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Orlov [11]
3 years ago
15

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

unction 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
Mathematics
1 answer:
nikklg [1K]3 years ago
7 0

The quadratic function given by:


f(x)=a(x-h)^2+k, \ \ \ a\neq 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 <em>11 units to the left:</em>

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


  • Then translating<em> 5 units down:</em>

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 <em>must be</em> less than 1, to make the graph fatter. So, according to our options, there are two factors 1/2 and 2.


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


SECOND.

  • Translating <em>8 units to the right:</em>

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


  • Then translating<em> 1 unit down:</em>

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.


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

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