If L is a horizontal line in a coordinate plane and P is a parabola in the same plane, how many points could lie on both L and P? What if L is a vertical line? What if L is neither horizontal nor vertical?

Solving Answer for L:Vertical Line

How many times can the parabola cross the vertical line? = how many possible y-intercepts are there?

Lowest Amount of Possible X Intercepts:

Solving for L: Horizontal Line

**Parabola Problem of The Week**

**By Etoile Boots**

How many times can the parabola cross the horizontal line? = how many possible x-intercepts are there?

Lowest Possible y-intercepts

The lowest possible number of y intercepts are 0, since negative numbers of intercepts are impossible in graphing, but lets test out if 0 really is possible.

It is Impossible: Here are a few graphs to explain why

y=5(x-1000)^2+7

y=5(x-100)^2+7

y=5(x-10)^2+7

Every parabola has a slope, and is always expanding outward, this means that however far from the vertical line the (p) parabola is, whether negative or not, it will always eventually reach the vertical line and cross it, this also means that p can only have one crossing point on the y axis, or any vertical line. I know this because just as the p crossed the vertical line, it was always expanding, once it crosses that line, it will only keep expanding, never coming back to the line and intersecting again.

I proved this by imputing multiple, vastly different values for the vertex of a random vertex form equation, all of them eventually crossed the y axis.

Explanation:

I believe that the lowest number of possible times that p crosses a horizontal line is 0 times, for the same reasons as the vertical line, but lets test it.

y=2(x-2)^2+5

It is possible: Here are a few graphs to prove why

Doesn't touch the x axis

What I found:

p does not cross or touch the x axis when the vertex is positive and does not have a 0 for x in the vertex.

I found this out by experimenting with desmos, when I had the vertex x as 0 (y=2(x-2)^2), the parabola had one point of intersection, just touching the x axis. When I subtracted the 5, instead of adding it, the parabola had 2 points of intersection, x in the vertex on -5.

y=2(x-2)^2-5

y=2(x-2)^2

What We know Now:

number x intercepts = 0, 1, or 2

We know that the number of x intercepts cannot be less than 0, but can it 3 (more than 2)?

If you think about it, it is impossible to have more than 2 x intercepts, this is because of the way parabolas works, it will always extend outwards and either up or down, for there to be 3 intercepts, the parabola would have to look like a wave, and then it wouldn't be a parabola.

What if the Line is Neither Horizontal or Vertical?

If L, the line is neither horizontal or vertical, this means that it is a diagonal line, lets graph a vertical line to see where it intersects!

y=2x

y=2(x-2^2

With a random parabola and a diagonal line (y=2x) there are two points of intersection, and that is the maximum times that the parabola will cross the diagonal line, this is because the parabola will always expand outwards, and the diagonal line will increase by the same rate, there will be no other intersections.

What Else?

Of course, it is possible for there to be to be no intersections, and even just one intersection, as the graphs below show:

y=-2x

y=2=(x-2)^2

no points of intersection:

y=x^2+1

y=2x

1 point of intersection:

End Reflection: Part 1:

Real World Applications of Parabolas:

Parabolas are used in many different ways in real life! They can be unintentional, such as a ball being thrown into the air. Or they may be designed in that specific way to serve a particular service, such as car headlights, they are placed in front of a parabolic mirror, this cause the light rays to be strong and visible!

End Reflection: Part 2:

I feel as though my explanations made sense and were concise enough. One thing that I am struggling with though, is figuring out another way to prove my answer. I have already proved it through visuals (graphs) and logic, but I would like to find another

People should trust my answers because they are not only come up with lgoic but are proven by graphs, which are easy to comprehend