Algebra II 5 7 Study Guide Radical Functions RevisedII+5_7... · test. However, if we limit ... Algebra II 5-7 Study Guide Page !11 of 12 Example 4. Use the description to write the

Algebra II 5-7 Study Guide Page ! of !1 12

Radical Functions !Attendance Problems. Identify the domain and range of each function.

1. ! 2. ! !!!!!Use the description to write the quadratic function g based on the parent function f(x) = x2.

3. f is translated 3 units up. 4. f is translated 2 units left. !!• I can graph radical functions and inequalities. • I can transform radical functions by changing parameters.

!Common Core

CCSS.MATH.CONTENT.HSF.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* CCSS.MATH.CONTENT.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. !

Recall that exponential and logarithmic functions are inverse functions. Quadratic and cubic functions have inverses as well. The graphs show the inverses of the quadratic parent function and cubic parent function. !

f x( ) = x2 + 2 f x( ) = 3x3

Vocabulary

radical function square-root function


Notice that the inverses of f(x) = x2 is not a function because it fails the vertical line test. However, if we limit the domain of f(x) = x2 to x ≥ 0, its inverse is the function ! . !A radical function is a function whose rule is a radical expression. A square-root function is a radical function involving ! . The square-root parent function is ! . The cube-root parent function is ! Video Example 1. Graph each function identifying its domain and range.

A. ! !!!!!!!!!

f −1 x( )= x

xf x( )= x f x( )= x3 .

f (x) = − x


B. ! !!!!!!!!

f x( ) = 2 x +13

Recall that exponential and logarithmic functions are inverse functions. Quadratic and cubic functions have inverses as well. The graphs below show the inverses of the quadratic parent function and the cubic parent function.

f(x) = x 2

y = x

42

-4

-2

4

2

y

x

-4 -2

f - 1 (x) = ± √ " x

y = x

42

-4

-2

4

2

y

x

-4 -2

f - 1 (x) = 3 √ " x

f(x) = x 3

Notice that the inverse of f (x) = x 2 is not a function because it fails the vertical line test. However, if we limit the domain of f (x) = x 2 to x ≥ 0, its inverse is the function f -1 (x) = √ # x .

A radical function is a function whose rule is a radical expression. A square-root function is a radical function involving √ # x . The square-root parentfunction is f (x) = √ # x . The cube-root parent function is f (x) = 3 √ # x .

1E X A M P L E Graphing Radical Functions

Graph the function, and identify its domain and range.

A f (x) = √ " x Make a table of values. Plot enough ordered pairs to see the shape of the curve. Because the square root of a negative number is imaginary, choose only nonnegative values for x.

x f (x) = √ " x (x, f (x) )

0 f (0) = √ # 0 = 0 (0, 0)

1 f (1) = √ # 1 = 1 (1, 1)

4 f (4) = √ # 4 = 2 (4, 2)

9 f (9) = √ # 9 = 3 (9, 3)

4 6 8 102-2

4

0

y

x

(1, 1)(4, 2)

(9, 3)

(0, 0)

The domain is ⎧

⎨

⎩ x | x ≥ 0

⎫

⎬

⎭ , and the range is

⎧

⎨

⎩ y | y ≥ 0

⎫

⎬

⎭ .

Radical Functions

ObjectivesGraph radical functions and inequalities.

Transform radical functions by changing parameters.

Vocabularyradical functionsquare-root function

Who uses this?Aerospace engineers use transformations of radical functions to adjust for gravitational changes on other planets. (See Example 5.)

When using a table to graph square-root functions, choose x-values that make the radicands perfect squares.

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5-7 Radical Functions 367

5-7CC.9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.* Also CC.9-12.F.BF.3, CC.9-12.F.IF.5, CC.9-12.A.CED.2, CC.9-12.A.CED.3

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Example 1. Graph each function identifying its domain and range.

A. ! !!!!!!!!

B. !

Guided Practice. 5. ! !!!!!!!!!

f x( ) = x − 3

f x( ) = 2 x − 23

f x( ) = x3


6. ! !!!!!!!The graphs of radical functions can be transformed by using methods similar to those used to transform linear, quadratic, polynomial, and exponential functions. This lesson will focus on transformations of square-root functions. !

f x( ) = x +1



Video Example 2. Using the graph of ! as a guide, describe the transformation and graph the function.

A. ! !!

B. !

Example 2. Using the graph of ! as a guide, describe the transformation and graph the function ! . !!!

f x( )= x

g x( ) = x + 3

g x( ) = 2 x

f x( )= x

g x( ) = x + 5

2E X A M P L E Transforming Square-Root Functions

Using the graph of f (x) = √ " x as a guide, describe the transformation and graph each function.

A g (x) = √ " x - 2 B g (x) = 3 √ " x g (x) = f (x) - 2 g (x) = 3 · f (x)

Translate f 2 units down.

4-2

4

2

0

y

x

(0, -2)

(0, 0)

f

g(4, 0)

(4, 2)

Stretch f vertically by a factor of 3.

2 4 6

2

4

6

0

y

x

(4, 6)g

f

(4, 2)(0, 0)


2a. g (x) = √ " x + 1 2b. g (x) = 1 _ 2

√ " x

Transformations of square-root functions are summarized below.

vertical translation

vertical stretch or compression factor reflection across the -axis

| | < 0

horizontal stretch or compression factor reflection across the -axis

| | < 0

horizontal translation

3E X A M P L E Applying Multiple Transformations


A g (x) = 2 √ """ x + 3 B g (x) = √ "" -x - 2 Stretch f vertically by a factor Reflect f across the y-axis,

of 2, and translate it 3 units left. and translate it 2 units down.

