Problem 63 - Precalculus : Building Concepts and Connections

Question

Graph each quadratic function by finding a suitable viewing window with the help of the TABLE feature of a graphing utility. Also find the vertex of the associated parabola using the graphing utility. $$y_{1}(x)=0.4 x^{2}+20$$

Step-by-Step Solution

Answer

The graph of \(y_{1}(x)=0.4 x^{2}+20\) is a parabola that opens upwards. Using a suitable viewing window we can see the vertex, which is the lowest point for this parabola, on the graphing utility.

Step 1: Initialize the Function on the Graphing Utility

Key in the function \(y_{1}(x)=0.4 x^{2}+20\) into your graphing utility. To input the function, replace \(y_{1}(x)\) with \(y\) or \(f(x)\) depending on the graphing utility you are using.

Step 2: Identify a Suitable Viewing Window

Using the graphing utility's table feature, observe the range of x-values and their corresponding y-values generated by the utility. From observation, determine the approximate values of \(x\) and \(y\) where the graph is most 'active', i.e., changing the most rapidly, along with a range of values where it seems to be relatively 'flat'.

Step 3: Graph the Function

After adjusting the scope of the viewing window, graph the function using the determined viewing window. You should get a upward-opening parabolic graph.

Step 4: Find the Vertex of the Parabola

Utilize the graphing utility's vertex finder feature, if available, to automatically highlight the vertex. Or visually identify where the graph reaches its lowest point (for an upward-opening parabola) or highest point (for a downward-opening parabola). For this function, since the parabola opens upwards, the vertex would be the minimum point of the graph.

Key Concepts

Finding the Vertex

The vertex of a quadratic function is the point where the parabola either reaches its maximum or minimum value. It's like the peak or the bottom of a hill. In the case of the function y_{1}(x)=0.4 x^{2}+20, because the coefficient of x^2 is positive, the parabola opens upwards and the vertex will be the lowest point on the graph. To find the vertex using a graphing utility, first graph the function.

Then, if your utility has a vertex finder feature, simply use it to locate the vertex automatically. If you have to find it manually, look for the point where the graph changes direction. For our function, note down the x-value at this point, which corresponds to the line of symmetry of the parabola. Then, find the associated y-value. Together, these values (x, y) represent the vertex's coordinates, providing a snapshot view of the most important point on the graph.

Graphing Utility

A graphing utility, such as a graphing calculator or software, is an invaluable tool when studying quadratic functions. It allows you to visualize the graph quickly and can often provide additional information like the vertex or intercepts. When using a graphing utility for the function y_{1}(x)=0.4 x^{2}+20, start by inputting the formula where y_{1}(x) is replaced by the standard y or f(x).

With the function entered, use the TABLE feature to explore different x-values and the corresponding y-values that the function produces. This facility enables you to grasp how the function behaves without plotting each point manually, saving time and providing a clear path to understanding the quadratic function’s graph.

Quadratic Function

A quadratic function can be expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, a symmetric curve that either opens upwards or downwards.

The function y_{1}(x)=0.4 x^{2}+20 is an example of a quadratic function, where a=0.4 and c=20, with b being 0 since it is not present. This function's graph will be a parabola opening upward due to the positive value of a. Key characteristics of a quadratic function include its vertex, axis of symmetry, intercepts, and the direction in which it opens, all of which can be determined and understood through careful graph analysis with the help of graphing utilities.

Viewing Window

Selecting the appropriate viewing window is crucial when graphing a function. This defines the range of x-values and y-values that you'll see on your graph. Think of it as adjusting the lens on a camera to capture the right picture of your graph. For our example function y_{1}(x)=0.4 x^{2}+20, finding the right window involves observing the function's output within the TABLE feature of a graphing utility.

Look at how the y-values change as you progress with different x-values. Choose x-min and x-max where the changes are most significant, ensuring you capture the vertex in your window. Similarly, adjust y-min and y-max values to see the whole parabola. Balancing these values lets you visualize the entire parabola, ensuring that the vertex and intercepts, if within the chosen range, are visible on-screen.