Question

Given the function f(x) = 2(x − 1)^2 + 1, indicate the shifts that will affect the location of the vertex and explain what effect they will have. Use complete sentences.

f(x+3)
f(x) + 3
f(3x)
3•f(x)

Answer 1

@phi

Answer 2

what is the vertex of
f(x) = 2(x − 1)^2 + 1,
?

Answer 3

the axis of symmetry is x = 4 I believe because the x of the vertex is 4

Answer 4

oh wait wrong question lol, working on a couple at once

Answer 5

your equation is in "vertex form"

y = a(x-h)^2 + k
(h,k) is the vertex

Answer 6

alright the vertex is 1,1 I think

Answer 7

yes, if you replace x with x+3 in
2(x − 1)^2 + 1,
and simplify
what do you get ?

Answer 8

2(3 - 1)^2 + 1
distribute...
6 - 2
3
3^3
9 + 1
10?

Answer 9

3^2, not

Answer 10

3^3

Answer 11

no, we keep the x
replace x with x+3

Answer 12

oh

Answer 13

2(x − 1)^2 + 1
2(x - 3)^2 + 1
distribute...
2x - 6 = -4x
-4x^2 = 16
+ 1
17

Answer 14

we will not get a number, just a different equation
\[ 2(x − 1)^2 + 1 \]
replace x with (x+3)
\[ 2(x+3 − 1)^2 + 1 \]
simplify
\[ 2(x +2 )^2 + 1 \\ 2(x - (-2) )^2 + 1
\]
what is the vertex of this parabola ?

Answer 15

-2, 1

Answer 16

so we moved from (1,1) to (-2,1)
it has the same shape because the 2 out front (the "a" parameter) did not change.
how would you describe the change ?

Answer 17

the graph changed by moving to the left 3 units

Answer 18

Yes. in general f(x PLUS N) shifts to the LEFT N steps.
(counter intuitive or backwards)

Answer 19

now do
f(x) + 3
where f(x) = 2(x − 1)^2 + 1

Answer 20

we would make (x) = 2(x − 1)2 + 1 into (x) = 2(x − 1)2 + 3 right?

Answer 21

almost. you do not change 1 into 3. you add 3 to what you started with.

Answer 22

right...
(x) = 2(x − 1)2 + 1 + 3?
so 2(x - 1)^2 + 4?

Answer 23

which would take the cordinates from 1,1 to 1,4?

Answer 24

yes, and how does the graph change ?

Answer 25

It moves upwards 3 units

Answer 26

yes, the rule is f(x)+N shifts up N steps. (this one is easier to remember)

Answer 27

For f(3x)
replace x with 3x
\[ f(x) = 2(x − 1)^2 + 1 \\
f(x) = 2(3x-1)^2 + 1 \\
f(x) = 2\cdot \sqrt{3}(x-\frac{1}{3})^2 + 1 \]
now two things have changed. the vertex shifted (how ?)
and the "a" got bigger. A bigger "a" (number out front) makes the parabola "zoom up" faster

Answer 28

uhh I'm a bit confused on this one..

Answer 29

the graph is now at 1/3, 1 cordinates??

Answer 30

Here it is

Answer 31

yes, the vertex is at ⅓,1

Answer 32

I see so now the spread is much thinner

Answer 33

it goes up faster, which makes it narrower.

if we changed a to ¼, the zoom up would be slower, and the parabola would be "fatter" or "flatter"

Answer 34

ok i gotcha

Answer 35

last one, 3•f(x)

Answer 36

start with
f(x) = 2(x − 1)^2 + 1
multiply both sides by 3

Answer 37

so...
3f(x) = 5(x-1)^2 + 1?

Answer 38

err not 5...

6

Answer 39

3f(x) = 6(x-1)^2 + 1?

Answer 40

you multiply *everything* by 3, not just your favorite terms.

Answer 41

ok..
3f(x) = 6(x - 3)^2 + 3

Answer 42

example, with numbers:
5 = 4+1
3*5 = 3*1 + 3*4
check: 3*5 =15 and 3*1 + 3*4= 3+12= 15
if we just did
3*5 = 3*4 + 1, it does not work.

Answer 43

3f(x) = 6(x - 3)^2 + 3

what happened to the vertex ?
and how does the shape change ?

Answer 44

the vertex moves from 1,1 to 3,3

Answer 45

so up two units and to the right 2 units

Answer 46

oops, wait.
f(x) = 2(x − 1)^2 + 1
3f(x) = 3*(2(x − 1)^2 + 1 )

try again

Answer 47

you should get
3*2*(x-1)^2 + 3

Answer 48

so then 6(x-1)^2 + 3?

Answer 49

much better

Answer 50

so the vertex does not move?

Answer 51

the original
f(x) = 2(x − 1)^2 + 1
has a vertex at (1,1)

Answer 52

oh wait 1,3

Answer 53

so the vertex goes up 2 units

Answer 54

and the shape ?

Answer 55

uhh skinnier?

Answer 56

yes

Answer 57

Thanks phi! I have a few more qestions that are shorter... Could you help me with those?

Answer 58

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