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Chapter 7: Problem 71

Write the given equation either in the form \((y-k)^{2}=a(x-h)\) or in the form \((x-h)^{2}=a(y-k)\). $$ x^{2}-3 x+4=2 y $$

Short Answer

Expert verified
The equation in parabolic form is \((x-\frac{3}{2})^{2}=2(y-\frac{7}{8})\).

Step by step solution

01

Examine the Equation Structure

The given equation is \(x^{2}-3x+4=2y\). This equation contains an \(x^2\) term on the left side, which suggests it might fit the form \((x-h)^{2}=a(y-k)\), which is the standard form for a parabola opening either upwards or downwards.
02

Rearrange the Equation

Rearrange the equation to isolate the \(y\) terms on one side. We have:\[ 2y = x^2 - 3x + 4 \]Now, divide through by 2 to express \(y\) in terms of the other variables:\[ y = \frac{1}{2}(x^2 - 3x + 4) \]
03

Completing the Square

To convert the quadratic equation \(x^2 - 3x + 4\) into a perfect square form, we complete the square:1. Take the coefficient of \(x\), which is -3, halve it to get \(-\frac{3}{2}\), and square it to get \(\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\).2. Rewrite the quadratic expression as:\[ x^2 - 3x + \frac{9}{4} + 4 - \frac{9}{4} \]3. Combine the constant terms:\[ x^2 - 3x + \frac{9}{4} + \frac{16}{4} - \frac{9}{4} = \left(x - \frac{3}{2}\right)^2 + \frac{7}{4} \]
04

Form the Equation in Parabolic Form

Now substitute back into the equation for \(y\):\[ y = \frac{1}{2}\left(\left(x - \frac{3}{2}\right)^2 + \frac{7}{4}\right) \]Distribute the \(\frac{1}{2}\):\[ y = \frac{1}{2}\left(x - \frac{3}{2}\right)^2 + \frac{7}{8} \]
05

Write in Standard Parabolic Form

The equation is now in the form \((x-h)^{2}=a(y-k)\) by rearranging:Multiply the entire equation by 2 to clear the fraction:\[ 2y = (x - \frac{3}{2})^2 + \frac{7}{4} \]Then,\[ (x - \frac{3}{2})^2= 2(y - \frac{7}{8}) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This process simplifies the expression and often helps in solving quadratic equations or in rewriting them in a more useful form, like a standard form of a parabola. Here鈥檚 how you complete the square:
  • Take the coefficient of the linear term, halve it, and square the result. For example, with a term like \(x^2 - 3x\), halve \(-3\) to get \(-\frac{3}{2}\), and square it to get \(\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\).
  • Add and subtract this squared term within the quadratic expression. This step helps in restructuring the expression into a square.
  • Rewrite the expression as a square of a binomial plus any remaining constant terms. For example, \(x^2 - 3x + \frac{9}{4} + 4 - \frac{9}{4} = \left(x - \frac{3}{2}\right)^2 + \frac{7}{4}\).
By completing the square, the equation is easier to solve and manipulate, especially when working with parabolas.
Standard Form of a Parabola
The standard form of a parabola gives insight into its shape and position in a coordinate system. By expressing a quadratic equation in this form, you can deduce the vertex of the parabola and its direction of opening. There are two standard forms based on the orientation of the parabola:
  • For parabolas opening upwards or downwards: \( (x-h)^2 = a(y-k) \). This form highlights that the parabola is centered horizontally with vertex at \(h,k\).
  • For parabolas opening to the right or left: \( (y-k)^2 = a(x-h) \), showing the parabola's vertex vertically centered.
In our exercise, the equation \( (x-\frac{3}{2})^2 = 2(y-\frac{7}{8}) \) represents a parabola that opens upwards or downwards, with vertex \( (\frac{3}{2}, \frac{7}{8}) \). Understanding this form helps in graphically representing and analyzing the parabola's properties.
Conic Sections
Conic sections are the curves obtained by intersecting a double-napped cone with a plane. Parabolas are one of the four types of conic sections, which also include circles, ellipses, and hyperbolas. Each conic section has unique geometric properties and equations:
  • **Circle:** A points equidistant from a center point, represented as \( (x-h)^2 + (y-k)^2 = r^2 \).
  • **Ellipse:** An extended circle with two focal points, expressed with \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).
  • **Parabola:** The path of all points equidistant from a focus and a directrix, often taking the form \( (x-h)^2 = a(y-k) \).
  • **Hyperbola:** A curve with two distinct branches, defined by \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).
Parabolas, specifically, are important in physics and engineering, modeling paths like those of projectiles or satellite dishes. Understanding parabolas as conic sections aids in visualizing and predicting their behavior in real-world applications.

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Most popular questions from this chapter

Shade the solutions set to the system. $$ \begin{aligned} x^{2}+y^{2} & \leq 4 \\ (x+1)^{2}-y & \leq 0 \end{aligned} $$

Halley's comet travels in an elliptical orbit with \(a=17.95\) and \(b=4.44\) and passes by Earth roughly every 76 years. Note that each unit represents one astronomical unit, or 93 million miles. The comet most recently passed by Earth in February 1986 (Source: M. Zeilik, Introductory Astronomy and Astrophysics.) (a) Write an equation for this orbit, centered at \((0,0)\) with major axis on the \(x\) -axis. (b) If the sun lies (at the focus) on the positive \(x\) -axis, approximate its coordinates. (c) Determine the maximum and minimum distances between Halley's comet and the sun.

Sketch a graph of the ellipse. Identify the foci and vertices. $$ \frac{x^{2}}{4}+\frac{(y-1)^{2}}{9}=1 $$

Write the equation in standard form for an ellipse centered at (h, k). Identify the center and the vertices. $$ 2 x^{2}+4 x+3 y^{2}-18 y+23=0 $$

Translate the ellipse with the given equation so that it is centered at the given point. Find the new equation and sketch its graph. $$ \frac{x^{2}}{2}+\frac{y^{2}}{9}=1 ;(-3,-4) $$
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