Problem 68 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation and then use a graphing utility to graph the related quadratic function in the standard viewing window. Discuss how the graph of the quadratic function relates to the solutions of the quadratic equation. Function \(y=-x^{2}+3 x-5\) Equation $$-x^{2}+3 x-5=0$$

Step-by-Step Solution

Answer

The solution to the equation are the roots / x-intercepts of the function. They are the points where the graph of the function intersects the x-axis. The vertex of the graph is located directly between these two points.

Step 1: Solve the quadratic equation

A quadratic equation can be solved by factoring, completing the square or using the quadratic formula. This equation \( -x^{2}+3x-5=0 \) isn’t easily factorable, so the quadratic formula, \[ x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \] will be used. Here, a=-1, b=3, and c=-5. Substituting these values into the quadratic formula provides the solutions

Step 2: Graph the function

After finding the solutions in Step 1, plug the x-values into the equation to find the corresponding y-values. Then use these points (x, y) to draw the graph using a graphing utility. The function will resemble a downward opening parabola since a, the coefficient of \(x^{2}\), is negative.

Step 3: Analyze the relationship

The solutions of the quadratic equation represent the x-intercepts of the graph, which are the points where the parabola crosses the x-axis. Notice that the vertex, or the highest or lowest point of the graph, is located directly between the two solutions (roots). This is generally the case with all quadratic functions.

Key Concepts

Graphing Utility

A graphing utility is an electronic tool or software used to graph mathematical functions and analyze their properties. For quadratic functions, using a graphing utility can help you to visualize the behavior of these functions quickly and accurately. When you input the quadratic equation, such as \( y = -x^2 + 3x - 5 \), a graphing utility will display the parabola's curve.

• Make sure to input the equation accurately and check if the graphing tool allows customization of the viewing window for better clarity.
• Different utilities might offer features such as zooming, dragging, or even calculating derivatives.
• By graphing, you can easily find key features of a quadratic function like intercepts and the vertex.

Using a graphing utility simplifies the process of analyzing quadratic functions, especially when solutions are non-integers or complex, making it an essential tool in visualizing mathematical concepts.

Quadratic Formula

The quadratic formula is a method for solving quadratic equations, which have the general form: \[ ax^2 + bx + c = 0 \]The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula can tackle any quadratic equation, even those that are not easily factorable. To use it:

• Identify coefficients \(a\), \(b\), and \(c\) from your quadratic equation.
• Substitute these coefficients into the formula.
• Solve for \(x\) using arithmetic operations to find the roots of the equation.

For instance, in the equation \( -x^2 + 3x - 5 = 0 \), where \(a = -1\), \(b = 3\), and \(c = -5\), the quadratic formula becomes an indispensable tool, yielding the values where the function intersects the x-axis: the x-intercepts.

Parabola

A parabola is a U-shaped curve that represents the graph of a quadratic function. It can open upwards or downwards depending on the sign of the leading coefficient \(a\) in the quadratic equation. For the function \( y = -x^2 + 3x - 5 \),the parabola opens downwards since \(a = -1\), which is negative. Parabolas have distinct characteristics:

• The vertex is the turning point, which is the highest or lowest point of the graph based on whether it opens up or down.
• The axis of symmetry is a vertical line passing through the vertex, helping divide the graph into two mirrored halves.
• The shape and width of the parabola are influenced by the absolute value of \(a\); a larger \(|a|\) means a narrower parabola.

Understanding the parabola helps in identifying important points such as maximum or minimum values and aids in drawing accurate graphs.

X-intercepts

X-intercepts are the points where a graph crosses the x-axis. These are crucial in solving quadratic equations because they represent the solutions of the equation. For a quadratic function such as \( y = -x^2 + 3x - 5 \)solving the corresponding quadratic equation \( -x^2 + 3x - 5 = 0 \)gives the x-intercepts.

• X-intercepts occur where the function's value \(y\) is zero.
• These points are also called "roots" or "zeros" of the quadratic function.
• Visually, on the graph, these are the points where the parabola touches or crosses the x-axis.

Finding x-intercepts using the quadratic formula involves calculating the roots of the quadratic equation, which directly tells us where the parabola intersects the x-axis. Understanding x-intercepts provides insight into the real-world applications of quadratic functions, since they often represent critical points like break-even points in business or projectile impacts in physics.