Example 1: Graphing a Radical Function
1.
Fill in the
table of values for the function f(x) = root(x).
2.
When
constructing a table of values for a square root function, it's a good
idea to
select x-values that are perfect squares to avoid decimals.
3.
The square root
of -1 is undefined.
4.
The square root
of zero is 0.
5.
The square root
of 1 is 1.
6.
The square root
of 4 is 2.
7.
And the square
root of 9 is 3.
8.
Draw the graph
of the function f(x) = root(x) and state the domain and range.
9.
The domain is
{x|x ≥ 0,xER}
10. The range is {y|y ≥ 0,yER}
11. State that the Radicand is greater than or equal to 0
12. The radicand is x greater than or equal to 0
13. State the endpoint and the x and y intercepts
14. The endpoint is @ (0,0)
15. The x-intercept @ x=0 and the y-intercept @ y=0
NOTE: when graphing radical functions using multiple
transformations graph in steps, starting with the basic function y= root(x)
then horizontal stretch, horizontal reflection in the y-axis, vertical stretch,
vertical reflection in the x-axis, and lastly translations.
Example 2: Reflections of Radical Functions
1.
Graph the
function of y = root(x) as a reference. In Part A, draw the graph of f(x) =
-root(x).
2.
A negative
coefficient tells you to reflect the graph about the x-axis.
3.
Reflect the
original graph about the x-axis to get the graph of f(x) = -root(x).
4.
In Part B, draw
the graph of f(x) = root(-x).
5.
A negative
coefficient within the radical tells you to reflect the graph about the y-axis.
6.
Reflect the
original graph about the y-axis to get the graph of f(x) = root(-x).
Example 3: Stretches of Radical Functions
1.
Graph the
function of y = root(x) as a reference. In Part A, draw the graph of f(x) =
2root(x).
2.
The coefficient
of the radical is the vertical stretch of the graph.
3.
Double the
height of the graph to draw f(x) = 2root(x).
4.
In Part B, draw
the graph of f(x) = 1/2root(x).
5.
The coefficient
of the radical is the vertical stretch of the graph.
6.
Halve the
height of the graph to draw f(x) = 1/2root(x).
7.
In Part C, draw
the graph of f(x) = root(2x).
8.
The reciprocal
of the internal coefficient is the horizontal stretch of the graph.
9.
The reciprocal
of 2 is 1/2, so the width of the graph is halved.
10.
In Part D, draw
the graph of f(x) = root(1/2x).
11.
The reciprocal
of the internal coefficient is the horizontal stretch of the graph.
12.
The reciprocal
of 1/2 is 2. The width of the graph is doubled.
Example 4: Translations of Radical Functions
1.
Graph the graph
of y = root(x) as a reference. In Part A, draw the graph of f(x) = root(x) - 5.
2.
The number
added or subtracted from the radical is the vertical translation of the graph.
3.
Move the
original graph down 5 units to get the graph of f(x) = root(x) – 5.
4.
In Part B, draw
the graph of f(x) = root(x) + 2.
5.
The number
added or subtracted from the radical is the vertical translation of the graph.
6.
Move the
original graph up 2 units to get the graph of f(x) = root(x) + 2.
7.
In Part C, draw
the graph of f(x) = root(x – 1).
8.
The OPPOSITE of
this number is the horizontal translation of the graph.
9.
The opposite of
-1 is +1. Move the original graph 1 unit right.
10.
In Part D, draw
the graph of f(x) = root(x + 7).
11.
The OPPOSITE of
this number is the horizontal translation of the graph.
12.
The opposite of
+7 is -7. Move the original graph 7 units left.
Example 5: Multiple Transformations of Radical Functions
1.
Graph the
function of y = root(x) as a reference. In Part A, draw the graph of f(x) =
root(x - 3) + 2.
2.
There is a
horizontal translation of three units right.
3.
There is also a
vertical translation of two units up.
4.
Move the graph
3 units right and 2 units up to get the graph of f(x) = root(x - 3) + 2.
5.
In Part B, draw
the graph of f(x) = 2root(x + 4).
6.
There is a
vertical stretch by a scale factor of 2.
7.
There is also a
horizontal translation of 4 units left.
8.
Double the
height of the graph and move it 4 units left. Note that stretches and
reflections must be performed before translations.
9.
In Part C, draw
the graph of f(x) = -root(x) - 3.
10.
There is a
reflection about the x-axis.
11.
There is a
vertical translation 3 units down.
12.
Reflect the
graph across the x-axis and move it 3 units down. Note that stretches and
reflections must be performed before translations.
13.
In Part D, draw
the graph of f(x) = root(-2x - 4).
14.
Before you
graph this radical function, you need to make sure the binomial within the
radical sign is fully factored.
15.
Factor out -2
to get f(x) = root(-2(x + 2)).
16.
The graph is
horizontally stretched by a factor of 1/2, reflected about the y-axis, and
horizontally translated 2 units left.
NOTE: to find the x-intercept plug in 0 for y into the original
equation and solve for x.
to
find the y-intercept plug in 0 for x into the original equation and solve for
y.
DETERMINING RADICAL
FUNCTIONS
When determining a radical function from a graph the very
first step is to draw the graph of y=root(x). Then you can compare the basic
graph with the transformed graph.
Determining the a value:
examine the graph and detect two x values that are the same for both of the
functions (basic function and transformed function). Determine the y values
with the corresponding x values. Making a table of values with the basic
function on one side and transformed function on the other side in order to compare
your values is recommended. Determine the difference between the two y-values
from the two different functions. The difference is the a value
Determining the b value:
examine the graph and detect two y values that are the same for both of the
functions (basic function and transformed function). Determine the x values
with the corresponding y values. Determine the difference between the two x-values from the two different functions. The difference is the b value.
= - 3
2.215
