Problem 61 - College Algebra

Question

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. \(y=-4 x^{2}+20 x+160\)

Step-by-Step Solution

Answer

The vertex of the parabola is at the point (2.5, 170). A reasonable viewing rectangle for the graph could be from x=0 to x=5 and y=0 to y=180.

Step 1: Find the x-coordinate of the vertex

The x-coordinate of the vertex can be calculated using the formula \(-\frac{b}{2a}\) where a and b are coefficients of the quadratic equation \(y=ax^{2}+bx+c\). In this case, the quadratic equation is \(y=-4x^{2}+20x+160\), so \(a=-4\) and \(b=20\). Plugging these values into the formula gives \(-\frac{b}{2a}=-\frac{20}{2*(-4)}=2.5\). So the x-coordinate of the vertex is 2.5.

Step 2: Find the y-coordinate of the vertex

Substitute the x-coordinate of the vertex (2.5) back into the quadratic equation to find the y-coordinate of the vertex. \(y=-4*(2.5)^{2}+20*(2.5)+160=170\). So the y-coordinate of the vertex is 170.

Step 3: Determine a viewing rectangle

Now, using the vertex at (2.5, 170) determine a reasonable viewing rectangle on the graph. For the x-values, a reasonable range might be from 0 to 5 to make sure the full width of the parabola is visible. For the y-values, choose a range that covers the vertex value plus some additional range above and below to show the curve of the parabola. Here, because the parabola opens downwards, choose a range from 0 to 180.

Step 4: Graph the function

Now, graph the function \(y=-4x^{2}+20x+160\) within the viewing rectangle determined in the previous step. It should be a downward opening parabola with its vertex at the point (2.5, 170). To graph the function, use a graphing utility and input the function as well as the viewing window.

Key Concepts

Parabola Vertex

Understanding the vertex of a parabola is crucial when graphing quadratic functions. The vertex represents the highest or lowest point on the parabola, depending on whether it opens upward or downward. The coordinates of the vertex can be found using the formula for the x-coordinate, where and are coefficients from the quadratic equation . For the given equation , we find that , which gives us an x-coordinate of 2.5. To find the y-coordinate, we substitute x back into the equation, yielding . Thus, the vertex of the parabola is at the point (2.5, 170).
Graphing a parabola accurately involves identifying this vertex, as it determines the way the rest of the graph is shaped and where it is positioned on the coordinate plane.

Quadratic Equation

The term 'quadratic equation' refers to a second-degree polynomial equation of the form . In our context, the equation provides a classic example of a quadratic equation. The coefficients , and influence the shape and position of the parabola on the graph. The coefficient affects the width and direction of the parabola’s opening, controls the slope at the y-intercept, and provides the y-intercept itself. By understanding these individual roles, students can predict the general shape of the parabola even before graphing it.

Graphing Utility Usage

A graphing utility, such as a graphing calculator or software, is an essential tool for visualizing functions. When it comes to quadratic functions, we input the equation into the utility to obtain the graph. For our example , you would enter into the graphing utility. It is important to familiarize oneself with the functionalities of these utilities, such as setting the viewing window and plotting points, to make the most out of the technology and to ensure that the graph is drawn to scale and accurately represents the function’s behavior.

Viewing Rectangle

Choosing an appropriate viewing rectangle is an important step in graphing any function. This 'rectangle' defines the range of x and y values visible on the graph. For the quadratic function at hand , we've determined that a reasonable range for the x-values would be from 0 to 5, as it comfortably encapsulates the full width of the parabola, centering on our vertex x-coordinate of 2.5. Regarding the y-values, selecting a range from 0 to 180 ensures that the vertex's y-coordinate of 170 is included, as well as some space below and above to capture the curvature. The goal of the viewing rectangle is indeed to provide the best possible visual representation of the function's behavior within the given frame, which allows for easier interpretation and analysis of its properties.