor of 5 17. Domain: {x | x ∈ }; range: {y | y ∈

, y ≥ −1}; Sample explanation: The function is defined for all real values of x, so the domain is all real numbers. The graph of f(x) opens upward, and its vertex is at (−4, −1).

−1 is the minimum value of f(x), so the range is all real numbers ≥ −1.

ADAPT

Check students’ answers to the Lesson Practice to ensure that they understand transformations of the absolute value parent function. If students have trouble graphing a function by using the values in the equation, allow them to graph by generating ordered pairs.

18. a.

–5 5

2 4

x y

–5 5

–2 2 4

x y

–5 –2 –4 2 4

x y

19. a. f(x) = 5|x + 9| − 23 b. f x( )= −1(x+ )

4 12

c. f(x) = 3|x − 2| − 4 or f(x) = |3(x − 2)| − 4 c.

–5 –2 –4 2 4

x y

ACTIVITY 4 PRACTICE Write your answers on notebook paper.

Show your work.

Lesson 4-1

1. Graph each of the following piecewise-defined functions. Then write its domain and range using inequalities, interval notation, and set notation.

2. Explain why the graph shown below does not represent a function.

A welder earns $20 per hour for the first 40 hours she works in a week and $30 per hour for each hour over 40 hours. Use this information for Items 3–5.

3. Write a piecewise function f(x) that can be used to determine the welder’s earnings when she works x hours in a week.

4. Graph the piecewise function.

5. How much does the welder earn when she works 48 hours in a week?

A. $990 B. $1040 C. $1200 D. $1440

6. The domain of a function is all real numbers greater than −2 and less than or equal to 8. Write the domain using an inequality, interval notation, and set notation.

7. The range of a function is [4, ∞). Write the range using an inequality and set notation.

8. Evaluate f(x) for x = −4, x = 1, and x = 4.

9. Write the equation of the piecewise function f(x) shown below. wallpaper steamer. If the time in days is not a whole number, it is rounded up to the next-greatest day. Customers are given the weekly rate if it is cheaper than using the daily rate.

a. Write the equation of a step function f(x) that can be used to determine the cost in dollars of renting a wallpaper steamer for x days. Use a domain of 0 < x ≤ 7.

Activity 4 • Piecewise-Defined Functions 71 continued ACTIVITY 4

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ACTIVITY 4

^Continued

ACTIVITY PRACTICE

because the input −2 has 2 outputs, the relationship is not a function.

12. A step function called the integer part function gives the value f(x) that is the integer part of x.

a. Graph the integer part function.

b. Find f(−2.1), f(0.5), and f(3.6).

13. A step function called the nearest integer function gives the value g(x) that is the integer closest to x.

For half integers, such as 1.5, 2.5, and 3.5, the nearest integer function gives the value of g(x) that is the even integer closest to x.

a. Graph the nearest integer function.

b. Find g(−2.1), g(0.5), and g(3.6).

14. Use the definition of f(x) = |x| to rewrite

f x x

( ) | |= x as a piecewise-defined function.

Then graph the function.

15. Consider the absolute value function f(x) = |x + 2| − 1.

a. Graph the function.

b. What are the domain and range of the function?

c. What are the x-intercept(s) and y-intercept of the function?

d. Describe the symmetry of the graph.

Lesson 4-3

16. Write the equation of the function g(x) shown in the graph, and describe the graph as a

transformation of the graph of f(x) = |x|.

17. Graph the following transformations of f(x) = |x|.

Then identify the transformations.

a. g(x) = |x + 3| − 1 b. g(x) = 1

3 |x| + 2 c. g(x) = −2|x − 1| − 1 d. g(x) = 5|x − 1| − 4

18. Write the equation for each transformation of f(x) = |x| described below.

a. translated right 7 units, shrunk vertically by a factor of 0.5, and translated up 5 units b. stretched horizontally by a factor of 5, reflected

over the x-axis, and translated down 10 units c. translated right 9 units and translated down

6 units Reason Abstractly and Quantitatively 20. Before answering each part, review them

carefully to ensure you understand all the terminology and what is being asked.

a. Describe how the graph of g(x) = |x| + k changes compared to the graph of f(x) = |x|

when k > 0 and when k < 0.

72 SpringBoard^® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions continued

ACTIVITY 4

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ACTIVITY 4

^Continued

12. a.

15. a. Check students’ graphs.

b. domain: {x | x ∈ }; range: reflection of the graph of f(x) across the x-axis, followed by a vertical stretch by a factor of 2, and then a translation 3 units up.

17. a. a horizontal translation 3 units left and a vertical translation 1 unit down

b. a vertical shrink by a factor of 1

3 followed by a translation 2 units up

c. a translation 1 unit right, followed by a reflection across the x-axis and a vertical stretch by a factor of 2, and then a translation 1 unit down d. a translation 1 unit right,

followed by a horizontal shrink by a factor of 1

5, and then a translation 4 units down ADDITIONAL PRACTICE

If students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital for additional practice problems. graph of g(x) is the graph of f(x) translated k units down.

b. When k > 1, the graph of h(x) is the graph of f(x) vertically stretched by a factor of k. When 0 < k < 1, the graph of h(x) is the graph of f(x) vertically shrunk by a factor of k.

c. When k < 0, the graph of j(x) is f(x) horizontally stretched or shrunk by a factor of 1

My Notes

Function Composition and Operations

In document Diseño y análisis de una estructura tubular para un vehículo de carreras todo terreno (página 93-100)