Problem 64 - College Algebra

Question

The function \(y=0.011 x^{2}-0.097 x+4.1\) models the number of people in the United States, \(y,\) in millions, holding more than one job \(x\) years after \(1970 .\) Use a graphing utility to graph the function in a \([0,20,1]\) by \([3,6,1]\) viewing rectangle. \([\text { TRACE }]\) along the curve or use your utility's minimum value feature to approximate the coordinates of the parabola's vertex. Describe what this represents in practical terms.

Step-by-Step Solution

Answer

The graph will be a parabola representing the number of people holding more than one job over the years. The vertex will show the minimum number of people holding more than one job since 1970 and the corresponding year. This highlights the point when multiple job holdings was at its least popular or necessary in the given period.

Step 1: Plotting the Function

Use a graphing utility to plot the function \(y=0.011 x^{2}-0.097 x+4.1\) in a viewing rectangle defined by \([0,20,1]\) by \([3,6,1]\). This means the x-axis should range from 0 to 20 in steps of 1, and the y-axis should range from 3 to 6 in steps of 1.

Step 2: Finding the Vertex

After plotting, trace along the graph or use your graphing utility's minimum value feature to identify the vertex of the parabola. The vertex is the point that defines the minimum or maximum point on a parabola. For this given function, our parabola will open upwards, implying that the vertex will represent the minimum point.

Step 3: Interpreting the Vertex

The vertex's coordinates represent the year when the least number of people held more than one job since 1970 and the corresponding number of those people in millions. Interpret and describe what this means in a real-life context.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key point that helps to understand the nature of the quadratic function described. In simpler terms, the vertex is either the highest or lowest point of the parabola. Whether it is a maximum or minimum point depends on how the parabola opens. When the parabola opens upwards, like in this exercise, the vertex is the lowest point, or the minimum. For a parabola opening downwards, it is the highest point, or the maximum.

To find the vertex of the parabola from our function, we can rely on a graphing utility. However, it’s also useful to know the formula for finding the vertex given a quadratic function in the form of \(y = ax^2 + bx + c\). The vertex \((h, k)\) can be calculated using:

• \(h = -\frac{b}{2a}\)
• Substitute \(h\) back into the function: \(k = a(h^2) + b(h) + c\)

Through graphing, tracing, or calculation, identifying the vertex offers concrete insights. Here, the vertex helps find the year and the number of people working multiple jobs, offering valuable real-world implications.

Graphing Utilities

Graphing utilities are invaluable tools when dealing with quadratic functions like the one in our exercise. These digital tools allow for precise plotting and analysis of equations. Rather than plotting each point manually, a graphing utility can quickly generate a visual representation of the function.

To use a graphing utility for our quadratic function, enter it into the program, ensuring the correct viewing range is selected:

• For the x-axis, choose a range from 0 to 20, stepping by 1.
• For the y-axis, choose a range from 3 to 6, stepping by 1.

Using features like tracing allows you to follow the curve and find key points such as the vertex easily. Additionally, most graphing tools have a function to calculate the minimum or maximum, making it straightforward to find the vertex of the parabola. By utilizing graphing utilities, one can gain a much clearer understanding of the behavior of the function.

Real-Life Applications of Functions

Quadratic functions like the one provided are not just abstract mathematical concepts; they have tangible real-world applications. In this exercise, the function models the number of people holding more than one job. It provides insights into workforce tendencies over time.

In practical terms, the function reveals trends and helps in planning and decision-making. For instance, understanding when the minimal number of people held multiple jobs can guide policymakers in understanding economic conditions or labor market dynamics during that period.

• This data can inform decisions on job creation strategies or economic interventions.
• Businesses may also use this information to align their workforce planning with anticipated economic conditions.

Therefore, functions are essential in various fields, such as economics, business, and social sciences, where interpreting them can have significant societal impacts.