Question

How do you sketch the graph of `y=x^2+8` and describe the transformation?

Answer 1

See below

Explanation:

The first thing you need to find is the axis of symmetry by using

-b/2a

after you find that, use it to find the vertex of `x^2+8`

In this case x should equal `0`

so `y=0^2+8`

giving just 8

the y-intercept is 8 as `c=8`

the graph should look like this

graph{x^2+8 [-11.5, 8.5, 0.84, 10.84]}

it is a parabola.

Answer 2

Detailed explanation given

The transformation is that of `x^2` but raised by 8 on the y-axis.

Vertex `->(x,y)=(0,+8)`

Axis of symmetry `->` y-axis

y-intercept = 8

x-intercept -> none

Explanation:

`color(blue)("General shape")`

The first thing you need to determine is the general shape.
This is a quadratic equation so it has a horse shoe type shape.

The `x^2` term is positive so the general form is that of `uu`
,..................................................................................................

`color(blue)("Axis of symmetry")`

Consider the standardised equation form of `y=ax^2+bx+c`

Write this as: `y=a(x^2+b/a x)+c` then we have we can use the rather nifty trick of:

`x_("vertex")=(-1)xx b/(2a)`

or the same thing in a different form:

`x_("vertex")=(-1/2)xx b/(a)`

Note that in the case of this question

`a=1 ->ax^2->1xx x^2 = x^2`

So `b/a->b/1=b`

Now compare `ax^2+bx+c color(white)(b)" to " color(white)(b)x^2+8`

`x^2+8` does not have a `bx` term. To make this happen `b=0`

so `x_("vertex") = (-1/2)xxb/a -> (-1/2)xx0/1 = 0`

Thus the vertex coincides with the y-axis as does the axis of symmetry .

'.........................................................................

`color(blue)("Determine the y intercept and x intercepts")`

As the axis of symmetry is the y-axis set `x=0` giving:

`y=x^2+8" "->" "y=0+8`

`y_("intercept")->"Vertex "->(x,y)=(0,+8)`

As the vertex is at `(0,8)` and the graph is of form `uu` the x-intercepts do not exist.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Drawing the "ul("sketch"))`

You do not need to work out a whole pile of points. Just make sure the general shape is `uu` central about the y-axis and has the vertex passing through the labeled point `(x,y)=(0,8)`

You do not need to draw to scale so it should only take about 5 to 15 seconds to complete ( some may take longer ). Do not forget to label your points and put a title on it.