Problem 65 - College Algebra

Question

Use a graphing utility to graph the parabolas.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 roots of the equation would be given by \(y = -1 \pm \sqrt{4(6x-9)}\). Two separate equations derived from these roots can be used to represent the final parabolic function in a graph.

Step 1: Convert the equation into a standard quadratic form

The given equation is \(y^{2}+2y-6x+13=0\). To convert it into a standard quadratic equation in the form \(ay^{2} + by + c = 0\), isolate \(y\) and rearrange the equation to yield \(y^{2}+2y = 6x-13\).

Step 2: Apply Quadratic Formula

The standard form of the quadratic equation now becomes \(y^{2} + 2y - (6x - 13) = 0\). Remember the quadratic formula: \(y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). In this case, \(a=1, b=2, c=-(6x-13)\). Substituting values, you get \(y = \frac{-2 \pm \sqrt{(2)^{2} - 4(1)(-(6x-13))}}{2(1)}\). Simplify this further to obtain the roots of \(y\).

Step 3: Simplify the roots

Simplify the equation: \(y = -1 \pm \sqrt{4+4(6x-13)}\). Consequently, \(y = -1 \pm \sqrt{4(6x-9)}\). This can be further broken down into 2 equations: for \(y_{1}\) and \(y_{2}\), based on the plus-minus sign.

Step 4: Graph the equations

With obtained roots, graph them separately on a graphing utility to create the complete parabolic equation for the given function.

Key Concepts

quadratic formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). This formula allows you to find the roots of the equation, which are the values of \(x\) that satisfy the equation. The formula is:\[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]To utilize this formula, you identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation. You will then substitute these values into the formula.
The term under the square root, \(b^2 - 4ac\), is known as the discriminant. This determines the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root.
• If \(b^2 - 4ac < 0\), the roots are complex numbers.

The quadratic formula simplifies the process of finding the roots, providing a straightforward method to tackle quadratic equations.

parabolas

Parabolas are U-shaped graphs that represent quadratic functions. A standard quadratic equation forms a parabola when graphed. The general form of a quadratic equation is \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
The value of \(a\) affects the shape and direction of the parabola:

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

The vertex of a parabola is its highest or lowest point, depending on the direction it opens. It's located at \((-\frac{b}{2a}, f(-\frac{b}{2a}))\). Parabolas are symmetric about their vertical line, called the axis of symmetry. Understanding these properties helps in graphing and analyzing the behavior of quadratic functions.

graphing utility

A graphing utility is a software or calculator that allows you to visualize mathematical functions by plotting them on a coordinate grid. Using a graphing utility can be particularly helpful when working with quadratic equations to sketch parabolas.
To graph a quadratic equation using a graphing utility:

• Input the equation in the utility.
• The graphing tool will calculate and display the curve.
• Observe points like the vertex, axis of symmetry, and intercepts.

Graphing utilities make it easy to explore the shape and position of parabolas without tedious calculations. They are useful for verifying solutions obtained algebraically, especially the roots found using the quadratic formula. With visual aids, you can gain deeper insights on how changing variables in the equation affects the parabola's graph.

roots of equations

The roots of a quadratic equation are the solutions that satisfy the equation \(ax^2 + bx + c = 0\). These roots, also called zeros or x-intercepts, are the points where the graph of the parabola intersects the \(x\)-axis.
Finding the roots can be done algebraically using the quadratic formula:

• The formula provides the precise x-values at which the y-value is zero.
• Graphing these roots helps illustrate their positions along the \(x\)-axis.

Understanding the concept of roots allows you to solve for unknowns in various contexts. When you plot the quadratic equation's graph, the roots serve as a visual representation of the solutions. Knowing how to find and interpret them is essential for solving equations in real-world scenarios and mathematical problems.