Problem 31 - Algebra 2

Question

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=x^{5}-6 x^{4}+4 x^{3}+17 x^{2}-5 x-6 $$

Step-by-Step Solution

Answer

Graph by plotting points; Real zero between identified integers; Estimate turning points on the graph.

Step 1: Create a Table of Values

To graph the function, we first need to make a table of values for the function \( f(x) = x^5 - 6x^4 + 4x^3 + 17x^2 - 5x - 6 \). Choose several values of \( x \) to get the corresponding \( f(x) \) values. It is usually helpful to choose both positive and negative integers for \( x \). For example, you can calculate \( f(x) \) for \( x = -2, -1, 0, 1, 2, 3, 4, 5 \).

Step 2: Plot the Points and Graph the Function

Use the table of values from Step 1 to plot the points on a coordinate plane. Connect the points smoothly to sketch the graph of the polynomial function. Since it is a degree 5 polynomial, the graph will have up to 4 turning points.

Step 3: Determine Consecutive Integer Values for Real Zeros

To find the zeros of the function, observe where the function changes sign from positive to negative or vice versa in the table of values. Identify the consecutive integer values of \( x \) between which a sign change occurs—this indicates a real zero is located between these \( x \)-values. For example, if \( f(x) \) changes from negative to positive between \( x = 2 \) and \( x = 3 \), then there is a real zero between these values.

Step 4: Estimate the Relative Maxima and Minima

Identify any turning points (where the graph changes direction) from the plotted points or graph. These points are where relative maxima or minima occur. Estimate the \( x \)-coordinates based on the graph where these high and low points appear. These estimates are typically in between the values where the function's growth or decline rate changes noticeably.

Key Concepts

Graphing Polynomials

Graphing polynomial functions is a useful way to understand the behavior of these mathematical expressions. When graphing a polynomial such as \( f(x) = x^5 - 6x^4 + 4x^3 + 17x^2 - 5x - 6 \), you start by creating a table of values. Choose a set of \( x \) values ranging on either side of the origin. This might include numbers like \(-2, -1, 0, 1, 2, 3, 4,\) and \( 5 \), to generate corresponding \( f(x) \) results.

• Use positive and negative integers to capture different sections of the graph.
• Calculate for each \( x \) and list its \( f(x) \) value.

Once you plot these points onto a graph and connect them, you’ll see the characteristic wavy line of a polynomial. The smooth curves and sharp turns tell a story about how the values of \( x \) interact with each other within the polynomial function. Since this particular polynomial is of degree 5, expect the graph to have up to 4 distinct turns.

Finding Zeros of Polynomials

The zeroes of a polynomial function are the values of \( x \) for which the polynomial equals zero. These are essentially the points where the graph of the polynomial crosses the x-axis.

• To locate these, look for sign changes between consecutive \( x \) values in your table.
• For instance, if \( f(x) \) changes from negative at \( x = 2 \) to positive at \( x = 3 \), a zero lies between those numbers.

This indicates there's a root or zero between \( x = 2 \) and \( x = 3 \). With polynomial functions, such observations come from the continuity principle, where a function must hit zero if it crosses the x-axis.

Relative Maxima and Minima

Relative maxima and minima are key points on a polynomial's graph, where the function changes its direction from increasing to decreasing or vice versa. By identifying these turning points analytically or visually, you'll understand the behavior of the polynomial.

• These occur where the slope of the function is zero, which means the tangent to the graph is horizontal at these points.
• You can estimate these from your plot by checking where the graph changes direction.

These points don't have to be the absolute highest or lowest points on the graph but are the highest or lowest in their immediate area. Finding them helps to predict the function's behavior in specific intervals.

Degree of a Polynomial

The degree of a polynomial gives insight into its structure and potential behavior. In essence, the degree indicates the highest power of \( x \) present in the polynomial expression. For the given polynomial \( f(x) = x^5 - 6x^4 + 4x^3 + 17x^2 - 5x - 6 \), the degree is 5.

• The degree tells us the maximum number of real zeros the polynomial can have, which is equal to the degree itself.
• It also predicts the total number of turning points in the graph—usually one fewer than the degree because the graph can change directions a maximum of \( n-1 \) times.

Understanding the degree helps us estimate the graph's end behavior; for odd degrees, the graph will extend in opposite directions at each end, implying one tail goes to infinity while the other goes to negative infinity.