Describe the transformation that map the function y = 2^{x} on each of the following functions.
Problem 1 :
y = 2^{x} - 2
Solution :
Comparing the given with the standard form of exponential function
y = a (b)^{x - h} + k
a = 1, b = 2, h = 0 and k = -2
The value of k tells us about the vertical translation. Here the value of k is -2 < 0, so we have to move the curve down 2 units.
Required transformation is,
shifting 2 units down
Problem 2 :
y = 2^{x+3}
Solution :
Comparing the given with the standard form of exponential function
y = a (b)^{x - h} + k
y = 2^{x - (-3)}
a = 1, b = 2, h = -3 and k = 0
The value of k tells us about the horizontal translation. Here the value of h is -3 < 0, so we have to move the curve left 3 units.
Required transformation is,
shifting 3 units left
Problem 3 :
y = 4^{x}
Solution :
The given function is not in terms of the parent function.
y = (2^{2})^{x}
The value of x is being multiplied by 2, so it should be horizontal stretch or compression.
Considering the value of a = 2 > 1, this should be horizontal compression.
Required transformation is,
horizontal compression by the factor of 2.
Problem 4 :
y = 3(2^{x - 1}) + 1
Solution :
Comparing the given with the standard form of exponential function
y = a (b)^{x - h} + k
a = 3, b = 2, h = 1 and k = 1
Transformations should be done in the following order.
1) compression or stretches
2) Reflection
3) translation
So, the required transformations are,
a = 3 >1, so vertical stretch with the factor of 3.
No reflection
Translation of 1 unit right and 1 unit up.
Describe the transformation that map into the function
y = 8^{x}
on to each function.
Problem 5 :
y = (1/2) 8^{x}
Solution :
Comparing the given with the standard form of exponential function
y = a (b)^{x - h} + k
a = 1/2
comparing with the parent function, the value of b is the same and no translation.
The value of a tells us the vertical stretch or compressions.
Considering the value of a = 1/2 ==> 0 < b < 1
So, the required transformation is,
vertical compression by the factor of 1/2.
Problem 6 :
y = -8^{x}
Solution :
y is changed as -y, so reflection across x-axis.
So, the required transformation is,
reflection across x-axis.
Problem 7 :
y = 8^{-2x}
Solution :
Scale factor is 2, horizontal compression with the factor of 2.
x is changed as -x, so reflection across y-axis.
So, the required transformation is,
horizontal compression with reflection across y-axis.
Using the parent graph of f(x) = 4^{x}, describe the transformation of each function.
Problem 8 :
f(x) = -2(4^{x + 3}) - 5
Solution :
f(x) = -2(4^{x - (-3)}) - 5
Here, a = -2, b = 4 (same), h = -3 and k = -5
Vertical stretch with the factor with the of 2 along with reflection across x-axis.
Horizontal translation of 3 units left and vertical translation of 5 units down.
So, the required transformations are,
Vertical stretch of factor 2 ==> reflection across x-axis => move left 3 units and down 5 units.
Problem 9 :
f(x) = (4^{2x + 6}) + 2
Solution :
f(x) = (4^{2x - (-6)}) + 2
Here, a = 1, b = 4 (same), h = -6 and k = 2
So, the required transformations are,
Horizontal compression of 2 units. No reflection. Move horizontally 6 units left and 2 unit up.
Problem 10 :
f(x) = (4^{-3x + 12}) + 1
Solution :
f(x) = (4^{-3x - (-12)}) + 1
Here, a = 1, b = 4 (same), h = -12 and k = 1
So, the required transformations are,
horizontal factor of 3, 3 > 1
Horizontal compression of 3 units. Reflection across y-axis. Move horizontally 12 units left and 1 unit up.
Problem 11 :
f(x) = (4^{(1/2)x - 2}) + 3
Solution :
f(x) = (4^{(1/2)x - 2}) + 3
Here, a = 1, b = 4 (same), h = 2 and k = 3
So, the required transformations are,
horizontal factor of 1/2, 0 < HF < 1
Horizontal stretch of 1/2 units. No reflection. Move horizontally 2 units right and 3 units up.
Problem 12 :
f(x) = (1/3)(4^{(1/3)x + 3}) - 4
Solution :
f(x) = (1/3)(4^{(1/3)x + 3}) - 4
Here, a = 1/3, b = 4 (same), h = -3 and k = -4
So, the required transformations are,
horizontal factor of 1/3, 0 < HF < 1
Horizontal stretch of 1/3 units. No reflection. Move horizontally 3 units left and 4 units down.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM