Problem 71 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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:\[ y = \frac{1}{2}(x^2 - 3x + 4) \]

Step 3: 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:\[ x^2 - 3x + \frac{9}{4} + \frac{16}{4} - \frac{9}{4} = \left(x - \frac{3}{2}\right)^2 + \frac{7}{4} \]

Step 4: 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 \(\frac{1}{2}\):\[ y = \frac{1}{2}\left(x - \frac{3}{2}\right)^2 + \frac{7}{8} \]

Step 5: 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,\[ (x - \frac{3}{2})^2= 2(y - \frac{7}{8}) \]

Key Concepts

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’s 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.