Question

The function f(x)= x^2 is similar to g(x)= -3(x-5)^2+4. Describe the transformations. Show Graphs.

Answer 1

The reflected across x-axis, stretched vertically by factor 3 and shifted 5 units right and 4 units up.

Explanation:

The parent function is

`f(x)=x^2`

The given function is

`f(x)=-3(x-5)^2+4` ... (1)

The transformations of a quadratic function is defined as

`f(x)=k(x+a)^2+b` .... (2)

Where, k is vertical stretch or compression factor, a is horizontal shift and b is vertical shift.

If |k|>1, then graph stretch vertically if 0<|k|<1, then graph compressed vertically. If k<0 or negative, then the graph reflected across 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.

From (1) and (2) it is clear that

`k=-3,a=-5,b=4`

k=-3<0, it means graph reflected across x-axis. |k|=3>1, so graph stretched vertically by factor 3.

a=-5<0, so the graph shifts 5 units right.

b=4>0, so the graph shifts 4 units up.

Therefore the reflected across x-axis, stretched vertically by factor 3 and shifted 5 units right and 4 units up.