Problem 50 - 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. $$f(x)=-x^{4}+3 x-1$$

Step-by-Step Solution

Answer

Upon the completion of these steps, there will be an estimate of where the x-intercepts are, the intervals where the function is positive or negative, the x-values for local maximum and minimum, and whether the function exhibits any symmetries.

Step 1: Graph the function

Using a graphing utility, plot the function \(f(x)=-x^{4}+3x-1\). Analyze the shape and its intersections with the x-axis, as well as the points where it changes directions.

Step 2: Approximate the x-intercepts

Observe where the graph intersects the x-axis. These points are the x-intercepts. Approximate these values to whenever the function crosses the x-axis. The solution may contain more than one x-intercept.

Step 3: Establish where the function is positive or negative

Identify the intervals where the graph lies above the x-axis (meaning the function is positive), and where it lies below the x-axis (indicating the function is negative).

Step 4: Approximate local maximum and minimum

Identify the points at which the function reaches a peak or a trough - these correspond to local maximum and minimum values respectively. Approximate these x-values by observing the highest and lowest points of the plotted function.

Step 5: Determine Symmetries

Analyze the graph to determine if the function is even (has y-axis symmetry), odd (has origin symmetry), or neither. If the right half of the graph is a mirror image of the left half, the function is even, if the graph is rotated 180 degrees about the origin and remains unchanged, then the function is odd.

Key Concepts

x-intercepts

When graphing polynomial functions, finding the x-intercepts is one of the crucial steps. X-intercepts are the points where the graph of the function crosses the x-axis. At these points, the value of the function, \(f(x)\), is zero because the y-value is zero at the x-intercepts. For the polynomial \(f(x) = -x^4 + 3x - 1\), you need to observe the graph closely to approximate where it intersects the x-axis.

These intersections may not always land on whole numbers, so it's important to use a graphing utility for precise approximation. From the graph, you can find these values visually by noting the points where the curve crosses the axis. A polynomial function of degree four like this one can have up to four x-intercepts. So take a close look at the plotted graph for these important values.

local maxima and minima

Local maxima and minima are points where the function reaches a local peak or trough, respectively. These points are important because at a local maximum, the function stops increasing and starts decreasing. Conversely, at a local minimum, the function stops decreasing and starts increasing.

To approximate these points for \(f(x) = -x^4 + 3x - 1\), look for high and low points on the graph that are not at the endpoints of your graphing window. By carefully analyzing the shape of the curve on your graphing utility, you can identify these peaks and troughs. Remember, a maximum or minimum is "local" if there are higher or lower values elsewhere on the graph.

• Local Maximum: A point where the function is bigger than neighboring points within a small interval.
• Local Minimum: A point where the function is smaller than neighboring points within a small interval.

symmetry in functions

Symmetry in mathematical functions makes graphs easier to read and understand. There are common types of symmetry: y-axis symmetry (even functions) and origin symmetry (odd functions).

For the function \(f(x) = -x^4 + 3x - 1\), check for symmetry by analyzing the graph or using algebraic methods. \(f(x)\) is even if \(f(x) = f(-x)\) and odd if \(f(x) = -f(-x)\).

Look at the graph to see if one-half is a mirror image of the other across the y-axis or rotated 180 degrees about the origin. Identifying symmetry can simplify understanding the nature of the graph and predict its behavior for different values of \(x\). However, this particular function does not show y-axis or origin symmetry; it is neither even nor odd. Each type of symmetry offers insights into the function's characteristics and simplifies the graphing process.

positive and negative intervals

Determining positive and negative intervals involves identifying where the function’s graph is above or below the x-axis. On a graph, where the curve is above the x-axis, the function's values are positive. Where it is below, the values are negative.

For \(f(x) = -x^4 + 3x - 1\), carefully observe the sections of the graph that are above the x-axis, indicating positive intervals. Similarly, find where the graph dips below the x-axis to identify negative intervals. Clearly marking these intervals on a number line can help visualize the function's behavior over different domains.

This step is useful to understand the function’s range and can aid in solving inequality-based problems where you want to know when \(f(x) > 0\) or \(f(x) < 0\). It provides insights into how the function behaves across its domain and helps in deciding the context where each interval will be applied.