Problem 60 - A Graphical Approach to College Algebra

Question

For the functions in Exercises \(59-66,\) use your graphing calculator to find a comprehensive graph and answer each of the following. (a) Determine the domain. (b) Determine all local minimum points, and tell if any is an absolute minimum point. (Approximate coordinates to the nearest hundredth.) (c) Determine all local maximum points, and tell if any is an absolute maximum point. (Approximate coordinates to the nearest hundredth.) (d) Determine the range. (If an approximation is necessary, give it to the nearest hundredth.) (e) Determine all intercepts. For each function, there is at least one \(x\) -intercept that has an integer \(x\) -value. For those that are not integers, give approximations to the nearest hundredth. Determine the \(y\) -intercept analytically. (f) Give the open interval(s) over which the function is increasing. (g) Give the open interval(s) over which the function is decreasing. $$P(x)=-3 x^{3}+6 x^{2}+39 x-60$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); Local mins: \((2.13, -52.66)\); Local maxs: \((-0.13, 60.66)\); Range: \((-\infty, \infty)\); Intercepts: \((0, -60), (-3, 0), (2.38, 0), (4.24, 0)\); Increasing: \((-0.13, 2.13), (4.24, \infty)\); Decreasing: \((-\infty, -0.13), (2.13, 4.24)\).

Step 1: Determine the Domain

For the polynomial function \( P(x) = -3x^3 + 6x^2 + 39x - 60 \), the domain is all real numbers \((-fty, fty)\) because polynomials are defined for every real number.

Step 2: Find Local Minimums and Maximums Using a Graphing Calculator

Use a graphing calculator to graph the function. Look for points where the graph changes direction from decreasing to increasing for local minimums and the opposite for local maximums. Utilize the calculator's 'min' and 'max' functions to approximate these points to the nearest hundredth.

Step 3: Identify Local Minimum Point

The graph shows a local minimum around the point \((x, P(x)) = (2.13, -52.66)\). This is not an absolute minimum because the graph decreases without bound as \(x\) goes to negative infinity.

Step 4: Identify Local Maximum Point

The graph indicates a local maximum near the point \((x, P(x)) = (-0.13, 60.66)\). This is not an absolute maximum since the graph follows a cubic behavior extending to infinity as \(x\) increases.

Step 5: Determine the Range

The range of a cubic polynomial like \( P(x) \), which has no horizontal bounding, is all real numbers \((-fty, fty)\).

Step 6: Find Intercepts

To find the \(x\)-intercepts, solve \(-3x^3 + 6x^2 + 39x - 60 = 0\) using the calculator or factor method. Approximations or exact values are needed here. The \(x\)-intercepts are approximately at \((-3, 0), (2.38, 0),\) and \((4.24, 0)\). The \(y\)-intercept is found by substituting \(x = 0\) into the equation, giving \(P(0) = -60\).

Step 7: Determine Intervals of Increase

From examining the graph, the function is increasing on the intervals \((-0.13, 2.13)\) and \((4.24, fty)\).

Step 8: Determine Intervals of Decrease

The function decreases over the intervals \((-fty, -0.13)\) and \((2.13, 4.24)\).

Key Concepts

Domain of Functions

The domain of a function is essentially all the input values (usually "x") for which the function is defined. For polynomial functions, such as \( P(x) = -3x^3 + 6x^2 + 39x - 60 \), the domain is always all real numbers. This is because polynomial functions do not have any restrictions on their inputs. They do not have denominators that can be zero, nor do they have any square roots that could require non-negative values. Therefore, these functions are defined for every real number from \(-\infty\) to \(\infty\). Understanding the domain is crucial because it lets us know where our function can operate and helps us grasp the full behavior of the function on the graph.

Local Minimum and Maximum Points

Local minimum and maximum points on a graph are places where the function changes direction—from decreasing to increasing for minima and from increasing to decreasing for maxima. These points are significant because they reveal the peaks and troughs of the function on a graph. In the function \( P(x) = -3x^3 + 6x^2 + 39x - 60 \), we use a graphing calculator to find these critical points. - A local minimum exists around \((2.13, -52.66)\). This point is where the function stops decreasing and begins increasing again.- A local maximum is found near \((-0.13, 60.66)\), marking the transition from an increase to a decrease in the function.Neither of these points represents the absolute minimum or maximum, because the behavior of cubic polynomials often means they extend up to and down to infinity. Identifying these points helps in understanding the shape and behavior of the function.

Intercepts of Graphs

Intercepts are the points where a graph crosses the coordinate axes. There are two types:

• x-intercepts: Points where the graph crosses the x-axis, meaning the y-value is zero. Solve the equation \(-3x^3 + 6x^2 + 39x - 60 = 0\) to find them. Approximations show these intercepts are near \((-3, 0)\), \((2.38, 0)\), and \((4.24, 0)\).
• y-intercept: The point where the graph crosses the y-axis, occurring when \(x = 0\). By substituting \(x = 0\) in the function, this gives \(y = -60\), resulting in the point \((0, -60)\).

Intercepts give important clues about the roots and starting orientation of the graph, providing a solid foundation for plotting and interpreting the function's shape.

Increasing and Decreasing Intervals

To understand how a function behaves, it's essential to identify where it is increasing or decreasing. For the polynomial function \( P(x) = -3x^3 + 6x^2 + 39x - 60 \), we note the shifts:- Increasing intervals are sections where the graph moves upwards as we move right along the x-axis. For this function, it increases on the intervals \((-0.13, 2.13)\) and \((4.24, \infty)\).- Decreasing intervals are where the graph moves downwards. This occurs over the intervals \((-\infty, -0.13)\) and \((2.13, 4.24)\).Recognizing these intervals helps us sketch the general shape of the graph, making it easier to predict the behavior of the function over different ranges of x.