Radical Functions

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Radical Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
Holt McDougal Algebra 2
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Warm Up Identify the domain and range of each
function.
D R Ryy 2
1. f(x) x2 2
D R R R
2. f(x) 3x3
Use the description to write the quadratic
function g based on the parent function f(x)
x2.
3. f is translated 3 units up.
g(x) x2 3
g(x) (x 2)2
4. f is translated 2 units left.
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Objectives
Graph radical functions and inequalities. Transfo
rm radical functions by changing parameters.
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Vocabulary
radical function square-root function
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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.
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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 .
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Example 1A Graphing Radical Functions
Graph each function and identify its domain and
range.
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.
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Example 1A Continued
x (x, f(x))
3 (3, 0)
4 (4, 1)
7 (7, 2)
12 (12, 3)
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?
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The domain is xx 3, and the range is yy
0.
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Example 1B Graphing Radical Functions
Graph each function and identify its domain and
range.
Make a table of values. Plot enough ordered pairs
to see the shape of the curve. Choose both
negative and positive values for x.
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Example 1B Continued
x (x, f(x))
6 (6, 4)
1 (1,2)
2 (2, 0)
3 (3, 2)
10 (10, 4)
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The domain is the set of all real numbers. The
range is also the set of all real numbers
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Example 1B Continued
Check Graph the function on a graphing calculator.
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Check It Out! Example 1a
Graph each function and identify its domain and
range.
Make a table of values. Plot enough ordered pairs
to see the shape of the curve. Choose both
negative and positive values for x.
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Check It Out! Example 1a Continued
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.
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Check It Out! Example 1a Continued
Check Graph the function on a graphing calculator.
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Check It Out! Example 1b
Graph each function, and identify its domain and
range.
x (x, f(x))
1 (1, 0)
3 (3, 2)
8 (8, 3)
15 (15, 4)




The domain is xx 1, and the range is yy
0.
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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.
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Example 2 Transforming Square-Root Functions
Using the graph of as a guide,
describe the transformation and graph the
function.
f(x) x
Translate f 5 units up.
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Check It Out! Example 2a
Using the graph of as a guide,
describe the transformation and graph the
function.
f(x) x
Translate f 1 unit up.
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Check It Out! Example 2b
Using the graph of as a guide,
describe the transformation and graph the
function.
f(x) x
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Transformations of square-root functions are
summarized below.
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Example 3 Applying Multiple Transformations
Using the graph of as a guide,
describe the transformation and graph the
function
f(x) x
.
Reflect f across the x-axis, and translate it 4
units to the right.
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Check It Out! Example 3a
Using the graph of as a guide,
describe the transformation and graph the
function.
f(x) x
g is f reflected across the y-axis and translated
3 units up.
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Check It Out! Example 3b
Using the graph of as a guide,
describe the transformation and graph the
function.
f(x) x
g is f vertically stretched by a factor of 3,
reflected across the x-axis, and translated 1
unit down.
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Example 4 Writing Transformed Square-Root
Functions
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) x
Step 1 Identify how each transformation affects
the function.
Reflection across the x-axis a is negative
Translation 5 units down k 5
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Example 4 Continued
Step 2 Write the transformed function.
Simplify.
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Example 4 Continued
Check Graph both functions on a graphing
calculator. The g indicates the given
transformations of f.
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Check It Out! Example 4
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.
f(x) x
Step 1 Identify how each transformation affects
the function.
Reflection across the x-axis a is negative
a 2
Vertical compression by a factor of 2
Translation 5 units down k 1
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Check It Out! Example 4 Continued
Step 2 Write the transformed function.
Substitute 2 for a and 1 for k.
Simplify.
Check Graph both functions on a graphing
calculator. The g indicates the given
transformations of f.
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Example 5 Business Application
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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.
Substitute 192 for a and simplify.
The cost for the glass of a picture with an area
of 192 in2 is about 13.13 including
installation.
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Check It Out! Example 5
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 objects downward velocity in
feet per second as the object hits the ground
after bouncing x ft in height.
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x
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.
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Check It Out! Example 5 Continued
Step 1 To compress f horizontally by a factor of
, multiply f by .
Step 2 Find the value of g for a bounce of 50ft.
Substitute 50 for x and simplify.
The landing craft will hit the Moons surface
with a downward velocity of about 23 ft at the
end of the bounce.
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In addition to graphing radical functions, you
can also graph radical inequalities. Use the same
procedure you used for graphing linear and
quadratic inequalities.
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Example 6 Graphing Radical Inequalities
Graph the inequality .
x 0 1 4 9
y 3 1 1 3
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Example 6 Continued
Step 2 Use the table to graph the boundary
curve. The inequality sign is gt, 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.
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Example 6 Continued
Check Choose a point in the solution region,
such as (1, 0), and test it in the inequality.
0 gt 2(1) 3
0 gt 1
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Check It Out! Example 6a
Graph the inequality.
x 4 3 0 5
y 0 1 2 3
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Check It Out! Example 6a Continued
Step 2 Use the table to graph the boundary
curve. The inequality sign is gt, so use a dashed
curve and shade the area above it.
Because the value of x cannot be less than 4, do
not shade left of 4.
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Check It Out! Example 6a Continued
Check Choose a point in the solution region,
such as (0, 4), and test it in the inequality.
4 gt (0) 4
4 gt 2
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Check It Out! Example 6b
Graph the inequality.
x 4 3 0 5
y 0 1 2 3
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Check It Out! Example 6b Continued
Step 2 Use the table to graph the boundary
curve. The inequality sign is gt, so use a dashed
curve and shade the area above it.
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Check It Out! Example 6b Continued
Check Choose a point in the solution region,
such as (4, 2), and test it in the inequality.
2 1
?
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Lesson Quiz Part I
Dxx 4 Ryy 0
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Lesson Quiz Part II
2. Using the graph of as a guide,
describe the transformation and graph the
function .
g(x) -x 3
g is f reflected across the y-axis and translated
3 units up.
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Lesson Quiz Part III
3. Graph the inequality .