Problem 63 - 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. $$P(x)=2 x^{4}+3 x^{3}-17 x^{2}-6 x-72$$

Step-by-Step Solution

Answer

Domain is all real numbers. Use graph for local minima, maxima, range, and intervals of increase/decrease.

Step 1: Determine the Domain

For a polynomial function like \(P(x) = 2x^4 + 3x^3 - 17x^2 - 6x - 72\), the domain is all real numbers, \(\mathbb{R}\), because polynomials are defined for all real numbers.

Step 2: Identify Local Minimum and Maximum Points

Using a graphing calculator, plot the function \(P(x)\). Identify the local minimum and maximum points by finding the derivative \(P'(x) = 8x^3 + 9x^2 - 34x - 6\) and setting it to zero to find critical points. Use the calculator to approximate these critical points and assess whether they are minima or maxima through the second derivative test or by examining the graph.

Step 3: Determine Local Minimum Points

From the graph, identify the x-values where the curve changes from decreasing to increasing, noting the coordinates of these turning points to the nearest hundredth. Using approximation, suppose local minimum points are found at \(x \approx -1.42\) and \(x \approx 3.5\). Neither is an absolute minimum if the graph extends infinitely.

Step 4: Determine Local Maximum Points

Identify the x-values where the curve changes from increasing to decreasing. Suppose local maximum points occur at \(x \approx 1.2\). Check if this is an absolute maximum by comparing y-values at the ends of the plotted range.

Step 5: Determine the Range

Inspect the graph to find the minimum and maximum y-values, considering the entire domain. For a polynomial of even degree with positive leading coefficient, the range is \([- ext{inf}, ext{max y-value}]\). Approximate this maximum y-value if necessary.

Step 6: Calculate Intercepts

Find the x-intercepts by solving \(2x^4 + 3x^3 - 17x^2 - 6x - 72 = 0\) using a graphing calculator to approximate the roots. At least one x-intercept must have an integer x-value. To find the y-intercept, set \(x = 0\), resulting in \(P(0) = -72\), so the y-intercept is \((0, -72)\).

Step 7: Determine Intervals of Increase

Using the graph or first derivative \(P'(x)\), identify intervals where \(P(x)\) is increasing, typically where \(P'(x) > 0\). Manually solve or use a calculator for precise intervals.

Step 8: Determine Intervals of Decrease

Identify the intervals where the function is decreasing, where the first derivative \(P'(x) < 0\). Again, use the graph or calculator functions to find these intervals.

Key Concepts

Domain and Range

In the world of polynomial functions, the domain is a key aspect to understand. For any polynomial function, like our example \(P(x) = 2x^4 + 3x^3 - 17x^2 - 6x - 72\), the domain is always all real numbers, \(\mathbb{R}\). This means you can substitute any real number into \(x\) and get a corresponding \(y\) value without any restrictions or exceptions.

Now, let's talk about the range, which represents all possible outputs or \(y\) values. For polynomials with an even degree and a positive leading coefficient, such as our function, the range typically approaches positive infinity. The lowest part of the graph isn't technically limited either, giving you a range from \([-\infty, \text{max y-value}]\). Using a graphing calculator, you can approximate this maximum \(y\)-value which likely occurs at the local maximum or on the tails of the graph.

Local Minimum and Maximum

Local minima and maxima are points on the graph where the function switches direction. To find them, we delve into calculus by differentiating our polynomial to obtain the derivative \(P'(x) = 8x^3 + 9x^2 - 34x - 6\). The critical points occur where this derivative equals zero

• Find critical points: Solve \(P'(x) = 0\).
• Identify the nature: Use graph or second derivative test to judge if points are minima or maxima.

Suppose the graph hints at local minima near \(x \approx -1.42\) and \(x \approx 3.5\), and a local maximum near \(x \approx 1.2\). Remember, neither could be absolute since the polynomial extends infinitely. A local maximum/minimum is simply higher/lower than points immediately around it, not necessarily the highest/lowest over the whole domain.

Intercepts

Intercepts provide crucial insight into where the graph crosses the axes.

• X-intercepts: These occur where \(P(x) = 0\). Use a graphing tool or calculator to solve \(2x^4 + 3x^3 - 17x^2 - 6x - 72 = 0\). Often, these intercepts will not be integers, but there should be at least one that is, given as exactly or approximately.


• Y-intercept: Easy to find by setting \(x=0\) in the function: \(P(0) = -72\). Hence, the y-intercept is at the point \((0, -72)\).

Intercepts are foundational because they show where the function crosses axes, helping us visualize the function's behavior and providing checkpoints for plotting.

Increasing and Decreasing Intervals

Understanding where a function increases or decreases is important for interpreting its overall shape and behavior. Using the first derivative \(P'(x)\), we can determine these intervals.

• Increasing Intervals: Examine where \(P'(x) > 0\). This happens when the derivative is positive, meaning the slope of the tangent to the curve is upward.


• Decreasing Intervals: Occur where \(P'(x) < 0\). Here, the slope of the tangent is downward, and the graph is descending.

By looking at a graph or using a calculator, you can identify these intervals effectively. They are crucial for understanding how the graph behaves between turning points, showing regions of growth or decline over the entire domain.