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Chapter 6: Problem 21

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: \(\left(0, \frac{1}{2}\right)\)

Short Answer

Expert verified
The standard form equation of the parabola with the provided characteristics is \( y = x^2 \).

Step by step solution

01

Determine Position of Parabola

First, understand that the focus lies on the axis of the parabola. Since the focus is \( (0, \frac{1}{2}) \), with the vertex at the origin, the axis of the parabola must be parallel to the y-axis, and so the parabola opens either upwards or downwards. Because the y-coordinate of the focus is positive, this particular parabola opens upwards.
02

Use Vertex Form of Equation

As the parabola opens upwards, use the standard form for a parabola that opens up or down, which is \( y = a(x - h)^2 + k \). In this case, the vertex (h,k) is at the origin (0,0), so the formula simplifies to \( y = ax^2 \).
03

Use the Focus to Determine 'a'

Next, determine the value of exact 'a' using the given focus. The distance from the vertex to the focus is \( \frac{1}{4a} \), so \( \frac{1}{2} = \frac{1}{4a}\). Solving for 'a', you get \( a = 1 \).
04

Write Final Form of Equation

Substitute the value of 'a' into the equation \( y = ax^2 \), giving the final standard form equation of the parabola: \( y = x^2 \).

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Most popular questions from this chapter

A projectile is launched at a height of \(h\) feet above the ground at an angle of \(\boldsymbol{\theta}\) with the horizontal. The initial velocity is \(v_{0}\) feet per second, and the path of the projectile is modeled by the parametric equations \(x=\left(v_{0} \cos \theta\right) t\) and \(y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}\) Use a graphing utility to graph the paths of a projectile launched from ground level at each value of \(\boldsymbol{\theta}\) and \(v_{0} .\) For each case, use the graph to approximate the maximum height and the range of the projectile. (a) \(\theta=60^{\circ}, \quad v_{0}=88\) feet per second (b) \(\theta=60^{\circ}, \quad v_{0}=132\) feet per second (c) \(\theta=45^{\circ}, \quad v_{0}=88\) feet per second (d) \(\theta=45^{\circ}, \quad v_{0}=132\) feet per second

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