Question

Plot the graph of the function \(f\) in (a) the standard viewing window and (b) the indicated window. $$ f(x)=x^{3}-20 x^{2}+8 x-10 ; \quad[-20,20] \times[-1200,100] $$

Short answer

In summary, to plot the given function \(f(x)=x^3-20x^2+8x-10\), we first analyzed the function to understand its behavior, created a table of values, and plotted the graph in both the standard viewing window and the indicated viewing window, \([-20,20] \times [-1200,100]\). The graph of the cubic function shows a local maximum and a local minimum, giving us a better understanding of how the function behaves in different viewing windows.

Step-by-step solution

Analyze the function

The given function is a cubic function with the general form \(f(x) = ax^3 + bx^2 + cx + d\), where \(a=1, b=-20, c=8\) and \(d=-10\). Since the leading coefficient a is positive, we know that the graph will end up going to positive infinity as x increases, and will end up going to negative infinity as x decreases. We can analytically find the roots of the function but it's not required for graphical representation.

Create a table of values

To create a table of values, start by selecting a range of x-values in the domain of the function, and then calculate the corresponding y-values using the function \(f(x)\): x | f(x) ----|------ -20 | -6700 -15 | -2675 -10 | -470 -5 | 795 0 | -10 5 | -885 10 | -190 15 | 1935 20 | 6180 (Note that more values can be added to the table to improve the precision of the graph.)

Step 3a: Plot the graph in the standard viewing window

Using a graphing tool (like a graphing calculator or software), enter the function \(f(x)=x^3-20x^2+8x-10\) and set the viewing window to its default, which usually displays x-values from -10 to 10 and y-values from -10 to 10. Plot the graph using the table of values from step 2. You should see a curve that changes direction twice, with a local maximum and a local minimum.

Step 3b: Plot the graph in the indicated viewing window

Now, adjust the viewing window to the specified range of x-values ([-20, 20]) and y-values ([-1200, 100]). Plot the graph using the same table of values from step 2. You should see the graph of the cubic function more accurately with the adjusted viewing window, showing the same features such as the local maximum and minimum but also giving a better understanding of how the function behaves in the specified range.

Key concepts

Cubic Function Analysis

Understanding the behavior of cubic functions is essential for graphing them accurately. A cubic function is expressed as f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are real numbers, and a ≠ 0. The term ax^3 determines the function's end behavior, meaning how the function behaves as x approaches positive or negative infinity.

When analyzing a cubic function such as f(x) = x^3 - 20x^2 + 8x - 10, we look at its coefficients to predict its graph's overall shape. Since a = 1 is positive, this particular function will rise to positive infinity as x goes to infinity, and fall to negative infinity as x goes to negative infinity. This information gives us a starting point for sketching the general direction of the graph.

Other features to look for include the locations of any local maxima or minima, which occur where the slope of the function changes from increasing to decreasing, or vice versa. These can often be found by calculating the derivative of the function and finding its zeros, although this may not be necessary for a basic sketch. The x-intercepts (where the function equals zero) can be found by solving for when f(x) = 0. For more complex cubic functions, using calculus to find inflection points where the concavity changes can provide additional insight into the graph's shape.

Creating a Table of Values

An effective way to start graphing a cubic function is by creating a table of values. This process involves choosing specific x-values, plugging them into the function, and calculating the corresponding y-values (outputs).

To illustrate, with the function f(x) = x^3 - 20x^2 + 8x - 10, you would select a range of x-values that effectively capture the behavior of the function. It's wise to include x-values where significant changes occur, such as turns in the graph or points of intersection with the axes.

Choosing Values

Select values strategically to display key features of the graph, like where the curve bends or crosses the axis. Also, include values around any points where the function's value is zero (roots or x-intercepts) and include regular intervals to capture the shape between these features.

Calculating Outputs

Using the function, calculate f(x) for each selected x value. These calculations give you points that you can plot on a coordinate plane. Efficiently constructed, this table can reveal the nuances of the cubic function's curve and even disclose patterns, like symmetry or repeating intervals, which further simplify graphing.

Adjusting the Viewing Window

To accurately graph a function, particularly one with a broad range of values like a cubic function, it's imperative to adjust the viewing window on your graphing tool. The viewing window controls which part of the coordinate plane you're currently observing.

When you first input the function f(x) = x^3 - 20x^2 + 8x - 10 into a graphing calculator or software, the default viewing window might not display all important features of the graph. Initially, it’s likely set to show only a small portion of the x and y axes, such as x-values and y-values between -10 and 10. In this state, important characteristics like turning points or intercepts might be outside the visible area.

To rectify this, adjust the window's range based on the values determined to be significant from your analysis or table of values. For example, the given range [-20, 20] × [-1200, 100] was chosen to ensure that the curve's major features are visible. By setting the x-range from -20 to 20 and the y-range from -1200 to 100, the graph's behavior is fully within view. This custom viewing window allows for a comprehensive visualization of the function, highlighting key areas such as where the function increases or decreases, where it reaches local maximums or minimums, and where it crosses the x-axis.