VOCABULARY
Radical
function
A function
containing a radical such as y =
PARENT
·
The parent function for the family
of square root functions is f(x) = . The domain is x _³ 0_, and the
range is y _³ 0_.
·
The parent
function for the family of cube root functions is g(x) = . The domain and range are _all
real numbers _.
Example 1
Graph a
square root function
Graph y = 2 ,
and state the domain and range. Compare the graph with the graph of
y = .
Solution
Make a table of values and sketch
the graph.
|
x |
0 |
l |
2 |
3 |
4 |
|
y |
_0_ |
_2_ |
_2.83_ |
_3.46_ |
_4_ |
The radicand
of a square root is always nonnegative. So, the domain is x_³_0. The
range is y _³ _ 0.
The graph of
y = 2 is a vertical _stretch_
of the parent graph of y = .
Example 2
Graph a
cube root function
Graph y = - , and state the domain and range.
Compare the
graph with the graph of y = .
Solution
Make a table of values and sketch
the graph.
|
x |
-2 |
-l |
0 |
|
y |
_0.63_ |
_0.5_ |
_0_ |
|
x |
1 |
2 |
|
y |
_-0.5_ |
_-0.63_ |
The domain
and range are _all real numbers_.
The graph of y = - is a vertical _shrink_
of the parent graph of y = by
a factor of followed by a
reflection in the x-axis.
Graph
the function. Then state the domain and range.
1.
y = 2
The domain
and range are all real numbers.
2.
y = -2
domain x ³ 0, range y £ 0
GRAPHS OF
RADICAL FUNCTIONS
To graph y
= a + k or y = + k, follow these steps:
Step 1 _Sketch _
the graph of y = a or y = a .
Step 2 Translate
the graph _h_ units horizontally and _k_ units
vertically.
Example 3
Graph a
translated square root function
Graph y
= 3 + 2. Then state the domain
and range.
Solution
1. Sketch the graph of y = 3 . Notice that it begins at the origin and
passes through the point (1, _3_).
2. Translate the graph. For y = 3 + 2, h = _1_ and k
= _2_. So, shift the graph right 1 unit _ and _up 2 units_.
The resulting graph starts at (_1_, _2_) and passes through (_2_,
_5_).
From the graph, you can see that the
domain of the function is _x ³ 1_ and the
range of the function is _y ³ 2_.
Example 4
Graph a
translated cube root function
Graph y
= -2 -2. Then state the domain and
range.
Solution
1. Sketch the graph of y = -2 . Notice that it
passes through the origin and the points (_-1_, _2_) and (_1_, _-2_).
2. Note that for y = -2 - 2, h = _-3_ and k
= _-2_. So, shift the graph _left 3 units_ and _down 2 units_.
The resulting graph passes through the points (_-4_,_0_), (_-3_, _-2_), and (_-2_, _-4_).
From the graph, you can see that the
domain and range of the function are both _all real numbers_.
Graph the
function. Then state the domain and range.
3.
y = - + 2
domain x ³ -3, range y
£ 2
4.
y = 3 + 2
The domain
and range are all real numbers.