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(solution) 1. A parabola has a very specific geometric definition: Given a


Hello! I'm taking Pre-Calculus and I have trouble with giving examples of the parabola as well as finding formula for them. I have attached the problems in the attachment.


1. A parabola has a very specific geometric definition:

 

Given a focus (a point in the plane) and a directrix (a line not passing through the

 

focus), a parabola is defined as the set of all points whose distance to the focus is

 

the same as its distance to the directrix.

 

In the image below, we can see that this amounts to the length of A being equal to the

 

length of B. (a) Find the formula for the parabola that has a focus of (2, 1) and a directrix of y =

 

?1. (Hint: You will need to figure out how to calculate the distance between a point

 

(x, y) and the focus and the distance between a point (x, y) and the directrix.)

 

(b) When you define a parabola in this way, you can also define a tangent line for

 

each point on the parabola. If you look at the image above, you can see that if the

 

point (x, y) is on the parabola, its tangent line is the line passing through (x, y) that is

 

perpindicular to C.

 

Tangent lines are useful because they allow us to quantify the slope of the parabola at

 

a given point: The slope of the tangent line is the slope of the parabola at that

 

particular point.

 

Find the slope of the parabola from part (a) at the point where x = 0.

 

2. Suppose that p(x) is a degree 5 polynomial. For each of the following, determine

 

if it is possible for p(x) to have these features. If it is possible, give an example of

 

a p(x) that has these features (provide a graph and a formula for p(x)). If it is not

 

possible, explain why. p(x) has 3 turning points p(x) has 4 turning point

 


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