Question

Help please 1. Describe the transformations necessary to transform the graph of

Answer 1

Here are all the steps, and what they do:

STEP 1: HORIZONTAL TRANSLATION

We transform

$x^2\mapsto (x-1)^2$

The general transformation is

$f(x)\mapsto f(x+k)$

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

$(x-1)^2\mapsto 3(x-1)^2$

The general transformation is

$f(x)\mapsto kf(x)$

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

$3(x-1)^2\mapsto 3(x-1)^2+4$

The general transformation is

$f(x)\mapsto f(x)+k$

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 $f(x)=x^2$ 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 $g(x)=3(x-1)^2+4$