Problem 15 - Algebra 2

Question

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 \(f(x)=-3 x^{3}+20 x^{2}-36 x+16\)

Step-by-Step Solution

Answer

The real zeros are at \(x = 2\) and \(x = 4\). The relative maximum is at \(x \approx 3\).

Step 1: Create a Table of Values

To graph the function, we start by selecting a range of x-values and calculating the corresponding y-values (or f(x)) using the given cubic function. Let's choose integer x-values from 0 to 5 for simplicity.- For \( x = 0 \), \( f(0) = -3(0)^3 + 20(0)^2 - 36(0) + 16 = 16 \)- For \( x = 1 \), \( f(1) = -3(1)^3 + 20(1)^2 - 36(1) + 16 = -3 + 20 - 36 + 16 = -3 \)- For \( x = 2 \), \( f(2) = -3(2)^3 + 20(2)^2 - 36(2) + 16 = -24 + 80 - 72 + 16 = 0 \)- For \( x = 3 \), \( f(3) = -3(3)^3 + 20(3)^2 - 36(3) + 16 = -81 + 180 - 108 + 16 = 7 \)- For \( x = 4 \), \( f(4) = -3(4)^3 + 20(4)^2 - 36(4) + 16 = -192 + 320 - 144 + 16 = 0 \)- For \( x = 5 \), \( f(5) = -3(5)^3 + 20(5)^2 - 36(5) + 16 = -375 + 500 - 180 + 16 = -39 \)This table of values will help us sketch the graph of the function.

Step 2: Sketch the Graph

Using the table of values from Step 1, plot the points \((x, f(x))\) on a Cartesian plane: - (0, 16)- (1, -3)- (2, 0)- (3, 7)- (4, 0)- (5, -39)Now, connect these points with a smooth curve, keeping in mind that the function is a cubic polynomial. It should display a single inflection point and have ends that head off to infinity in opposite directions.

Step 3: Identify Real Zeros

Real zeros occur where the graph of the function crosses the x-axis. From the table of values and the graph, we notice:- \( f(2) = 0 \) indicating a zero at \( x = 2 \)- \( f(4) = 0 \) indicating a zero at \( x = 4 \)These values indicate the x-coordinates where real zeros are located, which means there are zeros exactly at \( x = 2 \) and \( x = 4 \).

Step 4: Estimate Relative Extrema

Relative maxima and minima are points where the graph changes direction from increasing to decreasing or vice versa. Looking at our graph:- The function increases to a highest point around \( x = 3 \), this is our relative maximum approximately at \( (3, 7) \).There is no relative minimum from the values computed. The curve decreases from left to right and then, again after \( x = 3 \), decreasing particularly sharply after \( x = 5 \).

Key Concepts

Graphing Polynomials

Graphing cubic functions can be a fascinating journey. Understanding the behavior of these graphs begins with plotting points from a table of values. Start by selecting a range of integers for your x-values. In our example, we chose x-values from 0 to 5. For each x, plug the value into the cubic function to find the corresponding y-value, which is expressed as \( f(x) \). By calculating

• \( f(0) = 16 \)
• \( f(1) = -3 \)
• \( f(2) = 0 \)
• \( f(3) = 7 \)
• \( f(4) = 0 \)
• \( f(5) = -39 \)

you obtain a series of coordinates: \((0, 16), (1, -3), (2, 0), (3, 7), (4, 0), (5, -39)\). Plot these points on a Cartesian plane.
Connect these points with a smooth curve. For a cubic function, expect the curve to have an inflection point and for its ends to extend towards opposite infinities. Cubic graphs often represent polynomial growth and change direction at least once.

Real Zeros

Real zeros of a polynomial occur where the graph intersects the x-axis. These points represent the solutions to \( f(x) = 0 \). In our example with the cubic function, we find real zeros when:

• \( f(2) = 0 \)
• \( f(4) = 0 \)

This tells us that the graph crosses the x-axis at \( x = 2 \) and \( x = 4 \). Determining the real zeros of a cubic function involves looking for intervals where the sign of the y-values transitions from positive to negative or vice versa. This change indicates the presence of a zero.
Graphically, observing where the curve cuts through the x-axis solidifies your understanding, offering a visual guide to these pivotal points.

Relative Extrema

Relative extrema of a cubic function are the peaks and troughs in its graph, known as relative maxima and minima. These points occur where the graph's slope changes from positive to negative (a maximum) or negative to positive (a minimum).
In the given example, there is a relative maximum near \( x = 3 \), where the graph reaches the point \( (3, 7) \). While no clear relative minimum exists in the plotted points, changes around these extrema indicate shifts in the graph's direction. Observing the graph closely can be crucial to identifying such extrema,
as they highlight key transitions in the graph's behavior. Mastering the identification of relative extrema helps uncover the underlayer of polynomial functions, revealing how they fluctuate and stabilize in real-world phenomena.