Respuesta :
Given:
The parent function is
[tex]f(x)=x^2[/tex]
Consider the transformed function is g(x) instead of f(x) because both functions are different.
[tex]g(x)=-5(x-4)^2-8[/tex]
Step-by-step explanation:
We have,
[tex]f(x)=x^2[/tex]
[tex]g(x)=-5(x-4)^2-8[/tex]
It can be written as
[tex]g(x)=-5f(x-4)-8[/tex] ...(i)
The translation is defined as
[tex]g(x)=kf(x+a)+b[/tex] .... (ii)
Where, k is stretch factor, a is horizontal shift and b is vertical shift.
If 0<|k|<1, then the graph compressed vertically by factor k and if |k|>1, then the graph stretch vertically by factor k.
If k is negative, then f(x) is reflected across the x-axis to get g(x).
If a>0, then the graph shifts a units left and if a<0, then the graph shifts a units right.
If b>0, then the graph shifts b units up and if b<0, then the graph shifts b units down.
On comparing (i) and (ii), we get
[tex]a=-4<0[/tex], f(x) shifts 4 units right.
[tex]b=-8<0[/tex], f(x) shifts 8 units down.
[tex]k=-5[/tex], it is negative so f(x) reflected across the x-axis.
[tex]|k|=|-5|=5>1[/tex], so f(x) stretched vertically by factor 5.
Therefore, the function f(x) reflected across the x-axis, stretched vertically by factor 5 and shifted 4 units right 8 units down to get g(x).
