Problem 15 - 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. $$ f(x)=-3 x^{3}+20 x^{2}-36 x+16 $$

Step-by-Step Solution

Answer

Zeros are between \(x=1\) to \(x=2\), and \(x=3\) to \(x=4\); maximum at \(x=3\).

Step 1: Make a Table of Values

To graph the function, we start by selecting a range of integer values for \(x\). For each selected \(x\), we calculate the corresponding \(f(x)\). Let's choose values of \(x\) from -1 to 6. Calculations: - \(f(-1) = -3(-1)^3 + 20(-1)^2 - 36(-1) + 16 = 3 + 20 + 36 + 16 = 75\) - \(f(0) = -3(0)^3 + 20(0)^2 - 36(0) + 16 = 16\) - \(f(1) = -3(1)^3 + 20(1)^2 - 36(1) + 16 = -3 + 20 - 36 + 16 = -3\) - \(f(2) = -3(2)^3 + 20(2)^2 - 36(2) + 16 = -24 + 80 - 72 + 16 = 0\) - \(f(3) = -3(3)^3 + 20(3)^2 - 36(3) + 16 = -81 + 180 - 108 + 16 = 7\) - \(f(4) = -3(4)^3 + 20(4)^2 - 36(4) + 16 = -192 + 320 - 144 + 16 = 0\) - \(f(5) = -3(5)^3 + 20(5)^2 - 36(5) + 16 = -375 + 500 - 180 + 16 = -39\) - \(f(6) = -3(6)^3 + 20(6)^2 - 36(6) + 16 = -648 + 720 - 216 + 16 = -128\) Plot these points on a graph to visualize the function.

Step 2: Determine Consecutive Integer Values for Real Zeros

The real zeros of the function are the \(x\)-values where \(f(x) = 0\). By checking the table values from Step 1, observe that: - Between \(x = 1\) and \(x = 2\), the function changes from -3 to 0 (indicating a zero). - Between \(x = 3\) and \(x = 4\), the function changes from 7 to 0 (indicating another zero).Thus, there are zeros between 1 and 2, and between 3 and 4.

Step 3: Estimate Relative Maxima and Minima

Relative maxima and minima occur where the rate of change (slope) of the function changes sign. Check the table: - From \(f(x)\) values, it appears the function increases from \(x=1\) to \(x=3\) and decreases afterward. Thus, there is a relative maximum around \(x=3\).- Additionally, since \(f(x)\) decreases after \(x=3\) and alters a zero at \(x=4\), a potential relative minimum might be slightly after \(x=3\) but is not distinctly apparent with our given table values.

Key Concepts

Graphing Functions

Graphing a function essentially involves creating a visual representation of how the function behaves for various inputs. This exercise starts by choosing a range of integer values as sample inputs for the variable, called the domain. Based on these, the corresponding outputs or function values are calculated, helping create a series of points where the x-coordinate is the chosen input and the y-coordinate is the function's output.

When we graph the function, these points are plotted, forming the curve that represents the entire function. For the given cubic function, values of \(x\) from -1 to 6 are chosen, yielding outputs that describe the function's behavior.

• Understanding the shape: Cubic functions like \(f(x) = -3x^3 + 20x^2 - 36x + 16\) tend to have points where they twist or curve sharply, which can be seen clearly in the plotted graph.
• Sketching the curve: You will connect these plotted values by a smooth curve, noting any points where the curve changes direction sharply, such as around relative maxima or minima.

Graphing helps visualize these characteristics and understand the overall function's behavior.

Zeros of a Function

Zeros of a function are critical points where the graph intersects the x-axis, indicating that the output value \(f(x)\) is zero. They are significant because they signify the solutions to the equation \(f(x) = 0\). By examining the table of values computed, we can identify intervals where the function changes from positive to negative or vice versa, which indicates the presence of a zero.

• Locating zeros: For the function \( f(x) = -3x^3 + 20x^2 - 36x + 16 \), we observe a change from \(f(1) = -3\) to \(f(2) = 0\), and from \(f(3) = 7\) to \(f(4) = 0\). This suggests zeros exist between \(x=1\) and \(x=2\), and between \(x=3\) and \(x=4\).
• Significance of zeros: These points are crucial for understanding the roots of the polynomial and for sketching the function accurately.

Knowing where zeros occur can also help in optimizing and maximizing conditions in various applied mathematical situations.

Relative Maxima and Minima

Relative maxima and minima are points on a graph where the function experiences a peak or a trough, respectively. These points are characterized by a change in the direction of the function's slope. Identifying these points can give deep insights into the function's local behavior.

• Relative maxima: In the provided function, the output values increase from \(x=1\) to \(x=3\) then decrease, suggesting a peak (or relative maximum) near \(x=3\). This means \(x=3\) is a point where the function briefly reaches a high point before dipping again.
• Relative minima: After this maxima, the function appears to change direction again as it approaches another zero around \(x=4\). However, without a smooth decrease apparent from the values, this minimum is less pronounced in the given range.

Recognizing relative extrema helps not just in plotting graphs, but also in understanding the evolution of a function's outputs and predicting future outcomes based on current trends in fields like economics or physics.