Download 6-2 Attributes of Square Root Functions

6-2
Attributes of Square Root Functions
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Additional TEKS (1)(A), (1)(D), (4)(E), (7)(I)
ESSENTIAL UNDERSTANDING
A square root function is the inverse of a quadratic function that has
a restricted domain.
Key Concept
Table
The Square Root Parent Function
Function
f (x) = 1x, when x Ú 0.
Graph
x
f(x) = x
0
0
1
4
1
2
2
9
3
O
6
y
4
Domain
Range
x-intercept
y-intercept
5
x
10
xÚ0
yÚ0
(0, 0)
(0, 0)
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Problem 1
P
TEKS Process Standard (1)(D)
Domain and Range of Square Root Functions
Graph each square root function. Give the domain and range
using inequalities.
A f (x) = 2x
The square root function is defined over nonnegative real numbers, so the
value of the expression under the square root symbol must be nonnegative.
Can the square root
function be negative?
No. Square root means
the positive square root.
A negative symbol in
front of the square root is
the negative square root.
Domain: x Ú 0
Range: f (x) Ú 0
4
2
y
f(x) = x
x
O
5
B f (x) = 2x + 2
Domain: x Ú 0
This function adds 2 to the square root of x, which must be nonnegative.
Therefore, the value of the function must be 2 or greater.
Range: f (x) Ú 2
y
4
2
O
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Lesson 6-2
f(x) = x + 2
x
5
Attributes of Square Root Functions
Problem 2
P
TEKS Process Standard (1)(F)
Intercepts of Square Root Functions
Graph each square root function. Identify the x- and y-intercepts.
A f (x) = 2x
How do you find the
x-intercept?
The x-intercept occurs at
the x-value that makes
f (x) = 0. Since 10 = 0,
you need to find the
x-value that makes the
expression under the
radical equal to 0.
4
The graph of this function crosses the x-axis
at x = 0, and it crosses the y-axis at y = 0.
y
f(x) = x
2
x-intercept: (0, 0)
x
O
y-intercept: (0, 0)
B f (x) = 2x + 4
4
The graph of this function crosses the x-axis at x = -4,
and it crosses the y-axis at y = 24 = 2.
5
y
2
x-intercept: ( -4, 0)
f(x) = x + 4
O
y-intercept: (0, 2)
x
5
Problem
P
bl
3
Maximum and Minimum Values of Square Root Functions
Graph each square root function. Find the maximum and minimum values
over the given interval.
A f (x) = 2x ; 1 " x " 4
How do you find
minimum and
maximum values?
First graph the function.
Then look at the portion
of the graph within
the range of x-values
specified. Find the
minimum and maximum
y-values.
4
Notice from the graph that the function increases
over the interval 1 … x … 4. Therefore, the minimum
and maximum values are found at the endpoints of
the interval. The minimum value is f (1) = 1, and the
maximum value is f (4) = 2.
B
f(x) = x
2
x=4
5
O x=1
x
y
f (x) = 2x − 3 − 1; 3 " x " 8
This function also increases over the interval 3 … x … 8.
The minimum value is f (3) = -1, and the maximum
value is f (8) = 25 - 1.
y
2
O
f(x) = x − 3 − 1
x=3
5
x
x = 8 10
-2
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PRACTICE and APPLICATION EXERCISES
For additional support when
completing your homework,
go to PearsonTEXAS.com.
Scan page for a Virtual Nerd™ tutorial video.
1. Graph the function f (x) = 1x. Analyze the key attributes of the function,
including domain, range, and intercepts. Find the maximum and minimum
values of the function over the interval [4, 6].
Graph each function. Write the domain and range as inequalities.
2. f (x) = 2x + 4
3. f (x) = 3 - 2x
4. f (x) = 2x + 1 - 2
5. f (x) = 22x
6. f (x) = - 2x - 3
Match each function with its domain and range.
