Problem 66 - Precalculus : Building Concepts and Connections

Question

Graph each quadratic function by finding a suitable viewing window with the help of the TABLE feature of a graphing utility. Also find the vertex of the associated parabola using the graphing utility. $$h(x)=x^{2}+5 x-20$$

Step-by-Step Solution

Answer

The vertex of the function \(h(x) = x^2 + 5x - 20\) is \((-2.5, -2.5)\). For the graphing part, choose a viewing window that includes the vertex and plot the function.

Step 1: Identify the quadratic function

The function to graph is \(h(x) = x^2 + 5x - 20\).

Step 2: Find the Vertex

The vertex of a quadratic function in the form \(f(x) = ax^2 + bx + c\) can be found using the formula \((-b/2a, f(-b/2a))\). In this case, \(a = 1\), \(b = 5\), and \(c = -20\). Applying the formula, the x-coordinate of the vertex is \(-5 / (2*1) = -2.5\). Substituting \(x = -2.5\) into the function \(h(x)\) gives the y-coordinate: \(h(-2.5) = (-2.5)^2 + 5*-2.5 -20 = -2.5\). Therefore, the vertex of the function \(h(x) = x^2 + 5x - 20\) is \((-2.5, -2.5)\).

Step 3: Determine the viewing window and plot the function

Determine a suitable viewing window for the graphing utility, typically the vertex will be at the center of the window. Here, the vertex is \((-2.5, -2.5)\), choose a range that includes values less and greater than these coordinates.

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial when dealing with quadratic functions. A quadratic function is generally expressed in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The vertex forms a specific point on the parabola which serves as its peak or lowest point, depending on the parabola's orientation. This pivotal point is significant because it helps in determining the graph's shape and orientation.

To find the vertex, you can use the formula \((-b/2a, f(-b/2a))\). This involves calculating the x-coordinate using \(x = -b/(2a)\). By substituting this x-value back into the quadratic equation, you can find the y-coordinate, \(f(-b/2a)\).

For example, given a function \(h(x) = x^2 + 5x - 20\), we calculate:

• The x-coordinate of the vertex: \(-5/(2*1) = -2.5\)
• The y-coordinate is calculated as \(h(-2.5) = (-2.5)^2 + 5(-2.5) - 20 = -2.5\)

Thus, the vertex is located at \((-2.5, -2.5)\). Knowing this vertex point, provides a key feature to draw and understand the parabola.

Graphing Utility

A graphing utility is an essential tool for visualizing mathematical functions, especially quadratic functions, which graph as parabolas. These digital tools can offer a dynamic way to explore graph behaviors by providing features such as tables, graphs, and interactions with the plotted data.

When using a graphing utility, you can visualize how changes in the equation's constants affect the parabola's size and position. Typically, the utility allows you to input a range for the x-axis and y-axis, which can help find a suitable viewing window by centering around significant points like the vertex.

For quadratic functions such as \(h(x) = x^2 + 5x - 20\), graphing utilities can:

• Quickly graph the function for better comprehension
• Provide a table of values to see how the function behaves across different x-values
• Allow for manipulation via sliders to instantly see how changes affect the graph

Leveraging a graphing utility thus simplifies the task of plotting and analyzing quadratic functions by offering an intuitive interface to discover and verify calculations.

Finding the Vertex

Finding the vertex of a parabola is a systematic process and can be done manually or with the help of a graphing utility. The vertex formula \((-b/2a, f(-b/2a))\) facilitates this process by giving an exact point of the parabola.

The steps to find the vertex manually for \(h(x) = x^2 + 5x - 20\) are as follows:

• Calculate the x-coordinate using \(x = -b/(2a)\).
• Substitute the x-coordinate into the function to find the y-coordinate, \(f(-b/2a)\).

In our example, the x-coordinate is \(-2.5\) calculated as \(-5/(2 \cdot 1)\). Substituting \(x = -2.5\) yields the y-coordinate, which is also \(-2.5\). Hence, the vertex is \((-2.5, -2.5)\).

Utilizing a graphing utility can also expedite this process by visually showing the vertex as the lowest or highest point of the parabola. Entering the function into the utility will highlight the vertex and provide an immediate visual confirmation of calculations.

When you're tasked with finding the vertex, choosing the right tools and understanding the mathematical formula can enhance both accuracy and efficiency.