Problem 49 - Precalculus with Limits

Question

In Exercises 47-56, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a). \(g(s) = {s^2}{4}\)

Step-by-Step Solution

Answer

The function \(g(s) = \frac{s^2}{4}\) decreases over the interval \((- \infty, 0)\), increases over the interval \((0, \infty)\) and is not constant over any interval.

Step 1: Graph the Function

To graph \(g(s) = \frac{s^2}{4}\), you can use a graphing utility. By graphing the function, you will see that the graph opens upward and has its vertex at the origin (0,0). This means the function is increasing on the interval \((0, \infty)\) and decreasing on \((- \infty, 0)\) while there are no sections where the function is constant.

Step 2: Create a Table of Values

To validate the intervals over which the function is increasing, decreasing, or constant, a sample of values for \(s\) can be chosen and their corresponding \(g(s)\) values calculated. Choose values for \(s\) that cover the intervals \((- \infty, 0)\) and \((0, \infty)\), e.g., \(-2, -1, 0, 1, 2\). Then, \(g(s)\) will be calculated for each \(s\) value.

Step 3: Validate the Intervals

After applying the \(s\) values in the function, we will see that the function \(g(s)\) decreases for values from \(- \infty\) to \(0\) and increases for values from \(0\) to \(\infty\). There is no interval where the function is constant thus confirming that the function decreases over the interval \((- \infty, 0)\), increases over the interval \((0, \infty)\), and is not constant over any interval.

Key Concepts

Intervals of Increase and Decrease

To determine where a function increases or decreases, we need to understand its behavior across different intervals of its domain. For the function \(g(s) = \frac{s^2}{4}\), this is a parabola that opens upwards.

• Decreasing Intervals: The function is decreasing on the interval \((-\infty, 0)\). This means as \(s\) moves from left to right towards zero, the value of \(g(s)\) decreases.
• Increasing Intervals: After passing through zero, the parabola increases on the interval \((0, \infty)\). This means as \(s\) moves away from zero to the right, \(g(s)\)'s value increases.

To visually determine these intervals, it's helpful to graph the function and look for where the slope of the curve is positive (increasing) or negative (decreasing). Remember, there are no intervals where this particular function is constant.

Vertex of a Parabola

The vertex of a parabola is a crucial feature, particularly when identifying intervals of increase and decrease. For any quadratic function of the form \(y = ax^2 + bx + c\), the vertex represents the maximum or minimum point.For the function \(g(s) = \frac{s^2}{4}\), the equation simplifies to \(y = \frac{s^2}{4}\) with:

• The vertex is at the point (0,0). This is where the parabola changes direction from decreasing to increasing.
• The axis of symmetry, which is a vertical line running through the vertex, is \(s = 0\).

Knowing the vertex location helps in quickly identifying where the function shifts behavior and makes graphing more straightforward. It marks the position where the increasing intervals begin and the decreasing intervals end.

Table of Values

Creating a table of values is a fundamental step in understanding the behavior of a function over its domain. This involves selecting a range of inputs and calculating the corresponding outputs.For \(g(s) = \frac{s^2}{4}\):

• Select values such as \(-2, -1, 0, 1, 2\) for \(s\).
• For each value of \(s\), compute \(g(s)\) to see how the function behaves at those points.
• This will yield values like \(g(-2) = 1, g(-1) = 0.25, g(0) = 0, g(1) = 0.25, g(2) = 1\).

From this, it’s evident that as \(s\) moves from negative to positive past zero, the function's value decreases and then increases, verifying its behavior according to the graph. Using both graphical methods and a table of values provides a complete view of function behavior across different intervals.