7. f (x) = 2x - 3 + 7
A. Domain: x Ú -3, Range: f (x) Ú -7
8. f (x) = 2x + 7 + 3
B. Domain: x Ú -7, Range: f (x) Ú 3
9. f (x) = 2x + 3 - 7
C. Domain: x Ú 3, Range: f (x) Ú 7
Graph each function. Give the x- and y-intercepts.
10. f (x) = 2x - 1
11. f (x) = 2x - 3
12. f (x) = 22 - x
13. f (x) = 2x + 5 - 2
14. f (x) = 3 - 2x + 4
15. f (x) = 2x + 1
Determine whether each statement is always, sometimes, or never true.
16. A square root function has no x- or y-intercepts.
17. The domain of a square root function includes positive and negative real
numbers.
18. If a 7 0, then the graph of f (x) = 21 + x + a is entirely in Quadrant I of the
coordinate plane.
19. A square root function has two x-intercepts.
20. For any real number a, the range of f (x) = 2x - a includes negative numbers.
21. If a 6 0, the minimum of f (x) = a - 2x in the interval b … x … c is f (c).
240
Lesson 6-2
Attributes of Square Root Functions
Analyze Mathematical Relationships (1)(F) Write a square root function with
the following intercepts.
22. x-intercept: 0
23. y-intercept: 2
24. x-intercept: -1
25. y-intercept: -5
Graph each function. Give the minimum and maximum values
over the specified interval.
26. f (x) = 2x + 3, 0 … x … 5
27. f (x) = 2x - 2, 1 … x … 9
28. f (x) = 2x + 7, -7 … x … 0
29. f (x) = - 2x + 1, 0 … x … 8
30. f (x) = 2x - 1 - 1, 3 … x … 4
31. f (x) = 23 - x + 4,-5 … x … 0
32. Use Multiple Representations to Communicate Mathematical Idea (1)(D) A
company has found that the number of new customers who shop at their store is
modeled by the equation C = 1.25 + 21.8x, where x is the number of years since
2005, and C is the number of new customers in thousands. The graph of this
function is shown below.
Number of New Customers
(thousands)
y
5
4
3
2
1
O
x
2
4
6
8
Years since 2005
a. What is the y-intercept of the graph? What does the y-intercept mean in
this context?
b. What is the x-intercept of the graph? What does the x-intercept mean in
this context?
c. What are the minimum and maximum numbers of new customers between
2008 and 2011?
d. Assuming that the function continues to model the number of new customers
into the future, will the number of new customers continue to rise? Explain
your answer using the equation and graph.
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33. Explain Mathematical Ideas (1)(G) A student said that for the function
f (x) = a - 2x + b, where a 7 0, b 7 0, and c 7 0, the maximum value in
the interval 0 … x … c and the y-intercept are the same. Do you agree with the
student? Why or why not?
34. Use a Problem-Solving Model (1)(B) Mariana is making a square quilt out of
36 square quilt blocks. The area of the quilt is 3969 square inches. What is the
side length of the square blocks?
35. A rectangular room has an area of 324 square feet. The length and width of the
room have a ratio of 4 : 9. What are the dimensions of the room?
36. Analyze Mathematical Relationships (1)(F) Write a square root function with
a domain of x Ú 4 and an x-intercept of 5.
37. A fence company makes gates for their
fences. The gates can be made in any width,
but each gate has a height equal to its width.
The gates also include a diagonal brace. If x is
the width of the gate, what function gives the
length of the diagonal brace?
TEXAS Test Practice
T
38. What is the x-intercept of the function f (x) = 2x - 2?
A. -4
C. 2
B. -2
D. 4
39. Which function has a domain of x Ú -3?
F. f (x) = 2x - 3
H. f (x) = 2x - 3
G. f (x) = 2x + 3
J. f (x) = 2x + 3
40. Which function has a y-intercept of 2?
A. f (x) = 2x + 2
C. f (x) = 2 - 2x + 1
B. f (x) = 2x - 4
D. f (x) = 4 - 2x + 4
41. What is the maximum value of f (x) = 7 + 2x on the interval 0 … x … b?
Explain how you know.
42. What is the inverse of f (x) = x2 - 5?
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Lesson 6-2
Attributes of Square Root Functions