The given square root function can be considered as
a - Vertical stretch / compression by the factor of a
b - Horizontal stretch / compression by the factor of b.
h - Horizontal move towards left or right
k - Vertical move towards up or down.
Note :
Sign of a and b will decide if there is any reflection or not.
Describe the transformations from the graph of f(x) = √x to the graph of h. Then graph h.
Problem 1 :
h(x) = 4√(x + 2) - 1
Solution :
h(x) = 4√(x + 2) - 1
Considering the function with y = a√b(x - h) + k
h(x) = 4√(x - (-2)) - 1
a = 4, h = -2 and k = -1
Problem 2 :
h(x) = (1/2)√(x - 6) + 3
Solution :
h(x) = (1/2)√(x - 6) + 3
Considering the function with y = a√b(x - h) + k
a = 1/2, h = 6 and k = 3
Problem 3 :
h(x) = (1/3)√(x + 3) - 3
Solution :
h(x) = (1/3)√(x + 3) - 3
h(x) = (1/3)√(x - (-3)) - 3
Considering the function with y = a√b(x - h) + k
a = 1/3, h = -3 and k = -3
Problem 4 :
h(x) = √(-x - 4)
Solution :
h(x) = √(-x - 4)
h(x) = √-(x + 4)
h(x) = √-(x - (-4))
Considering the function with y = a√b(x - h) + k
b is negative , h = -4 and k = 0
Problem 5 :
How is the graph of y = √x - 5 translated from the graph of y = √x?
A. shifted 5 units left B. shifted 5 units right
C. shifted 5 units up D. shifted 5 units down
Solution :
y = √x - 5
Considering the function with y = a√b(x - h) + k
k = -5
So, the graph should be moved 5 units down. Option D is correct.
Problem 6 :
The parent function f(x) = √x is compressed vertically by a factor of 1/10, translated 4 units down, and reflected in the x-axis.
Solution :
Parent function :
f(x) = √x
Vertical compress with the factor of 1/10.
f(x) = (1/10) √x
Reflection about x-axis.
Put y = -y
f(x) = -(1/10) √x
Translating down 4 units, so k = -4
f(x) = -(1/10) √x - 4
So, the required function after the transformation is
f(x) = -(1/10) √x - 4
Problem 7 :
The parent function f (x) = √x is compressed horizontally by a factor of 7.5 and translated 2 units up.
Solution :
Parent function :
f(x) = √x
Horizontal compression with the factor of 1/7.5 or 2/15
f(x) = √(2/15)x
Translation of 2 units up :
f(x) = √(2/15)x + 2
Problem 8 :
The parent function f(x) = √x is translated 1/2 unit left and stretched vertically by a factor of 3.
Solution :
Vertical stretch = 3 units
Translation of 1/2 units left
f(x) = 3√(x + 1/2)
Problem 9 :
The parent function f (x) = √x is stretched vertically by a factor of 10, translated 5 units down, and reflected in the y-axis.
Solution :
Vertical stretch = 10, a = 10
translating down = 5 units, k = -5
reflection across y-axis. Then x = -x
f(x) = 10√-x - 5
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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