Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the function, approximate the relative minimum or maximum of the function, and estimate the open intervals on which the function is increasing or decreasing. \(f(x)=-x^{2}+6 x+3\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = -x^{2} + 6x + 3\) is a downward-opening parabola. The relative minimum point of function is at (3, 12). The function is increasing on the interval \(-\infty, 3\) and decreasing on the interval \(3, +\infty\).

Step 1: Graphing the Function

Using a graphing utility, input the function \(f(x)=-x^{2}+6 x+3\). This will generate a downward-opening parabola because the coefficient of \(x^2\) is negative.

Step 2: Find the Relative Minimum or Maximum

For a quadratic function in the form of \(ax^{2}+bx+c\), the x-coordinate of the vertex can be found using the formula \(-b/2a\). In this case, a=-1 and b=6, so the x-coordinate of the vertex is \(-b/2a = -6 / -2*1 = 3\). Substituting 3 back into the function will give the y-coordinate of the vertex. This vertex is the relative minimum point because it's a downward-opening parabola.

Step 3: Find the Intervals

The open intervals on which the function is increasing or decreasing can be found by looking at the graph. For the given function, since it opens downwards, it will be increasing before it reaches the vertex and decreasing after the vertex. We find that the function is increasing on the interval \(-\infty, 3\) and decreasing on the interval \(3, +\infty\)

Key Concepts

Understanding Relative Minimum and Maximum

When we discuss the relative minimum and maximum of a quadratic function, such as in the function \(f(x)=-x^{2}+6x+3\), we're looking for the highest or lowest point on the graph of the function—also known as the vertex. In the context of a parabola, which is the graph of a quadratic function, a relative maximum occurs at the vertex if the parabola opens downward. Conversely, a relative minimum occurs at the vertex if the parabola opens upward.

In the given exercise, \(f(x)\) is a downward-opening parabola, indicated by the negative coefficient of the \(x^2\) term. Remember that the coordinate of the vertex, and thusly the relative minimum or maximum, can be found using the formula \(x = -\frac{b}{2a}\). After calculating the x-coordinate of the vertex, plug it back into the original function to determine its y-coordinate. The result is the relative minimum point for the function \(f(x)\) provided in this case. Identifying this is essential as it signifies the turning point on the graph from which the function starts increasing or decreasing.

Open Intervals of Increase and Decrease

The open intervals on which a function is increasing or decreasing reveal how the function behaves at various segments along the x-axis. For a quadratic function such as the one given, \(f(x)=-x^{2}+6x+3\), the function increases on one interval and decreases on another, due to the nature of a parabola.

An open interval is expressed without brackets, suggesting it does not include the endpoints. If you picture the graph of a parabola, the interval of increase is where the function values (y-values) ascend as you move from left to right up to the vertex. Similarly, the interval of decrease is where the function descends from the vertex as you continue moving to the right.

In our case, since the parabola opens downward, the interval on which the function is increasing is \(\left(-\infty, 3\right)\), and the function is decreasing on \(\left(3, +\infty\right)\). This means that as \(x\) values approach the vertex from the left, the function's value increases, and as they move away from the vertex to the right, the function's value decreases.

Using Graphing Utilities

Graphing utilities, such as graphical calculators or computer software, are invaluable tools for visualizing functions and aiding in the analysis of graphs. These tools manage to plot the exact shape of a quadratic function and allow us to approximate the relative minimum or maximum, as well as determine the intervals where the function increases or decreases.

Integrating the use of graphing utilities from the start, by entering the function's formula, you obtain a visual output that clarifies many of the function's properties. In the context of \(f(x)\) from our exercise, this means seeing the downward-opening parabola and identifying its vertex. Additionally, you can zoom in on key areas of the graph to get a better estimate of crucial points and intervals. It's important to couple this graphical insight with analytical methods to fully comprehend the function's characteristics.

Keep in mind though, the graphing utility is a tool to supplement your understanding, and the ability to solve problems analytically reinforces your mathematical intuition.