8-7
Radical Functions
Objectives
Graph radical functions and
inequalities.
Transform radical functions by
changing parameters.
Holt Algebra 2
8-7
Radical Functions
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
8-7
Radical Functions
2
Notice that the inverses of f(x) = x 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
8-7
Radical Functions
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.
Holt Algebra 2
8-7
Radical Functions
Example 1A Continued
x
(x, f(x))
3
(3, 0)
4
7
12
(4, 1)
(7, 2)
(12, 3)
●
●
●
●
The domain is {x|x ≥3}, and the range is {y|y ≥0}.
Holt Algebra 2
8-7
Radical Functions
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.
Holt Algebra 2
8-7
Radical Functions
Example 1B Continued
x
(x, f(x))
–6
(–6, –4)
1
(1,–2)
2
(2, 0)
3
(3, 2)
10
(10, 4)
●
●
●
●
The domain is the set of all real numbers. The range
is also the set of all real numbers
Holt Algebra 2
●
8-7
Radical Functions
Example 1B Continued
Check Graph the function on a graphing calculator.
Holt Algebra 2
8-7
Radical Functions
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.
Holt Algebra 2
8-7
Radical Functions
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.
Holt Algebra 2
8-7
Radical Functions
Check It Out! Example 1a Continued
Check Graph the function on a graphing calculator.
Holt Algebra 2
8-7
Radical Functions
Check It Out! Example 1b
Graph each function, and identify its domain
and range.
x
(x, f(x))
–1
(–1, 0)
3
8
15
(3, 2)
(8, 3)
(15, 4)
•
•
•
•
The domain is {x|x ≥ –1}, and the range is {y|y ≥0}.
Holt Algebra 2
8-7
Radical Functions
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
8-7
Radical Functions
Holt Algebra 2
8-7
Radical Functions
Example 2: Transforming Square-Root Functions
Using the graph of f(x) = x as a guide, describe
the transformation and graph the function.
g(x) =
x+5
Translate f 5 units up.
•
•
Holt Algebra 2
8-7
Radical Functions
Check It Out! Example 2a
Using the graph of f(x)= x as a guide, describe
the transformation and graph the function.
g(x) = x + 1
Translate f 1 unit up.
Holt Algebra 2
•
•
8-7
Radical Functions
Check It Out! Example 2b
Using the graph of f(x) = x as a guide, describe
the transformation and graph the function.
g is f vertically compressed
by a factor of
Holt Algebra 2
1
2
.
8-7
Radical Functions
Transformations of square-root functions are
summarized below.
Holt Algebra 2
8-7
Radical Functions
Example 3: Applying Multiple Transformations
Using the graph of f(x)= x as a guide, describe
the transformation and graph the function
.
Reflect f across the
x-axis, and translate it
4 units to the right.
Holt Algebra 2
•
•
8-7
Radical Functions
Check It Out! Example 3a
Using the graph of f(x)= x as a guide, describe
the transformation and graph the function.
g is f reflected across the
y-axis and translated 3
units up.
●
●
Holt Algebra 2
8-7
Radical Functions
Check It Out! Example 3b
Using the graph of f(x)= x as a guide, describe
the transformation and graph the function.
g(x) = –3 x – 1
g is f vertically stretched
by a factor of 3, reflected
across the x-axis, and
translated 1 unit down.
Holt Algebra 2
●
●
8-7
Radical Functions
Example 4: Writing Transformed Square-Root
Functions
Use the description to write the square-root
function g. The parent function f(x)= x is
reflected across the x-axis, compressed vertically
by a factor of 1 , and translated down 5 units.
5
Step 1 Identify how each transformation affects the
function.
Reflection across the x-axis: a is negative
Vertical compression by a factor of
Translation 5 units down: k = –5
Holt Algebra 2
1
5
a=– 1
5
8-7
Radical Functions
Example 4 Continued
Step 2 Write the transformed function.
 1
g(x) =  -  x + (- 5) Substitute – 1 for a and –5 for k.
5
 5
Simplify.
Holt Algebra 2
8-7
Radical Functions
Example 4 Continued
Check
Holt Algebra 2
Graph both functions on a graphing
calculator. The g indicates the given
transformations of f.
8-7
Radical Functions
Check It Out! Example 4
Use the description to write the square-root
function g.
The parent function f(x)= x 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
Vertical compression by a factor of 2
Translation 5 units down: k = 1
Holt Algebra 2
a = –2
8-7
Radical Functions
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.
Holt Algebra 2
8-7
Radical Functions
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
8-7
Radical Functions
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.
Holt Algebra 2
8-7
Radical Functions
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) = 64x
approximates an object’s downward velocity in
feet per second as the object hits the ground
after bouncing x ft in height.
The downward velocity function for the Moon is
a horizontal stretch of f by a factor of about 25
4
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.
Holt Algebra 2
.
8-7
Radical Functions
Check It Out! Example 5 Continued
Step 1 To compress f horizontally by a factor of
multiply f by 4 .
25 ,
4
25
h(x)
=
4 
f  x =
25 
4
256
 64x =
25
25
Step 2 Find the value of g for a bounce of 50ft.
256
h (x) =
 50  23
25
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.
Holt Algebra 2
8-7
Radical Functions
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
8-7
Radical Functions
Example 6: Graphing Radical Inequalities
Graph the inequality
.
Step 1 Use the related equation y =2 x -3 to
make a table of values.
x
y
Holt Algebra 2
0
–3
1
–1
4
1
9
3
8-7
Radical Functions
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
8-7
Radical Functions
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
8-7
Radical Functions
Check It Out! Example 6a
Graph the inequality.
Step 1 Use the related equation y =
make a table of values.
x
y
Holt Algebra 2
–4
0
–3
1
0
2
5
3
x+4 to
8-7
Radical Functions
Check It Out! Example 6a 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 less than –4,
do not shade left of –4.
Holt Algebra 2
8-7
Radical Functions
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 > (0) + 4
4>2
Holt Algebra 2

8-7
Radical Functions
Check It Out! Example 6b
Graph the inequality.
Step 1 Use the related equation y =
make a table of values.
x
y
Holt Algebra 2
–4
0
–3
1
0
2
5
3
3
x - 3 to
8-7
Radical Functions
Check It Out! Example 6b 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.
Holt Algebra 2
8-7
Radical Functions
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
Holt Algebra 2

8-7
Radical Functions
Lesson Quiz: Part I
1. Graph the function
range and domain.
and identify its
D:{x|x≥ –4}; R:{y|y≥ 0}
•
Holt Algebra 2
8-7
Radical Functions
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.
•
•
Holt Algebra 2
8-7
Radical Functions
Lesson Quiz: Part III
3. Graph the inequality
Holt Algebra 2
.