1. a) Complete a table for each of the absolute-value functions shown in Figure 1. y = |x|
y = |x + 4|
y = |x – 2|
x
y
x
y
x
y
–5
5
–6
2
–3
5
–2
2
–5
1
–1
3
–1
1
–4
0
0
2
0
0
–3
1
1
1
2
2
–1
3
2
0
3
3
0
4
3
1
6
6
2
6
5
3
Figure 1.
b) Use graph paper and graph all three functions on a single set of axes. c) Write the coordinates of the x-intercept and y-intercept for y = |x + 4|.
d) Write a piecewise-defined representation for y = |x + 4|.
e) Explain how the 4 in y = |x + 4| transforms the graph of y = |x|.
f) What change(s) would you need to make in the equation y = |x – 2| to cause its graph to shift upward 3 units?
g) In the context of Hermitville, what is the meaning of y = |x – 2|?
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SUPPLEMENTAL ACTIVITY
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S1.1 page 2 of 5
2. a) Complete a table for each of the absolute-value functions listed in Figure 2. y = | x + 3|
y = |x|
y = |x| + 3
x
y
x
y
x
y
–5
5
–6
3
–3
6
–2
2
–4
1
–1
4
–1
1
–3
0
0
3
0
0
–2
1
1
4
2
2
0
3
2
5
3
3
2
5
3
6
6
6
3
6
5
8
Figure 2.
b) Use graph paper or the graphing calculator and create one graph showing all three functions. c) Write the coordinates of the x-intercept and y-intercept for y = |x| + 3.
d) Write a piecewise-defined representation for y = |x|+ 3.
e) Explain how the 3s in y = |x| + 3 and y = |x + 3| transform the graph of y = |x|.
f) What change(s) would you need to make in the function y = |x + 3| to cause the graph to shift right 2 units?
g) What is the meaning of y = |x + 3| in Hermitville?
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S1.1 page 3 of 5
3. a) Without plotting points or using a graphing calculator, predict what the graphs of y = |–2x| and y = –2|x| will look like.
b) Use graph paper or a graphing calculator and prepare one graph to display both functions. Compare the resulting graph with your predictions. c) Write a piecewise-defined representation for y = –2|x|.
d) Write a piecewise-defined representation for y = |–2x|.
e) Explain how the –2’s in y = –2|x| and y = |–2x| transform the graph of y = |x|.
f) What change(s) would you need to make in the equation y = |–2x| to cause its graph to shift downward 4 units?
g) If x represents location and y is distance, explain why the equation y = –2|x| does not make sense in the context of Hermitville.
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Unit 1: GRIDVILLE
Mathematics: Modeling Our World
ABSOLUTELY TRANSFORMED
page 4 of 5
4. a) Without plotting points or using a graphing calculator, predict what the graphs of y = |x – 1| and y = –|x + 1| will look like.
b) Graph the two functions on the same axes. Identify the graphing window you use and explain why you selected it. c) Explain the effect of the (–) in y = –|x + 1|.
d) Use the graph of y = –|x + 1| to solve the equation –3 = –|x + 1|. Explain your method.
e) What change(s) would you need to make in the equation y = |x – 1| to cause its graph to shift downward 4 units, to the left 2 units, and have slopes of 2 and –2 on the right and left segments of the graph, respectively?
f) Discuss the meaning of y = |x + 1| and y = –|x + 1| as distances in Hermitville.
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MAKING A ROUGH ESTIMATE
SUPPLEMENTAL ACTIVITY
S1.1 page 5 of 5
5. a) On graph paper, graph the function y = x2 in the window [–6, 8] x [–2, 10]. Prepare a table of values if needed. b) On the same axes, graph a function representing a horizontal translation of the graph of y = x2 by 3 units to the right. c) Write an equation to represent the graph that you sketched in part (b).
6. a) The function y = 2(x + 1)2 – 3 is a transformation of the function y = x2. Describe how the control numbers 2, 1, and –3 each change the shape or position of the graph of y = x2.
b) Describe how the control numbers 2, 3 and 5 in the equation y = 2(x + 3) + 5 transform the graph of the linear equation y = x.
c) Describe how the control numbers 1, 4, y=
1 ( x + 1)2 + 4 2
1 2
in the quadratic equation
transform the graph of the quadratic equation y = x2.
d) Predict how the control numbers in the equation y = 3(x – 4)3 – 5 transform the graph of the cubic equation y = x3. Graph y = x3 and y = 3(x – 4)3 – 5 on your graphing calculator to check your answer.
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SUPPLEMENTAL ACTIVITY
S1.2 page 1 of 2
Many graphing calculators use what are called Boolean functions in addition to the more usual kinds of functions you have studied. For the purposes of this activity, a Boolean function is one that has only two possible values, 0 and 1, having the value 1 when a particular condition is true, and having the value 0 when the condition is false. Thus the Boolean function amounts to a “yes” or “no” rule. To illustrate the use of this idea in graphing piecewise-defined functions, consider the absolute value function y = |x|. Recall that one characterization of this function is y = x when x ≥ 0 and y = –x when x < 0. Define Y1 = X*(X ≥ 0) and define Y2 = -X*(X
