Problem 38 - Beginning and Intermediate Algebra

Question

Rewrite each equation in the form \(x=a(y-k)^{2}+h\) by completing the square and graph it. $$x=-y^{2}-2 y-5$$

Step-by-Step Solution

Answer

The given equation is \(x = -y^2 - 2y - 5\). By completing the square, we rewrite the equation as \(x = -1(y + 1)^2 - 4\). To graph this, plot the vertex at \((-4, -1)\) and since the coefficient of the quadratic term is negative, the parabola opens horizontally to the left.

Step 1: Identify the equation

The equation we are given is \(x = -y^2 - 2y - 5\).

Step 2: Write the equation in the form x = a(y^2 + B*y)

Factor out the coefficient of \(y^2\) from the quadratic term and the linear term: \(x = -1(y^2 + 2y) - 5\) Now we have \(x = a(y^2 + By) - 5\), with \(a = -1\) and \(B = 2\).

Step 3: Complete the square for the expression inside the parentheses

To complete the square for the expression \(y^2 + 2y\), we will need to add and subtract the square of half the coefficient of the linear term inside the parentheses. First, evaluate \(\frac{B}{2}\) and square it: \(\frac{2}{2} = 1\) \(1^2 = 1\) Now add and subtract 1 inside the parentheses: \(x = -1(y^2 + 2y + 1 - 1) - 5\) The expression inside the parentheses is now a binomial square: \(x = -1((y + 1)^2 - 1) - 5\)

Step 4: Rewrite the equation in the form x = a(y - k)^2 + h

Now, we need to distribute the -1 to both terms inside the parentheses and simplify: \(x = -1(y + 1)^2 + 1 - 5\) \(x = -1(y + 1)^2 - 4\) Now our equation is in the form \(x = a(y - k)^2 + h\), where \(a = -1\), \(k = -1\), and \(h = -4\).

Step 5: Graph the equation

To graph the equation \(x = -1(y + 1)^2 - 4\): 1. Identify the vertex of the parabola as \((h, k) = (-4, -1)\) 2. Since \(a = -1 < 0\), the parabola opens horizontally to the left. 3. Plot the vertex and sketch the parabola accordingly. With these steps, the graph of the equation \(x = -1(y + 1)^2 - 4\) is created.

Key Concepts

Quadratic Equations

Quadratic equations are central to algebra and appear in various forms. They typically look like this: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants. The solutions of quadratic equations are the values of \(x\) that make the equation true. There are several methods to solve quadratic equations:

• Factoring
• Using the quadratic formula
• Completing the square

Completing the square is a technique particularly useful for rewriting quadratics in a form that makes graphing easy. In this exercise, we started with an equation involving \(y\), not \(x\) in the usual position, which may seem unusual. Yet, the basic principles of working with quadratics and completing the square still apply. Understanding how to manipulate these equations allows us to convert them into different forms, such as vertex form, which can reveal important properties of their graphs.

Graphing Parabolas

Graphing parabolas is an essential skill when working with quadratic equations. The graph of a quadratic equation is a u-shaped curve called a parabola. To graph a parabola effectively, you need to understand its main features:

• Vertex: The highest or lowest point on the graph, acting as a turning point.
• Axis of symmetry: A vertical line passing through the vertex, dividing the parabola into two mirror images.
• Opening direction: Determined by the sign of the coefficient in front of the \(y^2\) term. If negative, the parabola opens down or to the left, and if positive, it opens up or to the right.

In this exercise, the parabola is in the form \(x = -1(y + 1)^2 - 4\). The vertex can be found at \((-4, -1)\), representing the point where the parabola changes direction. We recognize that since the coefficient \(a = -1\), the parabola opens to the left. Understanding these features allows for accurate plotting and graphing of quadratic equations.

Vertex Form

The vertex form of a quadratic equation is a versatile representation that highlights the key features of a parabola. The general equation is: \[ x = a(y - k)^2 + h \]where \((h, k)\) is the vertex of the parabola. This format is helpful for both graphing and understanding the behavior of the parabola. ### Advantages of Vertex Form:- **Easy Graphing:** The vertex is immediately visible, simplifying the sketching process.- **Understanding Shifts:** Changes in \(h\) and \(k\) indicate horizontal and vertical shifts.- **Width and Direction:** The value of \(a\) determines the parabola's width and direction (whether it opens upwards/downwards or left/right).In this exercise, converting to vertex form helped us uncover that the vertex is \((-4, -1)\) and that the parabola opens to the left. Knowing how to rewrite equations in vertex form by completing the square is essential for both graphing and understanding the deeper properties of parabolas.