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