Problem 17 - 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. $$ f(x)=x^{4}-8 $$

Step-by-Step Solution

Answer

The real zeros are between \(-2\) and \(-1\), and \(1\) and \(2\); the relative minimum is at \(x = 0\).

Step 1: Create a Table of Values

To graph the function, first create a table of values by selecting specific values for \( x \) and calculating the corresponding \( f(x) \) values. Choose values like \( x = -3, -2, -1, 0, 1, 2, 3 \) and compute each \( f(x) \). For instance, for \( x = 0 \), \( f(0) = 0^4 - 8 = -8 \).

Step 2: Plot the Points and Draw the Graph

Using the table from Step 1, plot the points \( (x, f(x)) \) on a coordinate plane. For this function, the points will create a curve starting low when \( x \) is negative and increasing as \( x \) becomes positive.

Step 3: Identify the Real Zeros of the Function

The real zeros occur when \( f(x) = 0 \). Setting the equation equal to zero, we have \( x^4 - 8 = 0 \). Solving for \( x \), we find \( x = \pm \sqrt[4]{8} \). These values are roughly between \( x = -2 \) and \( x = -1 \), and \( x = 1 \) and \( x = 2 \).

Step 4: Estimate Relative Maxima and Minima

To find relative maxima and minima, observe the slope of the graph. For a polynomial function like \( x^4 - 8 \), the critical points would be at slopes of zero. By examining the graph or using calculus, identify that these occur symmetrically around the y-axis, specifically at \( x = 0 \), which is a relative minimum with \( f(0) = -8 \). The function has no relative maxima within the range of values typically considered without further analysis.

Key Concepts

Real Zeros

Understanding real zeros in a polynomial function is an important concept. Real zeros are the values of \( x \) for which the function \( f(x) \) equals zero. In simpler terms, they are the points where the graph of the function touches or crosses the x-axis.
To find the real zeros of the function \( f(x) = x^4 - 8 \), we set the equation equal to zero: \ \[ x^4 - 8 = 0 \].
Solving this equation, we find \( x = \pm \sqrt[4]{8} \). This expression signifies two real zeros on the positive side and two on the negative side of the x-axis. These values are approximately between \(-2\) and \(-1\), and \(1\) and \(2\).
Identifying real zeros helps students understand the behavior of the function around these points. It's crucial when sketching the graph, as it informs where the curve will intersect the x-axis.

Relative Maxima and Minima

Relative maxima and minima refer to the highest and lowest points in a particular section of the graph of a function. They help us understand the peaks and valleys along the curve.
To find these points for a polynomial like \( f(x) = x^4 - 8 \), we usually need to look for places where the slope changes from positive to negative, or vice versa, which mathematically means finding the derivative and solving for zero. However, for this specific function, the intuitive examination or the use of graphing can identify that the symmetry around the x-axis suggests \( x = 0 \) as a point of interest.
Once you evaluate \( f(0) = 0^4 - 8 = -8 \), you identify a relative minimum at \( x = 0 \), with no relative maximas within the examined region. Understanding the concept of these extrema plays a vital role in grasping the overall shape and important features of a function's graph.

Table of Values

Creating a table of values is one of the foundation steps in graphing any function, including polynomial functions.
This technique involves selecting several x-values and computing the corresponding y-values, or \( f(x) \), to gain a clearer picture of how the function behaves over a range. For the function \( f(x) = x^4 - 8 \), you can choose points such as \(-3, -2, -1, 0, 1, 2,\) and \(3\).
By calculating and organizing these into a table:

• For \( x = -3 \), \( f(-3) = -3^4 - 8 = -89 \)
• For \( x = -2 \), \( f(-2) = -2^4 - 8 = -24 \)
• For \( x = -1 \), \( f(-1) = -1^4 - 8 = -9 \)
• For \( x = 0 \), \( f(0) = 0^4 - 8 = -8 \)
• For \( x = 1 \), \( f(1) = 1^4 - 8 = -7 \)
• For \( x = 2 \), \( f(2) = 2^4 - 8 = 8 \)
• For \( x = 3 \), \( f(3) = 3^4 - 8 = 73 \)

With this information, you can plot these points on a coordinate grid and draw the curve. This visualization aids in identifying the behavior and trend of the function, such as how it rises or falls, and the general shape of its graph.