(1, 4)

(4, 2)(0, 0)(-3, 0)

42

4

2

6

0

y g

f

x

-4 -2

42-2

-4

2

0

y

g fx

(-4, 0)

(0, -2)

(0, 0)

(4, 2)


3a. g (x) = √ "" -x + 3 3b. g (x) = -3 √ " x - 1


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Guided Practice. Using the graph of ! as a guide, describe the transformation and graph the function.

7. ! !!

8. !

!!

f x( )= x

g x( ) = x +1

g x( ) = 12

x


Video Example 3. Using the graph of ! as a guide, describe the transformation and graph the function.

A. ! !!!!B. !

!

f x( )= x

g(x) = 3 x − 2

−x +1

2E X A M P L E Transforming Square-Root Functions


A g (x) = √ " x - 2 B g (x) = 3 √ " x g (x) = f (x) - 2 g (x) = 3 · f (x)

Translate f 2 units down.

4-2

4

2

0

y

x

(0, -2)

(0, 0)

f

g(4, 0)

(4, 2)

Stretch f vertically by a factor of 3.

2 4 6

2

4

6

0

y

x

(4, 6)g

f

(4, 2)(0, 0)


2a. g (x) = √ " x + 1 2b. g (x) = 1 _ 2

√ " x

Transformations of square-root functions are summarized below.

vertical translation

vertical stretch or compression factor reflection across the -axis

| | < 0

horizontal stretch or compression factor reflection across the -axis

| | < 0

horizontal translation

3E X A M P L E Applying Multiple Transformations


A g (x) = 2 √ """ x + 3 B g (x) = √ "" -x - 2 Stretch f vertically by a factor Reflect f across the y-axis,

of 2, and translate it 3 units left. and translate it 2 units down.

(1, 4)

(4, 2)(0, 0)(-3, 0)

42

4

2

6

0

y g

f

x

-4 -2

42-2

-4

2

0

y

g fx

(-4, 0)

(0, -2)

(0, 0)

(4, 2)


3a. g (x) = √ "" -x + 3 3b. g (x) = -3 √ " x - 1




Example 3. Using the graph of ! as a guide, describe the transformation and graph the function ! . !!!!!Guided Practice. Using the graph of ! as a guide, describe the transformation and graph the function.

9. ! !!!!!

10. !

Video Example 4. Use the description to write the square-root function g. the parent function ! is stretched horizontally by a factor of 3, reflected across the y-axis, and translated 2 units right. !!!!!!!

f x( )= x

g(x) = − x − 4

f x( )= x

g x( ) = −x + 3

g x( ) = −3 x −1

f x( )= x


Example 4. Use the description to write the square-root function g. The parent function ! is reflected across the x-axis, compressed vertically by a factor

of ! , and translated down 5 units.

!!!!!!

f x( )= x15

4E X A M P L E Writing Transformed Square-Root Functions

Use the description to write the square-root function g.

The parent function f (x) = √ " x is stretched horizontally by a factor of 2, reflected across the y-axis, and translated 3 units left.

Step 1 Identify how each transformation affects the function. Horizontal stretch by a factor of 2: ⎪b⎥ = 2

Reflection across the y-axis: b is negative ⎫

⎬

⎭ b = -2

Translation 3 units left: h = -3

Step 2 Write the transformed function.

g (x) = √ '''' 1 _ b

(x - h)

g (x) = √ '''''' 1 _ -2

⎡ ⎣ x- (-3) ⎤ ⎦ Substitute -2 for b and -3 for h.

g (x) = √ ''''' - 1 _ 2

(x + 3) Simplify.

Check Graph both functions on a graphing calculator. The graph of g indicates the given transformations of f.

Use the description to write the square-root function g. 4. The parent function f (x) = √ ' x is reflected across the x-axis,

stretched vertically by a factor of 2, and translated 1 unit up.

5E X A M P L E Space Exploration Application

Special airbags are used to protect scientific equipment when a rover lands on the surface of Mars. On Earth, the function f (x) = √ "" 64x approximates an object’s downward velocity in feet per second as the object hits the ground after bouncing x ft in height.

The corresponding function for Mars is compressed vertically by a factor of about 3 __ 5 . Write the corresponding function g for Mars, and use it to estimate how fast a rover will hit Mars’s surface after a bounce of 45 ft in height.

Step 1 To compress f vertically by a factor of 3 __ 5 , multiply f by 3 __ 5 .

g (x) = 3 _ 5

f (x) = 3 _ 5

√ '' 64x

Step 2 Find the value of g for a bounce height of 45 ft.

g (45) = 3 _ 5

√ ''' 64 (45 ) ≈ 32 Substitute 45 for x and simplify.

The rover will hit Mars’s surface with a downward velocity of about 32 ft/s at the end of the bounce.

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370 Chapter 5 Rational and Radical Functions



11. Guided Practice. Use the description to write the square-root function g. The parent function ! is reflected across the x-axis, stretched vertically by a factor of 2, and translated 1 unit up. !!!!!!5-7 Assignment (p 372) 25-41 odd, 49, 51-54, 57, 68, 73-77.

f x( )= x

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Algebra II 5 7 Study Guide Radical Functions RevisedII+5_7... · test. However, if we limit ... Algebra II 5-7 Study Guide Page !11 of 12 Example 4. Use the description to write the