Respuesta :
Here are all the steps, and what they do:
STEP 1: HORIZONTAL TRANSLATION
We transform
[tex]x^2\mapsto (x-1)^2[/tex]
The general transformation is
[tex]f(x)\mapsto f(x+k)[/tex]
These transformations translate the graph horizontally, k units to the left if k>0, k units to the right if k<0.
In this case, k = -1, so we translate the original graph 1 unit to the right.
STEP 2: VERTICAL STRETCH
We transform
[tex](x-1)^2\mapsto 3(x-1)^2[/tex]
The general transformation is
[tex]f(x)\mapsto kf(x)[/tex]
These transformations stretch the graph vertically. The graph expands if |k|>1, while it shrinks if 0<|k|<1. If k is negative, we also reflect the graph with respect to the x axis.
In this case, k = 3, so we stretch the graph vertically by a factor 3.
STEP 3: VERTICAL TRANSLATION
We transform
[tex]3(x-1)^2\mapsto 3(x-1)^2+4[/tex]
The general transformation is
[tex]f(x)\mapsto f(x)+k[/tex]
These transformations translate the graph vertically, k units up if k>0, k units down if k<0.
In this case, k = 4, so we translate the graph 4 units up.
So, we start from the original graph of [tex]f(x)=x^2[/tex] and we:
- Translate it 1 unit to the right
- Stretch it vertically by a factor 3
- Translate it 4 units up
(the order is important!)
to get the graph of [tex]g(x)=3(x-1)^2+4[/tex]
