Find the Vertex, Value of P, Axis of Symmetry, Focus, Directix, then graph the equation: y + 2 = 1/4(x-1)^2

Question

anonymous · Answer

If I could also get help on this question, it would be much loved. These are review questions for an upcoming exam and I would appreciate all the help I could get on it:

The shape of a park can be modeled by a circle with the equation x^2 + y^2 = 1600. A stretch of highway near the park is modeled by the equation y = 1/40(x - 40)^2. At what points does a car on the highway enter or exit the park?

campbell_st · Answer

ok... so the standard form of the parabola I use is 

(x - h)^2 = 4a(y - k)

the vertex is (h, k) and   a = focal length... 

so you're equation can be written as 

$$4(y + 2) = (x - 1)^2$$

can you identify the vertex and focal length a ?

anonymous · Answer

@campbell_st - I figured it out, thanks! 
But I still need help with the other question I mentioned. Do you think you could help me out with that one? And also this one if you can: 4cosθ = 1, for 0° ≥ θ  ≥ 360°

anonymous · Answer

I had figured it out ahead of time on my own. But I'm still stuck on the other two.

campbell_st · Answer

ok... so divide both sides by 4

$$\cos(\theta) = \frac{1}{4}$$

given the domain, there are 2 answers... 

1st quadrant and 4th quadrant.... this is where cos is positive

so find the 1st quadrant angle... 

and 4th quadrant is 

$$360 - \theta$$

hope it helps

anonymous · Answer

@campbell_st can you tell me how to figure out the quadrants? θ = 1/4, so would we do cos(1/4) to figure out the answer?

anonymous · Answer

I entered it in my calculator and got this: θ = 2πn - cos﻿^-1﻿(﻿1﻿/﻿4﻿) 
Is it right?

campbell_st · Answer

well there is this diagram 

|dw:1401748821663:dw|

it comes from the unit circle...

the diagram shows which trig ratios are positive in each quadrant...

campbell_st · Answer

so is 

cos(@) = 1/4

whats the value of @

normally its just   shift cos (1/4) =

anonymous · Answer

So it's 360° - 1/4?

anonymous · Answer

@campbell_st 
I got this answer after putting it in my calculator but I don't know if it's right: θ = 2πn - cos^﻿-1﻿(﻿1﻿/﻿4﻿)

campbell_st · Answer

no if

$$\cos(\theta) = \frac{1}{4} ....then..... \theta = \cos^{-1}(\frac{1}{4}) = 75.5^o$$

does that make sense... 

and that's the 1st quadrant angle

anonymous · Answer

@campbell_st - So then would the fourth quadrant be 224.5°?

campbell_st · Answer

no... its

360 - 75.5

anonymous · Answer

@campbelll_st - I mean 284.5°

campbell_st · Answer

that makes more sense