Holt Algebra 2 8-7 Radical Functions Graph radical functions and inequalities. Transform radical functions by changing parameters. Objectives.

Holt Algebra 2

8-7 Radical Functions

Graph radical functions and inequalities.

Transform radical functions by changing parameters.

Objectives

Holt Algebra 2


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 cubic parent function.

Holt Algebra 2


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 .

Holt Algebra 2


Graph each function and identify its domain and range.

Example 1A: Graphing Radical Functions

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 – 3.

Holt Algebra 2


Example 1A Continued

x (x, f(x))3 (3, 0)

4 (4, 1)

7 (7, 2)

12 (12, 3)

The domain is {x|x ≥3}, and the range is {y|y ≥0}.

●

●

●

●

Holt Algebra 2



Example 1B: Graphing Radical Functions

Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x.

Holt Algebra 2


x (x, f(x))–6 (–6, –4)

1 (1,–2)

2 (2, 0)

3 (3, 2)

10 (10, 4)

Example 1B Continued

●

●●

●●

The domain is the set of all real numbers. The range is also the set of all real numbers

Holt Algebra 2


Example 1B Continued

Check Graph the function on a graphing calculator.

Holt Algebra 2



Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x.

Check It Out! Example 1a

Holt Algebra 2


x (x, f(x))–8 (–8, –2)

–1 (–1,–1)

0 (0, 0)

1 (1, 1)

8 (8, 2)

•

• •

• •

The domain is the set of all real numbers. The range is also the set of all real numbers.

Check It Out! Example 1a Continued

Holt Algebra 2


Check Graph the function on a graphing calculator.


Holt Algebra 2


x (x, f(x))–1 (–1, 0)

3 (3, 2)

8 (8, 3)

15 (15, 4)

The domain is {x|x ≥ –1}, and the range is {y|y ≥0}.

•

••

•

Check It Out! Example 1b

Graph each function, and identify its domain and range.

Holt Algebra 2


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.

Holt Algebra 2


Holt Algebra 2


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

Example 2: Transforming Square-Root Functions

Translate f 5 units up.

•

•

g(x) = x + 5

f(x) = x

Holt Algebra 2



Translate f 1 unit up. ••


g(x) = x + 1

f(x)= x

Holt Algebra 2




g is f vertically compressed

by a factor of .1

2

f(x) = x

Holt Algebra 2


Transformations of square-root functions are summarized below.

Holt Algebra 2


Example 3: Applying Multiple Transformations

Reflect f across the x-axis, and translate it 4 units to the right.

••

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

f(x)= x

Holt Algebra 2


g is f reflected across the y-axis and translated 3 units up. ●

●



f(x)= x

Holt Algebra 2


g is f vertically stretched by a factor of 3, reflected across the x-axis, and translated 1 unit down.

●●



f(x)= x

g(x) = –3 x – 1

Holt Algebra 2


Example 4: Writing Transformed Square-Root Functions

Step 1 Identify how each transformation affects the function.

Reflection across the x-axis: a is negative

Translation 5 units down: k = –5

Vertical compression by a factor of 15

15

a = –

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.

15

f(x)= x

Holt Algebra 2


Example 4 Continued

Step 2 Write the transformed function.

Simplify.

Substitute – for a and –5 for k.15

15

g(x) = x + ( 5)

Holt Algebra 2


Example 4 Continued

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

Holt Algebra 2


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

Check It Out! Example 4

The parent function is reflected across the x-axis, stretched vertically by a factor of 2, and translated 1 unit up.

Step 1 Identify how each transformation affects the function.

Reflection across the x-axis: a is negative

Translation 5 units down: k = 1

Vertical compression by a factor of 2 a = –2

f(x)= x

Holt Algebra 2


Simplify.

Substitute –2 for a and 1 for k.

Step 2 Write the transformed function.

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

Check It Out! Example 4 Continued

Holt Algebra 2


Example 5: Business Application

A framing store uses the function to determine the cost c in dollars of glass for a picture with an area a in square inches. The store charges an addition $6.00 in labor to install the glass. Write the function d for the total cost of a piece of glass, including installation, and use it to estimate the total cost of glass for a picture with an area of 192 in2.

Holt Algebra 2


Example 5 Continued

Step 1 To increase c by 6.00, add 6 to c.

Step 2 Find a value of d for a picture with an area of 192 in2.

The cost for the glass of a picture with an area of 192 in2 is about $13.13 including installation.

Substitute 192 for a and simplify.

Holt Algebra 2


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

Check It Out! Example 5

The downward velocity function for the Moon is a horizontal stretch of f by a factor of about . Write the velocity function h for the Moon, and use it to estimate the downward velocity of a landing craft at the end of a bounce 50 ft in height.

25 4

64x

Holt Algebra 2


Step 2 Find the value of g for a bounce of 50ft.

Substitute 50 for x and simplify.

The landing craft will hit the Moon’s surface with a downward velocity of about 23 ft at the end of the bounce.

Check It Out! Example 5 Continued

Step 1 To compress f horizontally by a factor of , multiply f by .

25 4 4

25

4 25625 25

425

hxf x = 64x =

h x

25623

2550

Holt Algebra 2


In addition to graphing radical functions, you can also graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities.

Holt Algebra 2


Example 6: Graphing Radical Inequalities

x 0 1 4 9

y –3 –1 1 3

Graph the inequality .

Step 1 Use the related equation to make a table of values.

y =2 x 3

Holt Algebra 2


Example 6 Continued

Step 2 Use the table to graph the boundary curve. The inequality sign is >, so use a dashed curve and shade the area above it.

Because the value of x cannot be negative, do not shade left of the y-axis.

Holt Algebra 2


Example 6 Continued

Check Choose a point in the solution region, such as (1, 0), and test it in the inequality.

0 > 2(1) – 3

0 > –1

Holt Algebra 2


Graph the inequality.

x –4 –3 0 5

y 0 1 2 3



y = x+4

Holt Algebra 2



Because the value of x cannot be less than –4, do not shade left of –4.


Holt Algebra 2



4 > (0) + 4

4 > 2


Holt Algebra 2


Graph the inequality.

x –4 –3 0 5

y 0 1 2 3



3y x 3

Holt Algebra 2



Check It Out! Example 6b Continued

Holt Algebra 2



2 ≥ 1

Check It Out! Example 6b Continued

Holt Algebra 2


Lesson Quiz: Part I

D:{x|x≥ –4}; R:{y|y≥ 0}

1. Graph the function and identify its range and domain.

•

Holt Algebra 2


Lesson Quiz: Part II

g is f reflected across the y-axis and translated 3 units up.

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

•

•

gx = x + 3

Holt Algebra 2


Lesson Quiz: Part III

3. Graph the inequality .

Holt Algebra 2 8-7 Radical Functions Graph radical functions and inequalities. Transform radical functions by changing parameters. Objectives.

Documents