Problem 83 - Precalculus with Limits

Question

Graphical Analysis Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs. (a) \(y_{1}=2^{x}, y_{2}=x^{2}\) (b) \(y_{1}=3^{x}, y_{2}=x^{3}\)

Step-by-Step Solution

Answer

In both cases, the exponential functions grow considerably faster than the quadratic and cubic functions. Although these polynomial functions also grow with increasing x-values, their rise is slower. Furthermore, while y_1 in both cases always remains positive, the cubic function in case (b) can also yield negative y-values. However, quadratic and cubic functions exhibit a zero point, whereas exponential functions do not.

Step 1: Use a Graphing Utility

Use a graphing utility to plot the functions. Two functions will be graphed in each case: y_1=2^x and y_2=x^2, as well as y_1=3^x and y_2=x^3.

Step 2: Comparison of Graphs

Interpret the graphs and make precise comparisons. Differences and similarities should be noted.

Step 3: Graphical Analysis: Case (a)

For the case (a) set of functions: y_1=2^x, y_2=x^2, y_1 is an exponential function and y_2 is a quadratic function. The exponential function changes more rapidly as the value of x increases, resulting in a steeper slope. The quadratic function includes a minimum point, at x=0, after which it starts to increase, but at a slower rate than the exponential function.

Step 4: Graphical Analysis: Case (b)

For the case (b) set of functions: y_1=3^x, y_2=x^3, y_1 is an exponential function and y_2 is a cubic function. Similar to the case (a), the exponential function increases more rapidly than the cubic function. However, unlike the case (a), the cubic function also includes negative y-values (for x<0), while for y_1 function, y is always positive.

Key Concepts

Interpreting Function Graphs

The ability to interpret function graphs is an invaluable skill when studying mathematics. By analyzing the shape, direction, and intersection points of graphs, one can derive a wealth of information about the function's behavior. When comparing exponential and polynomial functions, as in the given exercise, one should address several aspects:

Rate of Change

Notice how the exponential function's rate of change increases drastically compared to the polynomial, which changes more steadily or with predictable intervals of increase and decrease.

Intersection Points

Identifying where graphs intersect with the axes or each other reveals solutions to equations and system of equations. In the example provided, one may observe that the exponential and polynomial functions intersect at specific points, hinting at common solutions.

Asymptotic Behavior

Exponentials have horizontal asymptotes, which represent values the function approaches but never reaches. Polynomials, depending on their degree, have different end behaviors but do not possess horizontal asymptotes.

In practical terms, students should take advantage of graph features such as tracing, which allows following the path of the function to determine exact values for any point, and should also experiment with varying the scale to better understand the graph's behavior in different ranges. By mastering these skills, students become much better equipped to tackle a variety of problems in calculus, physics, and beyond.