transformations

Question

	Write a function that transforms  in the following way:	vertical stretch by a factor of 2 and shift 7 units left.	a.		c.
b.		d.

Answer 1

f(x)=2x^4+4

a vertical stretch by a factor of 2 means 2(2x^4)+4

therefore we get f(x)=4x^4+4

a shift 7 units left means x+7 (a shift 7 units right means x-7; the sign is the opposite of what you would expect-left means "+" and right means"-")

f(x)=4(x+7)^4+4 which is A

for example: f(x)=x^2 has its vertex at (0,0) but f(x)=(x+2)^2 has its vertex at (-2,0), which is a shift 2 units to the left but the function says "x+2"

Answer 2

Well, you can easily tell that the vertical stretch will result in 2 turning into 4. The question, then, is how to get the shift to the left.

The easiest way to explain it is that if it goes with the x, it's a shift up and down. If not, it's left and right, but OPPOSITE the sign.

a. g(x) = 4(x + 7)⁴ + 4

The 7 is with the x. Instead of x⁴, we have (x + 7)⁴, so this will be a shift 7 units up.

b. g(x) = 4(x + 7)⁴ + 8

Well, this also shifts it up 7, but since 8 is 4 more than the original 4, it also shifts it LEFT 4 units.

c. g(x) = 4x⁴ + 11

Ok, we're back to just x⁴, so no vertical movement. Eleven is 7 more than 4, so we have a shift 7 to the left. This looks promising.

d. g(x) = 4(x - 7)⁴ + 4

Well, now we have (x - 7)⁴, so there's a shift down 7 units. Four is still 4, so it doesn't shift left or right.

Our answer is c.

I'm going to throw a couple more examples at you for effect.

g(x) = 4(x + 5)⁴ - 4

This (along with the vertical stretch of 2) would shift up 5 and right 8, since -4 is 8 less than 4.

g(x) = 6(x - 6)⁴ + 5

This would have a verticsl stretch of 3, since 6 is 3 times 2, move down 6 and left 1, since 5 is one more than 4.

Hope this helps!