Find the transformed version of y=1x when a horizontal translation of 3 units left, a vertical dilation of 2, and a vertical translation of 4 units down are applied.
A standard function form for a horizontal translation is y=f(x+h) where +h is a shift to the left movement along the x-axis.
A standard function form for a vertical translation is y=f(x)+c where +c is an upward movement along the y-axis.
A standard function form for a vertical dilation is kf(x) where k is the vertical dilation factor.
To transform y=1x with a horizontal translation of 3 units left, a vertical dilation of 2, and a vertical translation of 4 units down, first apply a horizontal translation moving to the left (negative) which means that +h is a positive. Do this by using y=f(x+h) and h=3.
y=
1x
Apply the horizontal translation of h=3. Remember y=f(x+h).
=
1x+3
Simplify
=
1x+3
Now apply the vertical dilation of k=2. Use kf(x).
y=
1x+3
Apply the vertical dilation of k=2. Use kf(x).
=
2×(1x+3)
Simplify
=
2x+3
Finally apply the vertical translation of 4 units down. Use y=f(x)+c where c is an upward movement along the y-axis. In this case c=-4 because we are going downward.
y=
2x+3
Apply the vertical translation of 4 units down. This means c=-4.
=
2x+3-4
Simplify
=
2x+3-4
y=2x+3-4
Question 2 of 6
2. Question
Find the equation when y=x3 is transformed with a horizontal dilation factor of 2 then a vertical translation of 3 units up.
The application of the horizontal dilation (factor) on a x variable is xfactor.
A standard function form for a vertical translation is y=f(x)+c where c is an upward movement along the y-axis.
To determine the transformed equation, start by applying the horizontal dilation factor of 2 first. Do this by using xfactor and factor=2.
y=
x3
Apply the horizontal dilation factor of 2. Remember xfactor.
=
(x2)3
Simplify
=
(x2)3
Apply the vertical translation of 3 units up. Use y=f(x)+c where c is an upward movement along the y-axis. This means c=3.
=
(x2)3+3
Simplify
=
(x2)3+3
Simplify.
y=
x38+3
Simplify.
y=x38+3
Question 3 of 6
3. Question
Find the equation when y=√x is reflected about the y-axis then transformed with a vertical dilation factor of 2, then a horizontal dilation factor of 13.
Reflections about the y-axis have the property where we replace x→-x.
A standard function form for a vertical dilation is kf(x) where k is the vertical dilation.
The application of the horizontal dilation (factor) on a x variable is xfactor.
To determine the transformed equation, start by applying the reflection around the y-axis by replacing x→-x.
y=
√-x
Replace x by -x.
=
2√-x
Apply the vertical dilation of k=2. Use kf(x).
=
2√-x
Apply the horizontal dilation factor of 13. Remember xfactor
=
2√-x13
Simplify
y=
2√-3x
y=2√-3x
Question 4 of 6
4. Question
Find the transformed equation when the original function y=logx is translated horizontally by 2 units to the right, then transformed with the horizontal dilation factor of 3.
A standard function form for a horizontal translation is y=f(x+h) where +h is a shift to the left movement along the x-axis.
The application of the horizontal dilation (factor) on a x variable is xfactor.
To determine the transformed equation, we start by applying the horizontal translation of 2 units right. Use y=f(x+h) where h is a shift to the left along the x-axis. This means h=-2.
y=
logx
Apply the horizontal translation of 2 units right. Use y=f(x-2).
=
log(x-2)
Simplify
=
log(x-2)
Apply the horizontal dilation factor of 3. Remember xfactor.
=
log((x3)-2)
Simplify
y=
log(13x–2)
y=log(13x–2)
Question 5 of 6
5. Question
Find the transformed version of y=3x when a vertical translation of 2 units up, a horizontal translation of 2 units to the right, and a vertical dilation factor of -1 are applied.
A standard function form for a horizontal translation is y=f(x+h) where +h is a shift to the left movement along the x-axis.
A standard function form for a vertical translation is y=f(x)+c where +c is an upward movement along the y-axis.
A standard function form for a vertical dilation is kf(x) where k is the vertical dilation.
To transform y=3x with a vertical translation of 2 units up, horizontal translation of 2 units right, and a vertical dilation of -1, first apply a vertical translation of 2 units up. This means c=2.
y=
3x
Apply the vertical translation of 2 units up. Use y=f(x)+c where +c is an upward movement along the y-axis. This means c=2.
=
3x+2
Simplify
=
3x+2
Next, remember +h (left) is a shift to the left along the x-axis. So moving to the right, h is a negative. Do this by using y=f(x+h) and h=-2.
y=
3x+2
Apply the horizontal translation of h=-2. Remember y=f(x+h).
=
3x-2+2
Simplify
=
3x-2+2
Now apply the vertical dilation of k=-1. Use kf(x).
y=
3x-2+2
Apply the vertical dilation of k=-1. Use kf(x).
=
-1[3x-2+2]
Simplify
=
-3x-2–2
y=-3x-2–2
Question 6 of 6
6. Question
Find the equation when y=√x is transformed with a vertical translation of 3 units up, a horizontal dilation factor of 2, a horizontal translation of 1 unit up, and a reflection about the x-axis.
A standard function form for a horizontal translation is y=f(x+h) where +h is a shift to the left movement along the x-axis.
A standard function form for a vertical translation is y=f(x)+c where +c is an upward movement along the y-axis.
The application of the horizontal dilationfactor on a x variable is xfactor.
Reflections about the x-axis have the property where we replace y→-y.
To determine the transformed equation, start by applying the vertical translation of 3 units up. Use y=f(x)+c where +c is an upward movement along the y-axis. This means c=3 .
y=
√x+3
Apply the vertical translation c=3.
=
√x2+3
Apply the horizontal dilation factor of 2. Remember xfactor.