Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window. Zoom out far enough to see the right-hand and left-hand behavior of each graph. Do the graphs of \(f\) and \(g\) have the same right-hand and Ieft- hand behavior? Explain why or why not. \(f(x)=-2 x^{3}+4 x^{2}-1, \quad g(x)=2 x^{3}\)

Step-by-Step Solution

Answer

No, the graphs of \(f\) and \(g\) do not have the same right-hand and left-hand behaviors. As x approaches positive infinity, \(f(x) = -2x^3 + 4x^2 -1\) decreases without bound (tends to negative infinity) while \(g(x) = 2x^3\) increases without bound (tends to positive infinity). Meanwhile, as x approaches negative infinity, \(f(x) = -2x^3 + 4x^2 -1\) increases without bound (tends to positive infinity) while \(g(x) = 2x^3\) decreases without bound (tends to negative infinity).

Step 1: Graph function f

To graph the function \(f(x) = -2x^3 + 4x^2 -1\), input the given formula into a graphing utility. Ensure the scale on the x-axis and y-axis is set wide enough to see the behavior as x approaches both negative infinity and positive infinity.

Step 2: Graph function g

Similarly, graph the function \(g(x) = 2x^3\) using the graphing utility. Superimpose this graph on top of the graph from step 1 to compare the two functions visually in the same viewing window.

Step 3: Compare right-hand behavior

To analyze the right-hand behavior, observe how each function behaves as x approaches positive infinity. The function \(f(x) = -2x^3 + 4x^2 - 1\) should decrease without bound (negative infinity) because it is a cubic function with a negative leading coefficient. On the other hand, the function \(g(x) = 2x^3\) should increase without bound (positive infinity) as it is a cubic function with a positive leading coefficient.

Step 4: Compare left-hand behavior

For the left-hand behavior, observe how each function behaves as x approaches negative infinity. Just like with the right-hand behavior, \(f(x) = -2x^3 + 4x^2 -1\) should increase without bound (positive infinity) and \(g(x) = 2x^3\) should decrease without bound (negative infinity).

Key Concepts

Graphing Functions

Graphing functions is a fundamental skill in understanding the visual representation of algebraic equations. When graphing a function like f(x) = -2x^3 + 4x^2 -1, which appears in our exercise, one must consider the shape and curvature of the graph across different values of x. Graphing utilities, such as calculators or computer software, can help us plot the points (x, f(x)) to reveal patterns and behaviors of the function.

For students aiming to improve, using a consistent scale and viewing window on your graphing utility is critical. Ensure that your viewing window extends sufficiently in both the positive and negative directions along the x-axis to capture the end behavior, which reveals important characteristics of the function. By doing so, you can more accurately compare and contrast functions by visually assessing their similarities and differences.

Right-Hand Behavior

Right-hand behavior is a term used to describe how a function behaves as the input value, or x, becomes very large or approaches positive infinity. In the context of our functions f(x) and g(x), we're asked to graph them and observe their behavior in the positive direction. For instance, the cubic function with a positive leading coefficient, g(x) = 2x^3, will rise indefinitely as x increases, indicating that the right-hand side of the graph extends upwards.

Understanding the right-hand behavior is essential for predicting the long-term trends of a function, such as population growth in biology or the limit of a sequence in mathematics. When studying these functions independently, it's important to recognize how their leading coefficients influence their trajectories as x grows larger.

Left-Hand Behavior

Conversely, left-hand behavior examines a function as x approaches negative infinity. This perspective is just as crucial when comparing functions, like what's asked in our exercise with f(x) and g(x). It is the reflection of the right-hand behavior but looking in the opposite direction along the x-axis.

For the function f(x) = -2x^3 + 4x^2 - 1, with its negative leading coefficient of -2, as x decreases, the value of f(x) increases – meaning the left-hand side of the graph extends upwards. On the other hand, for g(x), the left-hand side descends. Recognizing these trends is valuable in subjects such as physics for understanding the motion under constant acceleration or economics for modeling decreasing returns.