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## Algebra 1 (Eureka Math/EngageNY)

### Unit 4: Lesson 13

Topic C: Lessons 18-19: Translating graphs of functions

# Shifting parabolas

The graph of y=(x-k)²+h is the resulting of shifting (or translating) the graph of y=x², k units to the right and h units up. For example, y=(x-3)²-4 is the result of shifting y=x² 3 units to the right and -4 units up, which is the same as 4 units down.

## Want to join the conversation?

• For instance y=(x±9)^-1
Why does the sign before 9 have a counter effect on the parabola?
• wait, do you mean y=(x±9)^2 - 1? There is no squared value in the original question, just ^-1. Anyways, assuming that you mean y=(x±9)^2 - 1, then I would be happy to tell you how I think of the counter effect, as you put it. I pretend that I am trying to balance something on a seesaw, where the 0 is the pivot point. Imagine that you had a friend who weighed 9 kilos more than you. Your friend is x+9, and you are x. If you and your friend want to balance, you must shift the seesaw in your direction, or the heavier friend will tip it over. Basically, +9 means that it is 9 points too heavy on the positive side, so if the positive side is too heavy, what do you have to do? You have to shift the whole system to the left, so it can still balance. I hope this helps! I also hope that people still know what a seesaw is, even though people don't seem to play outside anymore.
• y=(x-h)^2+k How do negative values of h represent leftward shifts? I cannot get this one, Sal in the video explained that when we shift h units to the right we substract h units from the function.
• This is going to be true for all functions, so lets start with a linear equation y = x + 3. the y intercept is 3 (set x=0) and the x intercept is -3 (set y = 0). If we keep it as a change in y, we have y = x + 3, so it is easy to see the y intercept. By "making it a change in x" instead, we show it as y = (x + 3) + 0. So if we put in a negative 3 for x, we get y = 0 which gives us the correct x intercept. So we had to have the opposite sign for a change in x.
Similarly for the quadratic function such as y = (x + 3)^2 + 5, we would have to set x = -3 in order to make what is inside the parentheses to be 0, we have to change the sign.
SO a change in y follows the sign, a change in x has to be the opposite sign.
• How would you do this? Shifting f(x) 1 unit right then 2 units down. The equation is f(x)=x^2-2x-1.

My book says this

to shift 1 unit right its

g(x)=(x-1)^2-2(x-1)-1
=x^2-4x+2

for shift 2 units down

h(x)=x^2-2x-1-2
=x^2-2x-3

I need some help understanding this please
• Translations are often confusing at first glance. You just kind of have to memorize it.
Here are the basics.
Horizontal Translations (moving along x axis)
If we have f(x),
f(x-1) is the function moving to the RIGHT by 1.
f(x+1) is the function moving to the LEFT by 1.
confusing, I know
Vertical Translation (moving along y axis)
f(x)
f(x)+1 is the function moving UP by 1.
f(x)-1 is the function moving DOWN by 1.
Now how do we use these?
Let's say we have
f(x)=3x+5
and we want to move it to the right by 4 units.
that would make it f(x-4).
Replace all the x's inside the function with x-4.
So you get 3(x-4)+5
This equals to
3x-12+5=3x-7.
So 3x-7 is what we get when we move the function to the right by 4 units. *graph it and check!*
Let's say we have the same equation, f(x)=3x+5
but we want to make it go down by 3!
Remember,
f(x)-3 means it goes down by 3
f(x)-3=(3x+5)-3
f(x)-3=3x+2
The new function is 3x+2, which is the function where we moved it down by 3!
• would it be right to write it down like this?: g(x)+4=(x-3)squared
• Function notation always has the function name by itself.
g(x) = (x-3)^2 - 4 would be the correct form.
• So just to be clear:

Shifting to the right by 3 is (x-3)
Shifting to the left by 3 is (x+3)

Shifting down by 3 is (x)-3
Shifting up by 3 is (x)+3
• Yep! And that works with any function. the trick is just internalizing what is inside and what is outside the function.
• Does anyone know the mentioned videos that explain shifting more in depth?
• It's the video right before this one, in the Quadratic functions & equations unit of Algebra 1. The title is "Intro to parabola transformations"
• If moving the vertex to the right makes it (x-3), why, when I move the vertex down four, doesn't the equation then equal (x-3)+4?
(1 vote)
• If you have y = 2(x-5)^2 + 2, the 5 is with the x, so if you want to do the same with the ys, you have to subtract 2 on both sides to get y - 2 = 2(x-5)^2, in this case the y would also have to change signs (similar to the point slope form of a linear equation y-y1=m(x-x1). We could do the same thing with this, y = m(x-x1)+y1 where x1 changes sign and y1 would stay the same, So when the 2 is on the same side as the x (right side of equation), you do not change the sign. Hope this makes sense.
• if you minus by a number when you shift to the right, do you add by a number when you shift to the left?
(1 vote)
• Yes that is correct. If you have something like (x-5)^2 + 3, that negative shifts to the right because you need to have x=5 for the inside of parentheses to be 0 (5-5)^2 and if you have (x + 4)^2 - 3, you need to have x=-4 to had to have it be 0 because (-4+4)^2=0.
• Does it matter if we write `y + 4 = (x-3)^2` instead of `y = (x-3)^2 - 4` ? Should be the same but is one preferred over the other? Last video he used the former format, where the `4` was being added to the `y`.