SQUARE ROOT
Functions
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8-7: Square Root Graphs
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RADICAL FUNCTIONS
A radical function is a function whose rule is a
radical expression, which include the square-root
function, x
f  x  x
x
x
y
0
0
0
1
1
1
4
4
2
9
9
3
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Square Root Graphs
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Vertex form
Vertical stretch or compression
a<0 --- reflection across the x-axis
h----horizontal translation
f ( x)  a b  x  h   k
k---veritcal translation
b<0 --- reflection across the y-axis
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Square Root Graphs
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EXAMPLE 1
Graph each function and identify its domain
and range of f  x   2 x  1
x
f  x  2 x 1
f(x)
g(x)
1
2 1  1  2 0
0
2 0 1  2 1
3
2 3 1  2 2
8
2 8 1  2 3
Domain: x  1  1,  
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0
2
4
6
Range: y  0
 0,  
4
EXAMPLE 2
Using the graph, f(x) = x , as a guide, describe the
transformation, identify the domain and range, and
graph the function, g  x   x  4
Domain:
x4
 x x  4
g(x)
y0
 4,  
 y y  0
g(x) translates 4 units right
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Range:
8-7: Square Root Graphs
 0,  
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EXAMPLE 3
Using the parent function as a guide, describe the
transformation, identify the domain and range, and
graph the function, g  x   x  4
Domain:
Range:
x0
y  4
 0,  
g(x)
 4,  
g(x) translates 4 units down
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EXAMPLE 4
Using the parent function as a guide, describe the
transformation, identify the domain and range, and
graph the function, g  x   x  5  5
Range:
Domain:
y  5
x  5
g(x)
 5,  
 5,  
g(x) translates 5 units left and 5 units down
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EXAMPLE 5
Graph each function and identify its domain
and range of f  x   2 x  1  3


Domain: x  1 x x  1
1,  
Range: y  3
3, 
 y y  3
g(x) translates 1 unit right, 3 units up and stretches by
a factor of 2
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Example 6: Applying Multiple Transformations
Use the graph shown as a guide, write the
equation and describe the transformation
g ( x)   x  4
Reflect f across the
x-axis, and translate it
4 units to the right.
•
•
Example 7
Using the graph shown, write the equation and
describe the transformation
g is f reflected across the y-axis and translated 3
units up.
g ( x)   x  3
•
•
Example 8
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.
●
●
Example 9: 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.
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Example 10
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.
Example 11
Graph the function
and domain.
and identify its range
Range:
Domain:
•
D:{x|x≥ –4}; R:{y|y≥ 0}
EXAMPLE 12
Graph each function and identify its domain
and range of f  x   2 x  1
Domain:
x0
 x x  0  0,  
Range:
y 1
 y y  1  ,1
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