Question 120056
This question comes from a quiz that my teacher gives us. I've worked on it for two weeks and can not seem to get the answer right. Please help! 
Given y = -2(x - 1)^2 + 3,describe the transformations on y=x^2 that would produce y = -2(x - 1)^2 + 3. Write the equation in standard form.
<pre><font size = 5 face = "book antiqua"><b>
You must start with y = x² and end up with y = -2(x - 1)² + 3

Start with this red equation: 

<font color = "darkred">y = x²</font>

which has this red curve as its graph:

{{{graph(300,300,-10,10,-10,10,x^2)}}}

Then you multiply the right side of the red
equation by 2 to get the green equation:

<font color = "green">y = 2x²</font>

That stretches the red curve vertically by a 
factor of 2, giving the green curve below.

{{{graph(300,300,-10,10,-10,10,x^2,2x^2)}}}

Then you multiply the right side of the green
equation by -1 to get

<font color = "blue">y = -2x²</font>

That reflects the green curve across the x-axis, 
giving the blue curve below:

{{{graph(300,300,-10,10,-10,10,0,2x^2,-2x^2)}}}

Then you replace x by (x - 1) in the right side of
the blue equation to get

<font color = "indigo">y = -2(x - 1)²</font>

That shifts the blue graph 1 unit right, giving the
purple curve below:

{{{graph(300,300,-10,10,-10,10,0,0,-2x^2,-2(x-1)^2)}}}

Finally you add 3 to the right side of the purple equation
to get

<font color = "aqua">y = -2(x - 1)² + 3</font> 

That shifts the purple graph 3 units upward, giving the
light blue graph below:

{{{graph(300,300,-10,10,-10,10,0,0,0,-2(x-1)^2,0,-2(x-1)^2+3)}}}

That light blue graph is hard to see, so here it is in red:

{{{graph(300,300,-10,10,-10,10,-2(x-1)^2+3)}}} 

But as you see the transformations in order were:

1. Stretch vertically by a factor of 2.  (Multiply right side by 2)
2. Reflect across the x-axis.  (Multiply right side by -1)
3. Shift right 1 unit. [Replace x by (x-1) ]
4. Shift up 3 units. {Add 3 to the right side.]

Edwin</pre>