# Graph the following function using transformation. Be sure to graph all of the stages on one graph. State the domain and range. y=-3x^2-12x-8

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`y =ax^2+bx+c` is an equation of a parabola in general form. To graph by transformation, express the equation in standard form by completing the square.

`y = -3x^2 - 12x - 8`

`y = -3(x^2 + 4x) - 8`

`y = -3(x^2 + 4x + 4 )-8+12`

`y = -3(x+2)(x+2) + 4`

Hence, the standard equation is:

`y = -3(x+2)^2 + 4`

First, graph the basic equation of parabola which is:

>> `y=x^2`

Its vertex is (0,0) and opens upward as shown in the graph (Color - Green).From the graph, let's take three points (-2,4), (0,0) and (2,4) as our reference.

Then, graph:

>> `y = 3x^2`

Multiply all the values of y. So, our three points become (-2,12), (0,0) and (2,12). As shown below, the graph (Red) is stretched vertically. This transformation is referred as *vertical stretch*.

Next, graph:

>> `y =` `-3x^2`

Multiply the values of y by -1. The reference points, then, become (-2,-12) (0,0) and (2,-12). So, the graph changes direction. It opens downward as shown below (orange). The flipping of graph is called *reflection*.

Next, graph:

>> `y = -3(x+2)^2`

Subtract the values of x by 2. The reference points become (-4,-12), (-2,0) and (0,-12). The graph (Purple) is obtained by shifting two units to the left. This type of transformation is called *horizontal shifting*.

Last, graph:

>> `y = -3(x+2)^2 + 4`

Add values of y by 4. The reference points become (-4,-8) , (-2,4) and (0,-8). The new graph (Blue) is obtained by moving the points 2 units up and this is called *vertical shifting*.

**Hence, the graph of `y=-3x^2 - 12x - 8` opens downward and its vertex is (-2,4). Moreover, the domain is all real numbers `(-oo,+oo)` . And since the graph opens downward, the range is `(-oo,4]` .**

Graph: `y=-3x^2-12x-8`

This is a quadratic function. The base function is `y=x^2` . The possible transformations are given by:

`y=A(x-h)^2+k` where A determines whether the graph opens up or down (reflection over x-axis) and how narrow/wide (vertical stretch/compression), h determines a slide (horizontal translation), and k a slide (vertical translation).

Rewriting in vertex form we get:

`y=-3x^2-12x-8`

`y=-3(x^2+4x+8/3)` **To use completing the square we need the leading coefficient to be one**

`y=-3(x^2+4x+4-4+8/3)`

`y=-3(x^2+4x+4-4/3)`

`y=-3((x+2)^2-4/3)`

`y=-3(x+2)^2+4`

**Thus we take the base function `y=x^2` , reflect it over the x-axis, stretch it by a factor of 3, then move the vertex 2 units left and 4 units up.**

In the order given above, start with green, blue,yellow,purple,red.

The domain of the function is all real numbers (`x in RR` ) while the range is `y<=4` .

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