Problem 44 - Precalculus: Functions and Graphs

Question

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x+4=y^{2}+y$$

Step-by-Step Solution

Answer

The graph is a parabola with a horizontal axis.

Step 1: Rearrange the Equation

Given the equation is \(x + 4 = y^2 + y\). We notice that all the terms on one side are in terms of \(y\), while \(x\) is isolated. Let's rearrange the equation to make it easier to analyze:\[x = y^2 + y - 4\]

Step 2: Complete the Square for Y

Let's complete the square on the right side of the equation for the terms involving \(y\).- Consider the expression \(y^2 + y\).- To complete the square, take half of the coefficient of \(y\), square it, and add and subtract it within the equation.- Half of 1 (coefficient of \(y\)) is \(\frac{1}{2}\), and squaring it gives \(\frac{1}{4}\).Now add and subtract \(\frac{1}{4}\) within the equation:\[x = (y^2 + y + \frac{1}{4}) - \frac{1}{4} - 4\]

Step 3: Rewrite as a Perfect Square

Now rewrite the squared term:- \(y^2 + y + \frac{1}{4}\) can be rewritten as \((y + \frac{1}{2})^2\).So the equation becomes:\[x = (y + \frac{1}{2})^2 - \frac{1}{4} - 4\]- Simplify the constant terms on the right side:\[x = (y + \frac{1}{2})^2 - \frac{17}{4}\]

Step 4: Identify the Conic Section

The equation is now expressed as \(x = (y + \frac{1}{2})^2 - \frac{17}{4}\), which corresponds to the general form \(x = a(y-k)^2 + h\), indicating a parabola.- Since \(x\) is expressed in terms of \(y^2\), the parabola opens horizontally. Hence, it is a parabola with a horizontal axis of symmetry.

Key Concepts

Parabola

A parabola is a specific type of curve you encounter often in math. It’s a symmetrical shape that looks like the typical 'U' for a vertical parabola, but it can also open sideways. In mathematical terms, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It can be represented by different equations depending on its orientation, either vertical or horizontal. The key feature of a parabola is its axis of symmetry, which can either be vertical or horizontal. Parabolas have various applications, including physics, engineering, and computer graphics. Understanding how to identify and manipulate their equations is crucial for solving complex real-world problems.

Completing the Square

"Completing the square" is a method used to simplify quadratic equations, making them easier to analyze or solve. It transforms a quadratic expression into a perfect square trinomial. This helps in finding the vertex form of a parabola's equation.
Here's how you complete the square:

• Firstly, identify the quadratic expression you want to transform, such as \(y^2 + y\).
• Next, take half of the coefficient of the linear term \(y\), square it, and then add and subtract this square within the expression.
• In our case, half of 1 (the coefficient of \(y\)) is \(\frac{1}{2}\), and squaring it gives \(\frac{1}{4}\).
• Add and subtract \(\frac{1}{4}\) to transform \(y^2 + y\) into a perfect square: \(y^2 + y + \frac{1}{4} = (y+ \frac{1}{2})^2\).

This technique is extremely useful when rearranging equations to identify the geometric conic sections like parabolas, circles, ellipses, and hyperbolas.

Equation Rearrangement

Rearranging an equation is often a critical first step in simplifying or solving it. In the context of conic sections, rearranging helps us see the underlying geometric structures better. In our example, the equation \(x + 4 = y^2 + y\) was rearranged to \(x = y^2 + y - 4\). This separates variables on opposite sides, which is crucial for the next steps like completing the square.

• It isolates the variable of interest (here, \(x\)).
• This also helps us to reorganize terms to match standard forms of conic sections.

Being meticulous in this step is key as it lays the groundwork for further simplification techniques like completing the square. Getting the initial form right can save time and reduce errors in subsequent calculations.

Horizontal Axis of Symmetry

The axis of symmetry of a parabola is an imaginary line that divides it into two identical halves, acting as a mirror. In parabolas derived from the form \(x = a(y - k)^2 + h\), the axis of symmetry is horizontal. This means the parabola opens either to the left or the right.
The concept is straightforward but understanding which direction a parabola opens is vital for graphing and analyzing it.

• If \(x\) is squared and expressed in terms of \(y\), the symmetry is vertical.
• If \(y\) is squared, as in our scenario, the symmetry is horizontal.

This distinction also helps when solving real-world problems like satellite dish design or optics, where knowing the orientation affects how signals or light are focused. Thus, recognizing the axis of symmetry is key to using mathematical models effectively.