Problem 16 - 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. $$ f(x)=x^{3}-4 x^{2}+2 x-1 $$

Step-by-Step Solution

Answer

Zeroes are between \(x = -1\) and \(x = 0\), and between \(x = 3\) and \(x = 4\). Max at \(x = 0\), min at \(x = 3\).

Step 1: Create a Table of Values

Choose a range of integer values for \(x\) (for example, from -2 to 4) and substitute these into the function \(f(x) = x^3 - 4x^2 + 2x - 1\). Calculate the corresponding \(f(x)\) for each \(x\). This provides points \((x, f(x))\) to be plotted on the graph, such as:\[\begin{array}{c|c} x & f(x) \ \hline -2 & -27 \ -1 & -8 \ 0 & -1 \ 1 & -2 \ 2 & -3 \ 3 & -1 \ 4 & 15 \ \end{array}\]

Step 2: Graphing the Function

Using the table of values, plot the points on a coordinate grid. For example, plot the point \((-2, -27)\), \((-1, -8)\), and continue through \((4, 15)\). Connect these points smoothly to show the curve of the function. This forms a cubic graph shape as it has a degree of three.

Step 3: Identify Real Zeros

Examining the plotted graph, check where the curve crosses or touches the x-axis. For this function, it is evident that it crosses between -1 and 0, and again between 3 and 4 based on where 'f(x)' changes sign between values in the computed table. Hence, two segments exist where a real zero is located between these integer \(x\) values.

Step 4: Estimate Relative Maximum and Minimum

On the graph, find the peaks and troughs. By observation, the relative maximum occurs around \(x = 0\), as \(f(x)\) shows an increasing slope turning into a decreasing slope. Similarly, the relative minimum occurs around \(x = 3\), determined by a slope changing from decreasing to increasing. Use values in the table to verify these observations.

Key Concepts

Graphing Functions

Graphing cubic functions is a valuable skill for visualizing complex relationships. When you draw a graph of a function like \( f(x) = x^3 - 4x^2 + 2x - 1 \), you'll get a sense of its behavior over different intervals. Start by creating a table of values. Select a range for \( x \), such as from -2 to 4. Substitute these values into the function and calculate \( f(x) \). This gives you coordinates like \((-2, -27)\) and \((4, 15)\). These coordinates form key points to plot on a graph. When you plot the graph:

• Make sure to use consistent spacing on the axes.
• Plot each point from your table accurately.
• Connect the dots with a smooth curve.

The graph of a cubic function usually has a distinctive curve that rises and falls. Cubic graphs are characterized by a degree of three, which means that the highest power of \(x\) is cubic. This shape helps you see where the function increases or decreases as well as where it changes its behavior.

Polynomial Zeros

Finding polynomial zeros is crucial in understanding the function's x-intercepts. Zeros occur where a function equals zero, meaning the graph crosses the x-axis. For the cubic function \( f(x) = x^3 - 4x^2 + 2x - 1 \), zeros can be identified through a table of values and observation of the graph. When plotted, you find that the graph of \( f(x) \) crosses the x-axis between:

• -1 and 0
• 3 and 4

These intervals show where the function changes sign, from positive to negative or vice versa. Computing \(f(x)\) values at these points confirms if real roots (zeros) exist. Between these values, \( f(x) \) transitions across zero indicating an actual root presence. Often in practice, further methods like synthetic division or the Rational Root Theorem might be required for exact zero calculations, especially if zeros aren’t clear from the graph alone.

Relative Maxima and Minima

To determine a function's behavior, identifying relative maxima and minima on its graph is essential. For cubic functions such as \( f(x) = x^3 - 4x^2 + 2x - 1 \), these points are where the graph reaches a peak (maximum) or a valley (minimum) relative to nearby sections. By examining the connected curve:

• A relative maximum appears around \( x = 0 \). Here, the curve transitions from rising to falling, forming a peak.
• A relative minimum is observed near \( x = 3 \), where the curve shifts from falling to rising, creating a trough.

These points provide insight into the function's local extremes. For validation, construct a table, observing where function values increase or decrease, further confirming these observations. In calculus, derivative tests often support these findings, specifying slopes where maxima or minima occur.