Question

Solve.
5. The graph of f(x)= x is reflected across the x-axis. The graph is then
translated 11 units up and 7 units to the left. Write the equation of the
transformed function.

Answer 1

Given:

The function is:

$f(x)=x$

The graph of this function reflected across the x-axis. The graph is then  translated 11 units up and 7 units to the left.

To find:

The equation of the transformed function.

Solution:

The translation is defined as

$g(x)=kf(x+a)+b$                .... (i)

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<0, then the graph is reflected across the x-axis.

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.

The graph of this function reflected across the x-axis. The graph is then  translated 11 units up and 7 units to the left.  So, $k=-1, b=11, a=7$. Putting these value in (i), we get

$g(x)=(-1)f(x+7)+11$

$g(x)=-(x+7)+11$                     $[\because f(x)=x]$

$g(x)=-x-7+11$

$g(x)=-x+4$

Therefore, the required function is $g(x)=-x+4$.