# Use a sequence of function transformations to take y = x^2 to y = (2x + 1)^2 - 2

#### IshaanM8

##### New member
Q: For each of the following, find a sequence of transformations that takes $$\displaystyle y\, =\, x^2$$ to $$\displaystyle y\, =\, (2x\, +\, 1)^2\, -\, 2$$

By what I understand about transformations, there's 4 main ones:
1. Translation
2. Dilation
3. Reflection
4. Rotation
Now, with the equation...

We move -2 from the right side to the left side

Making it y' + 2 = (2x' + 1)2

This makes it Y = Y' + 2 & X = 2x' + 1

Now, Solving for Y' and x'

Y' = Y - 2
X' = (X - 1)/2

Now, to put it in words - A transformation is a dilation of 1 from Y axis, then a translation of 1/2 unit in the negative direction of the x-axis and 2 units in the negative direction of the Y-axis.

Is this correct guys? My answers say it's a dilation of a factor of 1/2 from the y axis... I don't get why..

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#### Harry_the_cat

##### Full Member
Q: For each of the following, find a sequence of transformations that takes $$\displaystyle y\, =\, x^2$$ to $$\displaystyle y\, =\, (2x\, +\, 1)^2\, -\, 2$$

By what I understand about transformations, there's 4 main ones:
1. Translation
2. Dilation
3. Reflection
4. Rotation
Now, with the equation...

We move -2 from the right side to the left side

Making it y' + 2 = (2x' + 1)2

This makes it Y = Y' + 2 & X = 2x' + 1

Now, Solving for Y' and x'

Y' = Y - 2
X' = (X - 1)/2

Now, to put it in words - A transformation is a dilation of 1 from Y axis, then a translation of 1/2 unit in the negative direction of the x-axis and 2 units in the negative direction of the Y-axis.

Is this correct guys? My answers say it's a dilation of a factor of 1/2 from the y axis... I don't get why..
$$\displaystyle y=(2x+1)^2 -2 = (2(x+\frac{1}{2}))^2 -2 = 4(x+\frac{1}{2})^2 -2$$

This represents a dilation of factor $$\displaystyle \frac{1}{4}$$ from the y-axis, a translation to the left of $$\displaystyle \frac{1}{2}$$ unit and down 2 units.

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#### Dr.Peterson

##### Elite Member
Q: For each of the following, find a sequence of transformations that takes $$\displaystyle y\, =\, x^2$$ to $$\displaystyle y\, =\, (2x\, +\, 1)^2\, -\, 2$$

By what I understand about transformations, there's 4 main ones:
1. Translation
2. Dilation
3. Reflection
4. Rotation
Now, with the equation...

We move -2 from the right side to the left side

Making it y' + 2 = (2x' + 1)2

This makes it Y = Y' + 2 & X = 2x' + 1

Now, Solving for Y' and x'

Y' = Y - 2
X' = (X - 1)/2

Now, to put it in words - A transformation is a dilation of 1 from Y axis, then a translation of 1/2 unit in the negative direction of the x-axis and 2 units in the negative direction of the Y-axis.

Is this correct guys? My answers say it's a dilation of a factor of 1/2 from the y axis... I don't get why..
You seem to be interchanging "dilation" and "translation". A dilation is a multiplication, not an addition or subtraction; a "dilation of 1" would mean no change at all. And a translation is an addition or subtraction, not a multiplication by 1/2 or division by 2, as you have here.

Can you explain how you came to your conclusion? You may want to copy what you were taught as a rule or example, so we can see where you are getting your idea.

Of course, there are more than one valid answer, depending on the order of the transformations. The horizontal transformations can either be a horizontal translation by 1 unit (to the left) followed by a dilation by a factor of 1/2, OR a dilation by 1/2 followed by a translation by 1/2.

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