Problem 47 - Precalculus : Building Concepts and Connections

Question

Graph the polynomial function using a graphing utility. Then (a) approximate the \(x\) -intercept(s) of the graph of the function; (b) find the intervals on which the function is positive or negative; (c) approximate the values of \(x\) at which a local maximum or local minimum occurs; and (d) discuss any symmetries. $$f(x)=-x^{3}+3 x+1$$

Step-by-Step Solution

Answer

The x-intercepts would approximately be -2.62, -0.79 and 1.41. The function would be positive in the interval (-2.62, -0.79) and (1.41, ∞) and negative otherwise. Local maxima occurs at x=-0.79 and local minima occurs at x=1.41. Lastly, \(f(x)=-x^{3} + 3x + 1\) is an odd function because it is symmetric about the origin.

Step 1: Graphing the Polynomial Function

To begin with, input the function \(f(x)=-x^3 + 3x + 1\) into a graphing utility tool like Desmos or Geogebra in order to get the graph of the polynomial function.

Step 2: Approximating the x-intercept(s)

The x-intercept(s) of the graph signifies the x-value(s) where the graph of the function intersects or touches the x-axis. Observing from the graph, approximate this x-value at which the y-value is zero.

Step 3: Determine the Intervals Where the Function is Positive or Negative

In the graph, you can deduce where the function is positive or negative. This is done by noting the x values where the function is above (positive) or below (negative) the x-axis.

Step 4: Approximating the local Minimum or Maximum

Study the points at which the graph has a peak or a trough. These represent local maximum or minimum points respectively. At these points, the function changes from increasing to decreasing or vice versa. Approximate these x-values by studying the graph.

Step 5: Identifying Symmetries

Finally, consider whether the function exhibits any symmetries – that is, if the function is symmetric about the y-axis (making it an even function), or if the function reflects symmetry about the origin (making it an odd function). In this case, \(f(x)=-x^3 + 3x + 1\) is an odd function given that it is symmetrical about the origin. You can realize this by noting that functions in the form \(f(x)=x^n\), where \(n\) is an odd number, are symmetrical about the origin.

Key Concepts

X-Intercepts

Understanding the x-intercepts of a polynomial function is essential because it reveals where the graph intersects the horizontal axis. In our example, the function is \(f(x) = -x^3 + 3x + 1\). To find the x-intercepts, we set the function equal to zero: \( -x^3 + 3x + 1 = 0 \). While this can be challenging to solve algebraically, using a graphing utility can provide a visual approximation. The points where the graph crosses the x-axis are the x-intercepts. In simpler cases, one might be able to factor the equation or apply the Rational Root Theorem to find intercepts exactly, but estimation is often necessary for complex polynomials.

When providing this explanation to students, it's beneficial to clarify that each x-intercept represents a real root of the polynomial. Highlighting how the graph intersects the x-axis can help visually inclined learners understand the concept of roots more intuitively.

Intervals of Positivity and Negativity

A polynomial function's y-values tell us whether the function is positive or negative over different intervals. As you graph \(f(x) = -x^3 + 3x + 1\), notice where the function lies above the x-axis (indicating positivity) and where it dips below (indicating negativity). To communicate this concept, we emphasize that the function's value is positive when the graph is in the upper region of the Cartesian plane and negative when it's in the lower region.

• If \(f(x) > 0\), the function is positive within that interval.
• If \(f(x) < 0\), the function is negative within that interval.

It is essential to ensure students understand that these intervals can vastly impact the behavior of the function, affecting computations in calculus such as integrations and optimizations.

Local Maxima and Minima

Local maxima and minima are high and low points on the graph of a function, respectively. They mark where a function changes its direction of increase or decrease. For \(f(x) = -x^3 + 3x + 1\), the graph will show a wavelike motion because of its cubic nature, and where it peaks or dips represent the local extrema. These points can be critical for understanding the overall shape and behavior of a polynomial function.

When explaining to students, we stress that local maxima and minima are not necessarily the absolute highest or lowest points of the function but are localized to their surrounding area. To identify these by graphing, look for points where the slope of the function appears to level out to a flat line momentarily before changing its slope direction.

Function Symmetry

A function's symmetry refers to its invariance under certain transformations, like reflection or rotation. For polynomial functions, we often look for two main types of symmetry: even and odd symmetry. Even symmetry means the function is unchanged when \(x\) is replaced by \( -x \) (symmetric about the y-axis), while odd symmetry means the function is the negation of itself when \(x\) is replaced by \( -x \) (symmetric about the origin).

The given cubic function \(f(x) = -x^3 + 3x + 1\) exhibits odd symmetry. It's crucial to explain to students that not all functions will display symmetry, but when they do, it can simplify calculations and offer insights into the algebraic structure of the function. For instance, an odd function integrates to zero over an interval symmetric about the origin, which is a valuable property in calculus.