1. Give an example of a quadtratic whose recirpocal has no vertical asymptotes

Ans. y = x^2 - 2x + 3

2. Compare the vertex (or turning point) of the quadratic and the corresponding feature of the reciprocal. Write down your observation.

What does the question mean by corresponding feature?

And what would be the observation be. I realised the x-coorindate remains the same.

your example would work.

As long as you pick a quadratic function which does not have any x-intercepts, that is , it does not cross the x-axis, it would work

That would mean that the corresponding equation f(x) = 0 would not have a solution in the real numbers

your resulting curve of the reciprocal would be a shallow bell curve whose maximum point has the same x coordinate as the vertex of the original function.
You could use something like
http://fooplot.com/
to see your two graphs

Ivé graphed the function, and would it be correct to say that if the vertex was (1,2), the corresponding feature's coordinates will be (1,1/2), the y coordinates will reciprocals of each other.

yes

Since you are basically taking 1/y for each y value, the height of the function would be the reciprocal of the original value

e.g.
if the height of the original was 5, then the corresponding height would be 1/5 or .2

if the height of the orignal is 1/10 then the new height wouldbe 10

etc.

To understand the term "corresponding feature" in this context, let's first review how to find the reciprocal of a quadratic equation.

To find the reciprocal of a quadratic equation, we interchange the roles of the x and y variables. In other words, we swap the x and y coordinates in the quadratic equation. For example, if the original quadratic equation is y = ax^2 + bx + c, its reciprocal would be x = ay^2 + by + c.

Now, let's consider the vertex (or turning point) of the original quadratic equation and its corresponding feature in the reciprocal equation.

The vertex of a quadratic equation in the form y = ax^2 + bx + c is given by the coordinates (h, k), where h is the x-coordinate and k is the y-coordinate. The x-coordinate of the vertex, h, can be found using the formula h = -b / (2a).

In the reciprocal equation x = ay^2 + by + c, the corresponding feature to the vertex is the y-coordinate of the vertex, which we'll call k'.

Observation:
When comparing the vertex of the quadratic equation and the corresponding feature (y-coordinate of the vertex) in the reciprocal equation, you are correct in noting that the x-coordinate remains the same.

The key observation here is that the y-coordinate of the vertex in the reciprocal equation may have different values than the y-coordinate of the original quadratic equation's vertex. The relationship between the two y-coordinates will depend on the specific quadratic equation and its reciprocal.

To make a more specific observation, you can find the vertex of the original quadratic equation and then substitute the x-coordinate into the reciprocal equation to find the corresponding y-coordinate. By comparing the two y-values, you can identify any patterns or relationships between the vertex and its corresponding feature.