Problem 86 - Algebra and Trigonometry

Question

Use a graphing utility to graph the parabolas in Exercises 86–87. Write the given equation as a quadratic equation in y and use the quadratic formula to solve for y. Enter each of the equations to produce the complete graph. $$ y^{2}+2 y-6 x+13=0 $$

Step-by-Step Solution

Answer

The quadratic equation in y becomes \(y^{2} + 2y - 13 = 6x\). After applying the quadratic formula, the values of y are \(y_{1,2}=-1\pm\sqrt{14}\). The graph of the equation \(y=-1\pm\sqrt{14}\) displays a parabola.

Step 1: Convert the given equation into quadratic equation in y

By simply rearranging the given equation \(y^{2}+2 y-6 x+13=0\), it becomes \(y^{2} + 2y - 13 = 6x\). On comparing with the standard form of quadratic equation \(ax^{2} + bx + c = 0\), the coefficients a, b, and c are identified as 1, 2, and -13 respectively.

Step 2: Use the quadratic formula to solve for y

The quadratic formula is defined as \(y = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Substituting the values of a, b, and c obtained from step 1 we find the values of y. A discriminator will be calculated as \(D=b^{2}-4ac=2^{2}-4*1*(-13)=4+52=56\). Hence the solutions for y will be \(y_{1,2}=\frac{-2\pm\sqrt{56}}{2}\) or \(y_{1,2}=-1\pm\sqrt{14}\)

Step 3: Graph the equation

Enter the equation \(y =\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\) into a graphing software, in this case it becomes \(y=-1\pm\sqrt{14}\). You should see a parabola in the visual output.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool used for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). To find the roots of a quadratic equation using the quadratic formula, you apply the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, the symbols \( a \) and \( b \) represent the coefficients of \( x^2 \) and \( x \) respectively, while \( c \) is the constant term.

When you substitute the coefficients into the formula, you have to calculate the discriminant (\( D = b^2 - 4ac \) ). The discriminant tells us important information about the roots:

• If \( D > 0 \) , the equation has two real and distinct roots.
• If \( D = 0 \) , the equation has exactly one real root (also called a repeated or double root).
• If \( D < 0 \) , the equation has no real roots but instead has two complex roots.

Applying this to the given exercise, \( a=1 \) , \( b=2 \) , and \( c=-13 \) , we get a discriminant of \( D=56 \) indicating two real and distinct roots. Solving for \( y \) gives us \( y_1 = -1 + \sqrt{14} \) and \( y_2 = -1 - \sqrt{14} \) .

Solving Quadratic Equations

Solving quadratic equations can be accomplished through several methods such as factoring, completing the square, using the quadratic formula, or graphing. The method choice depends on the specific equation and sometimes multiple methods can be used.

Applying the Quadratic Formula

Given a quadratic equation, the direct application of the quadratic formula tends to be straightforward, as seen in the textbook exercise. Converting the given equation into a standard quadratic form and identifying \( a \) , \( b \) , and \( c \) is requisite before applying the formula. It is crucial to understand each step, from rearranging the equation to graphing the parabolas for visual confirmation of the solution.

Checking Your Solution

It's always good practice to check your solution by substituting your roots back into the original equation to ensure that the left side equals the right side. If the initial equation contains a variable like \( x \) , as in the exercise, you can select a value for \( x \) to verify the corresponding \( y \) values.

Graphing Utilities

Graphing utilities, such as graphing calculators or computer software, are incredibly useful in visualizing quadratic equations. These tools can plot equations and instantly show the shape and position of a parabola, which is the graph formed by a quadratic equation. Understanding the visual representation of quadratic equations can reinforce the concept of roots and the axis of symmetry.

Creating a Complete Graph

To use a graphing utility effectively, you must be able to enter the correct form of the equation. In the case of the textbook exercise, the equation is rewritten in terms of \( y \) and \( x \) and then graphed. This reveals the parabolic curve, where the axis of symmetry and vertex can be identified, as well as the points where the parabola crosses the \( x \) and \( y \) axes—a vital aspect for understanding the roots and the overall behavior of the quadratic function.

Graphing utilities can also help you discover the characteristics of the parabola, such as direction of opening (upward or downward), width, and whether it is a maximum or minimum curve. This visualization supports and enhances the learning and problem-solving process for students.