Question

If the parent function f(x) = x^2 is fatter translated 11 units to the left, then translated 5 units down, write the resulting function g(x) in vertex form.

a. f(x) = 1/2(x - 11)^2 - 5
b. f(x) = 1/2(x + 11)^2 - 5
c. f(x) = 2(x - 11)^2 - 5
d. f(x) = 2(x + 11)^2 - 5

If the parent function f(x) = x^2 is fatter translated 8 units to the right, then translated 1 unit down, write the resulting function g(x) in vertex form.
a. g(x) = 1/3(x - 8)^2 - 1
b. g(x) = 1/3(x + 8)^2 - 1
c. g(x) = 3(x - 8)^2 - 1
d. g(x) = 3(x + 8)^2 - 1

Answer 1

The quadratic function given by:

`f(x)=a(x-h)^2+k, \ \ \ a\\eq 0`

is in vertex form. The graph of
`f` is a parabola whose axis is the vertical line
`x=h` and whose vertex is the point
`(h, k)`. So:

To translate the graph of a function to the right, left, upward or downward we have:

`For \ a \ positive \ real \ number \ c. \ \mathbf{Vertical \ and \ horizontal \ shifts} \\ in \ the \ graph \ of \ y=f(x) \ are \ represented \ as \ follows:\\ \\ \bullet \ Vertical \ shift \ c \ units \ \mathbf{upward}: \\ g(x)=f(x)+c \\ \\ \bullet \ Vertical \ shift \ c \ units \ \mathbf{downward}: \\ g(x)=f(x)-c \\ \\ \bullet \ Horizontal \ shift \ c \ units \ to \ the \ \mathbf{right}: \\ g(x)=f(x-c) \\ \\ \bullet \ Horizontal \ shift \ c \ units \ to \ the \ \mathbf{left}: \\ g(x)=f(x+c)`

By knowing this things, we can solve our problem as follows:

FIRST.

• Translating 11 units to the left:

`g(x)=f(x+11) \\ \\ \therefore g(x)=(x+11)^2`

• Then translating 5 units down:

`g(x)=f(x)-c \\ \\ \therefore g(x)=(x+11)^2-5`

Since the new function is fatter, the factor we need to multiply the term
`(x+11)^2` must be less than 1, to make the graph fatter. So, according to our options, there are two factors 1/2 and 2.

Therefore, the right answer is b. f(x) = 1/2(x + 11)^2 - 5

SECOND.

• Translating 8 units to the right:

`g(x)=f(x-8) \\ \\ \therefore g(x)=(x-8)^2`

• Then translating 1 unit down:

`g(x)=f(x)-c \\ \\ \therefore g(x)=(x-8)^2-1`

As explained in the previous case, there are two factors 1/3 and 3, so we choose the first one.

Therefore, the right answer is a. g(x) = 1/3(x - 8)^2 - 1