Problem 74 - 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. $$y=0.01 x^{2}+0.6 x+100$$

Step-by-Step Solution

Answer

The vertex of the parabola given by \(y=0.01x^{2}+0.6x+100\) is (-30, 10). The reasonable viewing rectangle for graphing this parabola on a graphing utility is '[-50, -10] x [-10, 30]'.

Step 1: Identify coefficients

First, identify the coefficients. In the equation \(y=0.01x^{2}+0.6x+100\), a is 0.01, b is 0.6 and c is 100.

Step 2: Find x-coordinate of the vertex

Use the formula \(-b/(2a)\) to find the x-coordinate \(\(-b/(2a)=-0.6/(2*0.01)=-30\). This means the x-coordinate of the vertex is -30.

Step 3: Find y-coordinate of the vertex

Substitute \(-30\) for \(x\) in the equation to get \(y = 0.01(-30)^{2}+0.6(-30)+100 = 10\). The y-coordinate of the vertex is 10.

Step 4: Determine a reasonable viewing rectangle

Since the parabola's vertex is negative for x, a '[b - 20, b + 20] x [c - 20, c + 20]' range for x and y values could provide a sufficiently detailed graph. Thus, the viewing rectangle would be '[-50, -10] x [-10, 30]'.

Step 5: Graph the parabola

Finally, Input the original equation \(y = 0.01x^{2}+0.6x+100\) into your graphing utility and set the viewing window using the rectangle determined in the previous step. The graph should display a parabola that opens upwards, with vertex at (-30,10).

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial when studying quadratic functions, as it represents the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. For the equation given in the problem, which is of the form \(y=ax^2+bx+c\), the vertex can be found using the formula \( (-b/(2a), f(-b/(2a)) ) \). In simpler terms, this formula calculates the x-coordinate by taking the opposite of 'b', dividing by twice 'a', and then plugging that result back into the original equation to find the y-coordinate.

For our example, \(y=0.01x^2+0.6x+100\), after computing the vertex, we found it to be (-30,10). This tells us not only where the parabola turns but also helps in graphing it accurately, as it provides a reference point around which we plot our curve.

Quadratic Functions

Quadratic functions are an essential aspect of algebra and they form parabolic graphs. They are written in the standard form \(y=ax^2+bx+c\), where 'a', 'b', and 'c' are constants. The value of 'a' determines whether the parabola opens up (a > 0) or down (a < 0), while 'b' and 'c' affect the location and shape of the parabola.

A key feature of these functions is their symmetry about the vertical line that passes through the vertex, known as the axis of symmetry. For our example function, since 'a' is positive, we know that the parabola opens upwards. Additionally, the graph's width, narrowness, and direction of opening can be determined by the quadratic coefficient 'a'. The smaller the value of 'a' (in absolute terms), the wider the parabola opens.

Graphing Utilities

Modern graphing utilities are invaluable tools for visualizing functions like parabolas. They allow for a dynamic learning experience where one can input the equation of the parabola and adjust the viewing window for the best visual representation. In our case, after determining the vertex, students can set up a reasonable viewing rectangle to graph \(y=0.01x^2+0.6x+100\).

The choice of window size determines how much of the graph is visible and can highlight different features of the parabola. The viewing rectangle '[-50, -10] x [-10, 30]' recommended in the solution helps in centring the graph around the vertex and providing a clear display of the parabola's shape. Encouraging students to experiment with window settings on graphing utilities also reinforces their understanding of the behavior of quadratic functions.