Problem 50 - 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). \(h(x) = x^2-4\)

Step-by-Step Solution

Answer

The function \(h(x) = x^2-4\) is decreasing for \(x < 0\), increasing for \(x > 0\) and is constant at \(x = 0\).

Step 1: Graph the function

Use a graphing utility to draw the graph of the function \(h(x) = x^2-4\). By looking at the graph, you can visually see the intervals of \(x\) where the function is increasing, decreasing, or constant.

Step 2: Identify the intervals visually

Upon observing the graph, it is visible that the function h(x) decreases for \(x < 0\), increases for \(x > 0\) and constant at \(x = 0\). So, these are the intervals over which the function is increasing, decreasing, and constant respectively.

Step 3: Create a table of values

To verify the intervals identified visually, create a table of values. Choose values for \(x\) that fall into the identified intervals and calculate the corresponding \(h(x)\) for each of them. By looking at whether \(h(x)\) increases or decreases as \(x\) increases in each interval, you can verify if the function is increasing, decreasing, or constant on these intervals.

Step 4: Verify the intervals

Upon checking the table, if for an interval, as \(x\) increases, \(h(x)\) also increases, then the function is increasing on that interval. If \(h(x)\) decreases as \(x\) increases, the function is decreasing on that interval. If \(h(x)\) remains the same as \(x\) increases, the function is constant on that interval. Verify that the observations from the table matches with the findings from the graph.

Key Concepts

Increasing and Decreasing Functions

Understanding increasing and decreasing functions can be made simple with a few core ideas. When you have a function, you want to know how the function behaves as the input, or "x" value, changes. Specifically, you look for where a function rises or falls on its graph.

- **Increasing Functions:** A function is increasing on an interval if the values of the function go up as the x-values move from left to right. That means, if you pick any two points within that interval, the point on the right will have a higher function value than the point on the left.

- **Decreasing Functions:** Conversely, a function is decreasing on an interval if the function values go down as the x-values increase. Graphically, these sections slope downwards as you move from left to right.

To find these intervals, graphing tools can be immensely helpful. They allow you to visually identify segments where the function rises or decreases. For instance, with the function \(h(x) = x^2 - 4\), it appears as a parabola that opens upwards. By observing, we see the curve decreases for \(x < 0\) and increases for \(x > 0\). At \(x = 0\), the function reaches its lowest point, known as the minimum, meaning it doesn't increase or decrease right at that point.

Table of Values

A table of values is an organized way of listing input values alongside their corresponding output values for a function. This helps verify the nature of a function's increasing or decreasing behavior that we observe graphically.

Here's how you can create a table of values:

• Select a handful of points, particularly those around where you expect changes in behavior (e.g., points around \(x = 0\)).
• Input these points into your function \(h(x) = x^2 - 4\) to get the respective values of \(h(x)\).
• Look at how the values change as x increases: if \(h(x)\) steadily rises, then it is part of an increasing interval; if it reduces, then it is a decreasing interval.

For example, calculate \(h(-1)\), \(h(0)\), and \(h(1)\). If \(h(-1) > h(0) < h(1)\), this table supports what we saw in the graph: the function decreases for \(x < 0\) and increases for \(x > 0\). There is no constant interval here since the function only pauses at \(x=0\) before changing direction.

Quadratic Functions

Quadratic functions are polynomial functions of degree two. They typically have the form \(h(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our exercise, the function \(h(x) = x^2 - 4\) is a simple quadratic function. These functions often graph as parabolas.

Key aspects of quadratic functions include:

• The parabola's direction: If \(a > 0\), it opens upward (like in our exercise). If \(a < 0\), it opens downward.
• The vertex: The point where the function changes direction (either from increasing to decreasing or vice versa). For \(h(x) = x^2 - 4\), the vertex is at \((0, -4)\). This is a minimum point since the parabola opens upwards.
• Symmetry: Quadratic functions are symmetric about a vertical line passing through their vertex.
• Zeros: The points where the parabola crosses the x-axis. These can be found by solving \(x^2 - 4 = 0\), giving us zeros at \(x = -2\) and \(x = 2\).

Understanding these characteristics allows you to predict and interpret the behavior of quadratic functions, making graphical analysis a handy method for visual learners. Quadratic functions, with their predictable shape, offer a perfect introduction to polynomial functions and their behavior.