Problem 24 - A Graphical Approach to College Algebra

Question

Graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. Over the interval specified, this function is __________. $$f(x)=-x^{3} ;(-\infty, \infty)$$

Step-by-Step Solution

Answer

Over the interval \((-\infty, \infty)\), this function is decreasing.

Step 1: Function Analysis

The function given is \( f(x) = -x^3 \). This is a cubic function with a negative leading coefficient. Cubic functions typically have an 'S' shape, but since the leading coefficient is negative, the 'S' shape is inverted.

Step 2: Graph the Function

Graph \( f(x) = -x^3 \) using your calculator. Enter the function formula in the calculator's function menu, and set the viewing window to the standard default, which is typically from \( x = -10 \) to \( x = 10 \) and \( y = -10 \) to \( y = 10 \) or similar.

Step 3: Examine the Graph

Observe the graph of \( f(x) = -x^3 \) in the calculator's standard window. Notice how the curve starts from the top left, moves through the origin, and inches downwards to the bottom right.

Step 4: Determine the Behavior over the Interval

Since we need to describe the behavior of the function over the interval \((-\infty, \infty)\), we observe that as \(x\) increases (from left to right), \( f(x) \) decreases.

Step 5: Fill in the Blank

Based on the graph analysis, over the interval \((-\infty, \infty)\), the function \( f(x) = -x^3 \) is decreasing.

Key Concepts

Function Graphing

Function graphing can be a powerful way to visualize and understand the behavior of a mathematical function. When graphing a cubic function like \( f(x) = -x^3 \), we typically begin by identifying key characteristics, such as the degree of the polynomial and the sign of the leading coefficient. These features offer insights into the graph's overall shape.

Cubic functions generally have an 'S' shaped curve. This particular graph begins by searching for the position of key points and structure. Setting up the graph on a calculator involves inputting the function formula \( f(x) = -x^3 \) and defining a viewing window. A standard window might range from \( x = -10 \) to \( x = 10 \) and \( y = -10 \) to \( y = 10 \) or similar. This farmework provides a clear domain and range to view the entire graph.

Once graphed, it's crucial to trace the function visually or with calculator tools. This helps you comprehend the direction in which the function moves and how it behaves over different intervals. Cubic graphs like this one are essential for understanding how changes to the equation influence its graphical representation.

Negative Leading Coefficient

A negative leading coefficient in a polynomial, such as in \( f(x) = -x^3 \), profoundly impacts the shape and direction of the graph. Where typical cubic functions appear as 'S' curves, this negative leading coefficient flips the curve vertically. Instead of starting from the bottom left and moving to the top right, it begins from the top left and descends to the bottom right.

Understanding how a negative leading coefficient modifies the graph allows for quicker predictions about function behavior without needing to graph every instance.

• The graph inverts, or "flips," compared to when the leading coefficient is positive.
• This inversion means that as \( x \) increases, the function value, \( f(x) \), decreases.

Paying attention to signs of coefficients assists in predicting function traits quickly and comprehensively, which is crucial for effective function analysis.

Function Behavior over Interval

Analyzing the behavior of a function over an interval involves evaluating how \( f(x) \) changes as \( x \) progresses. For \( f(x) = -x^3 \), which covers the entire real number line \((-\infty, \infty)\), the entire graph needs consideration.

In this instance, as \( x \) moves from negative infinity to positive infinity, \( f(x) \) exhibits a decreasing trend. Consequently, moving left to right across the graph:

• Function values drop continuously.
• There are no turning points that change the increasing or decreasing trend.

Recognizing that a function is decreasing over an entire interval provides insight into its rate of change.

Thus, such analysis not only describes what happens visually on an interval but also indicates potential applications where function decrease or increase is essential.