Problem 67 - Contemporary Precalculus

Question

Because of a calculator's small screen size, it is not always easy (or even possible!) to find a viewing window that displays what its user desires. (a) Graph \(f(x)=x^{2}-x-6\) in the standard viewing window. Let \(h(x)=f(x-1000) .\) What should \(h(x)\) look like? (b) Find an appropriate viewing window for the graph of \(h(x)\) (c) Try to find a viewing window that clearly displays both the graph of \(f\) and the graph of \(h .\) What makes this problem difficult? (d) Let \(g(x)=1000 f(x) .\) What should \(g(x)\) look like? (e) Find an appropriate viewing window for the graph of \(g(x) .\) Can you find a viewing window that clearly displays both the graph of \(f\) and the graph of \(g\) ?

Step-by-Step Solution

Answer

Answer: The difficulties in finding a viewing window that displays both graphs of different functions derived from a main function arise from the significant differences in their vertical and horizontal scales. In the given case, \(h(x)\) has a 1000-unit horizontal shift, making it difficult to view both \(f(x)\) and \(h(x)\) clearly in a single viewing window without squeezing the graphs. Meanwhile, \(g(x)\) has a vertical stretch by a factor of 1000, resulting in a graph with much larger y-values than \(f(x)\). Attempting to display both \(f(x)\) and \(g(x)\) in a single viewing window would likely render the \(f(x)\) graph as a flat line or not visible at all due to the dominating presence of the \(g(x)\) graph. These difficulties are further exacerbated by the small screen size of calculators or graphing tools.

Step 1: Graphing \(f(x)\)

To graph \(f(x)=x^{2}-x-6\), use a calculator or graphing tool in the standard viewing window.

Step 2: Finding the appearance of \(h(x)\)

Now let's determine what \(h(x)=f(x-1000)\) should look like. Since it is a transformation of the original function with a horizontal shift, we can expect \(h(x)\) to have the same shape and features as \(f(x)\), but with the entire function shifted 1000 units to the right. ##Part (b) Find the appropriate viewing window for the graph of \(h(x)\).

Step 3: Appropriate viewing window for \(h(x)\)

Given the observation in part (a) that \(h(x)\) is a horizontal shift of \(f(x)\), we can focus on finding the viewing window that highlights the similarities between the two graphs. Since \(h(x)\) is 1000 units to the right, an appropriate viewing window might be something like: \(x_{min} = 980\), \(x_{max} = 1020\), \(y_{min} = -50\), and \(y_{max} = 50\). ##Part (c) Find a viewing window that displays both graphs of \(f(x)\) and \(h(x)\).

Step 4: Attempting to view both graphs

To attempt viewing both graphs, we need to find a viewing window that covers x-values from the left-most portion of \(f(x)\), to the right-most portion of \(h(x)\). Try setting the viewing window as follows: \(x_{min} = -20\), \(x_{max} = 1020\), \(y_{min} = -50\), and \(y_{max} = 50\).

Step 5: Difficulty in viewing both graphs

The problem with this viewing window is that it squeezes the graphs due to the difference in x-values (with the 1000-unit shift). The small screen size makes it difficult to clearly see and compare both functions' graphs simultaneously. ##Part (d) Determine the appearance of \(g(x)\).

Step 6: Appearance of \(g(x)\)

The function \(g(x)=1000f(x)\) has the same shape as \(f(x)\), but amplified vertically by a factor of 1000. The points of the graph will have their y-values multiplied by 1000, resulting in a more "stretched" graph compared to \(f(x)\). ##Part (e) Find an appropriate viewing window for \(g(x)\), and try to display both \(f(x)\) and \(g(x)\) together.

Step 7: Appropriate viewing window for \(g(x)\)

Given the inspection that \(g(x)\) is a vertical stretch of \(f(x)\), we can focus on finding the viewing window that highlights this change in amplitude. With the y-values being multiplied by 1000, an appropriate viewing window might be: \(x_{min} = -10\), \(x_{max} = 10\), \(y_{min} = -50000\), and \(y_{max} = 50000\).

Step 8: Attempting to view both \(f(x)\) and \(g(x)\) graphs

To attempt viewing both graphs, we need to find a viewing window that accommodates the maximum y-values of both functions. However, due to the drastic difference in their y-range, it is nearly impossible to display both graphs clearly on a calculator's small screen. The \(g(x)\) graph will dominate the viewing window, and the \(f(x)\) graph will appear as a flat line or not visible at all.

Key Concepts

Viewing Window for Graphs

Choosing the right viewing window is essential to visualize the behavior of a function graphically, especially on the limited display of a calculator. A viewing window determines the range of x and y values displayed, essentially zooming in or out of a graph to better capture the important features of a function.

For instance, in graphing the function f(x) = x^2 - x - 6, the standard viewing window may reveal where the function intersects the x-axis and its vertex. However, if we apply a transformation, such as a horizontal shift, as seen with h(x) = f(x-1000), the function moves 1000 units to the right, necessitating a change in our viewing window to x_{min} = 980, x_{max} = 1020, y_{min} = -50, and y_{max} = 50 to observe similar features.

Attempting to find a single viewing window to capture both f(x) and the shifted h(x) can prove challenging due to the vast horizontal separation. A significant widening of the viewing window, such as from x_{min} = -20 to x_{max} = 1020, flattens the graphs visually, making comparative analysis on a small screen difficult.

Horizontal Shift in Functions

A horizontal shift occurs when every point of a function is moved left or right along the x-axis. This is often represented as f(x - h), where h is the number of units the function is shifted. Positive h shifts the graph to the right while negative h shifts it to the left.

In our example, the function h(x) = f(x-1000) is f(x) shifted 1000 units to the right. On a graph, this shift doesn't affect the shape or orientation of the curve; it simply translates the entire function horizontally. To visualize such a large shift on a graphing calculator, an expanded viewing window is necessary, emphasizing the new location of important features like intercepts and the vertex.

Vertical Stretch in Graphs

A vertical stretch is a transformation that affects the y-values of a function, causing the graph to appear elongated or compressed vertically. This is often represented by multiplying the function by a factor a, as in g(x) = af(x). A factor greater than 1 stretches the graph, while a factor between 0 and 1 compresses it.

In the exercise, the transformation g(x) = 1000f(x) indicates a vertical stretch by a factor of 1000. This increases the y-values so dramatically that viewing both g(x) and the original f(x) on the same screen becomes impractical. A viewing window such as x_{min} = -10, x_{max} = 10, y_{min} = -50000, and y_{max} = 50000 might be appropriate for g(x) alone, but it would render f(x) nearly invisible, illustrating the difficulties when dealing with drastic vertical stretches.