Problem 73 - Introductory Algebra for College Students

Question

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the parabola. $$y=5 x^{2}+40 x+600$$

Step-by-Step Solution

Answer

The vertex of the given parabola \(y=5x^{2}+40x+600\) is (-4, 440). A suitable viewing rectangle for graphing this parabola would be from x = -10 to x = 10 and y = 0 to y = 500.

Step 1: Determine the Vertex

The vertex of a parabola given by the equation \(y = ax^{2} + bx + c\) is given by the point \(-b / (2a)\), \(f(-b / (2a))\). For the given equation \(y=5x^{2}+40x+600\), a is 5 and b is 40. Therefore, the x-coordinate of the vertex is \(-b / (2a) = -40 / (2*5) = -4\). Substituting x = -4 into the equation gives y = 5*(-4)^{2}+40*(-4)+600 = 440. Therefore, the vertex of the parabola is (-4, 440).

Step 2: Determine Viewing Rectangle

The viewing rectangle for the graphing utility is determined based on the vertex of the parabola. Since the vertex of the parabola is (-4, 440), a suitable viewing rectangle might be from x = -10 to x = 10 and y = 0 to y = 500. The x values are chosen to be a little larger than the x-coordinate of the vertex to ensure the vertex and a reasonable amount of the parabola on either side will be visible. The y values are chosen from 0 (since because the parabola opens upwards it will not go below the vertex), up to a bit more than the y-coordinate of the vertex.

Step 3: Graph the Parabola

Finally, use your chosen viewing rectangle to graph the parabola with your graphing utility.

Key Concepts

Understanding Quadratic Functions

Quadratic functions form a basic component of algebra and are represented by the equation format of \(y = ax^2 + bx + c\). The graph of this equation is a curve called a parabola. Parabolas have various orientations, but the most commonly explored type is the one opening upwards or downwards. This nature is determined by the value of \(a\): if \(a\) is positive, the parabola opens upward, and if negative, it opens downward. Importantly, each quadratic function has a vertex, which is the highest or lowest point, depending on the direction in which the parabola opens. For the function \(y = 5x^2 + 40x + 600\), the vertex is key in determining how the curve appears and behaves on a coordinate axis. To identify the vertex, use the formula \(-b / (2a)\) for the x-coordinate and substitute back into the equation to find the y-coordinate. This results in the vertex \((-4, 440)\), marking the turning point or peak of this parabola, since it opens upwards.

Utilizing Graphing Utilities

A graphing utility is a tool or software designed to help graph equations. Calculators, computer applications, and graphing software are typical examples. They allow for the visualization of functions, especially helpful in understanding the behavior of functions like quadratic equations, which are not easily drawn by hand. Graphing utilities automate the process of plotting points by allowing input of the function’s equation. They use computational power to evaluate the function for several values of \(x\) and graph the results in seconds. For our quadratic function \(y = 5x^2 + 40x + 600\), a graphing utility plots this equation efficiently, showing a parabola with the calculated vertex point clearly visible. Students should learn how to input equations accurately and adjust settings to utilize these utilities aptly for various types of mathematical exploration.

Significance of Viewing Rectangle

The concept of a viewing rectangle is crucial for correctly displaying a function on a graphing utility. It dictates the section of the graph to be visible on your screen, essentially framing the portion of the coordinate plane you wish to explore.For the quadratic equation \(y = 5x^2+40x+600\), we determined a viewing rectangle with x-values ranging from -10 to 10 and y-values from 0 to 500. This choice ensures that not only is the vertex \((-4, 440)\) comfortably within view, but that the general shape and length of the parabola up to points of interest are also visible. Such considerations aid in effective analysis and comprehension of the function's graphical representation. When setting a viewing rectangle, it is wise to include areas beyond the vertex in its range to ensure comprehensive understanding of a parabola's extension and to capture its broader behavior.