Problem 14 - Algebra 2

Question

For Exercises \(11-18,\) complete each of the following. a. Graph each function 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. $$ f(x)=x^{3}+5 x^{2}-9 $$

Step-by-Step Solution

Answer

Real zero is between 1 and 2. Relative minimum near x=0; relative maximum near x=-2.

Step 1: Create a Table of Values

To graph the function, we start by selecting several consecutive integer values of \( x \). Choose a range from, say, \(-3 \) to \(3\). Then, calculate \( f(x) = x^3 + 5x^2 - 9 \) for each value of \( x \) and organize these calculations in a table.

Step 2: Calculate f(x) for Each x Value

Calculate the function for the chosen \( x \) values:- For \( x = -3 \), \( f(-3) = (-3)^3 + 5(-3)^2 - 9 = -27 + 45 - 9 = 9 \)- For \( x = -2 \), \( f(-2) = (-2)^3 + 5(-2)^2 - 9 = -8 + 20 - 9 = 3 \)- For \( x = -1 \), \( f(-1) = (-1)^3 + 5(-1)^2 - 9 = -1 + 5 - 9 = -5 \)- For \( x = 0 \), \( f(0) = 0^3 + 5(0)^2 - 9 = -9 \)- For \( x = 1 \), \( f(1) = 1^3 + 5(1)^2 - 9 = 1 + 5 - 9 = -3 \)- For \( x = 2 \), \( f(2) = 2^3 + 5(2)^2 - 9 = 8 + 20 - 9 = 19 \)- For \( x = 3 \), \( f(3) = 3^3 + 5(3)^2 - 9 = 27 + 45 - 9 = 63 \)Table:| \( x \) | \( f(x) \) ||-----|--------|| -3 | 9 || -2 | 3 || -1 | -5 || 0 | -9 || 1 | -3 || 2 | 19 || 3 | 63 |

Step 3: Plot the Graph

Using the table of values, plot the graph of \( f(x) = x^3 + 5x^2 - 9 \). Plot the points from the table and connect them smoothly to illustrate the curve of the function.

Step 4: Find Consecutive Integer Values of the Zero

Look at the table and find where \( f(x) \) changes sign, indicating a zero between consecutive integers. In this case, \( f(-1) = -5 \) and \( f(0) = -9 \) are both negative, but \( f(1) = -3 \), which is close to zero but still negative. This suggests there might be a real zero as the graph turns direction between \( x = 1 \) and some \( x > 1 \).

Step 5: Estimate Relative Maxima and Minima Locations

By analyzing the table and graph, identify where the function changes direction, indicating relative maxima or minima. From \( x = -2 \) to \( x = -1 \), \( f(x) \) goes down and continues until \( x = 0 \); it then starts increasing from \( x = 1 \). This suggests a relative minimum near \( x = 0 \). The next analysis shows the potential for a relative maximum near \( x = -2 \) because \( f(x) \) increases from \( x = -3 \) to \( x = -2 \) and decreases to \( x = -1 \).

Key Concepts

Zero of a Function

A zero of a function, also called a root of the function, is any value of the input, often signified as \( x \), that results in the function output being zero. When dealing with polynomial functions like \( f(x) = x^3 + 5x^2 - 9 \), determining where the function equals zero is crucial for graphing and understanding its behavior.

To find the zeros, look at the table of values where the function transitions from positive to negative or vice versa. In this case, examining the table indicates that the function changes direction between \( x = -1 \) and \( x = 0 \), as \( f(-1) = -5 \) and \( f(1) = -3 \); however, it doesn't switch signs here, showing no zero in this region. However, around \( x = 1 \), there are indications of change, guiding further investigation or more precise methods (such as analytical or graphical) to pinpoint the zero location.

Zerof a function play a vital role in graph plotting, where the curve intersects the x-axis, highlighting important features of a polynomial.

Relative Maxima and Minima

In any polynomial graph, relative maxima and minima identify the high and low points within a particular neighborhood or interval. For the function \( f(x) = x^3 + 5x^2 - 9 \), these points can be identified by observing where the slope of the function changes from increasing to decreasing (maxima) or decreasing to increasing (minima).

Using the table of values, you can approximate where these points occur. With this function, from \( x = -2 \) to \( x = -1 \), there is a decrease indicating a transition, suggesting a possible peak (maximum) near \( x = -2 \). From \( x = 0 \) to \( x = 1 \), the function begins to increase again, signaling a trough (minimum), possibly around \( x = 0 \).

Understanding these turning points is important because they help determine the overall shape and behavior of the graph, essential in real-world applications where such peaks and troughs may signify optimal values or levels.

Table of Values

Creating a table of values is a fundamental step in graphing polynomial functions, providing a structured way to understand the function's behavior over a set interval. The table involves choosing specific \( x \) values, calculating corresponding \( f(x) \) values, and organizing these results. This approach offers a clear view of how the function behaves across different points.

For \( f(x) = x^3 + 5x^2 - 9 \), selecting integers from \( -3 \) to \( 3 \) makes it manageable to compute manually:

• At \( x = -3 \), \( f(x) = 9 \)
• At \( x = -2 \), \( f(x) = 3 \)
• At \( x = -1 \), \( f(x) = -5 \)
• At \( x = 0 \), \( f(x) = -9 \)
• At \( x = 1 \), \( f(x) = -3 \)
• At \( x = 2 \), \( f(x) = 19 \)
• At \( x = 3 \), \( f(x) = 63 \)

Once plotted, these points provide a backbone for sketching the curve, illustrating how the function trends and bends, and pointing out zones for zeros and extremum values.

Polynomial Roots

Polynomial roots are synonymous with the zeros of the function—these are key values where the polynomial evaluates to zero. For any polynomial \( f(x) = ax^n + bx^{n-1} + \, \ldots \, + k \), the roots tell us where the graph touches or crosses the x-axis. Recognizing these points is crucial for understanding the graph's shape and behavior.

Polynomials can have real or complex roots, but graphing usually concentrates on real roots. The process involves:

• Setting the polynomial equal to zero: \( f(x) = 0 \)
• Solving for \( x \) to find where these intersection points occur: sometimes possible via factorization, or using the quadratic/cubic formulas for specific cases.
• Observing plotted points to affirm these intersections.

For \( f(x) = x^3 + 5x^2 - 9 \), zero locations might initially be guessed from the graph's behavior and supported with further analytical methods to confirm the precise roots. Knowing these roots can reveal valuable information, such as the function's relation to other mathematical entities or everyday phenomena.