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Chapter 2: Problem 65

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. $$h(x)=(\sqrt{2}) x^{2}+x+1$$

Short Answer

Expert verified
The graph of the quadratic function \(h(x) = (\sqrt{2})x^{2} + x + 1\) shows a parabolic curve that opens upward. The vertex of the parabola is located at point \((-1/(2\sqrt{2}), h(-1/(2\sqrt{2})))\).

Step by step solution

01

Setting up the equation

The quadratic function to graph is \(h(x) = (\sqrt{2})x^{2} + x + 1\).
02

Determine the viewing window

Use the table feature of the graphing utility to identify the input-output pairs. This can help to estimate the suitable viewing window for the graph. Choose an interval for \(x\) that provides a good overview of the behavior of the function.
03

Graph the function

Enter the equation \(h(x) = (\sqrt{2})x^{2} + x + 1\) into the graphing utility and plot the function using the determined viewing window.
04

Determine the vertex

Remember that the vertex of the quadratic equation is at \((-b/(2a), h(-b/(2a)))\). In this equation, \(a = \sqrt{2}\) and \(b = 1\), so we compute the vertex as: \((-1/(2\sqrt{2}), h(-1/(2\sqrt{2})))\). This can also be confirmed visually from the graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Utilities
Graphing utilities are valuable tools for visualizing mathematical functions, especially complex ones like quadratic equations. These tools help you to easily plot graphs that would otherwise require meticulous manual calculations. For the function \(h(x) = (\sqrt{2})x^{2} + x + 1\), you can input this expression directly into the graphing utility. The utility will automatically calculate the y-values for various x-values and plot them on the coordinate plane.

With a graphing utility, you can quickly see how the parabola behaves in different areas of the graph. This helps identify trends and characteristics like intercepts, peaks, or valleys without error-prone manual processes.

  • Input function directly to get a visual representation.
  • Helps in understanding function behavior.
  • Reduces manual graphing errors.
Vertex of a Parabola
In a quadratic function, the vertex is a crucial point. It is either the highest or lowest point of the parabola, depending on its orientation. For any quadratic function of the form \(ax^{2} + bx + c\), the vertex coordinates can be found using the formula \(-b/(2a)\) for the x-coordinate and substituting this back into the function for the y-coordinate.

For \(h(x) = (\sqrt{2})x^{2} + x + 1\), the coefficients are \(a = \sqrt{2}\) and \(b = 1\). Plugging these into the vertex formula gives the x-coordinate as \(-1/(2\sqrt{2})\). Simply evaluate \(h(-1/(2\sqrt{2}))\) to find the y-coordinate. This spot represents the minimum point on the graph since \(a\) is positive.

  • Vertex coordinates: \(-b/(2a)\) for x, \(h(x)\) for y.
  • Parabola's turning point or extremum.
  • Helps in determining the parabola's shape and direction.
Viewing Window
Selecting an appropriate viewing window is a vital part of graphing functions. It determines how much of the function's behavior you see on your screen. Choose too small a window, and you might miss important details. Pick too large a window, and critical points like the vertex might be lost in a sea of data.

Use the table feature of your graphing utility to see how the function reacts with different values of x. For \(h(x) = (\sqrt{2})x^{2} + x + 1\), look for the x-values where the function changes behavior significantly. These input-output pairs can guide you in setting a range for both x and y axes that captures the parabola's key features, such as vertex and intercepts.

  • View a function's full behavior within your due screen range.
  • Use input-output pairs to define meaningful ranges.
  • Balance between too little and too much data visibility.
Input-Output Pairs
Input-output pairs are fundamental when utilizing graphing utilities. They show the function's behavior through various values it can take. For the function \(h(x) = (\sqrt{2})x^{2} + x + 1\), by inputting various values of x, you generate corresponding y-values from the function. These pairs are critical for plotting the graph accurately.

By examining these input-output pairs in the graphing utility's table feature, you can predict how the parabola behaves and ensure that the graph's scale is set correctly. The better your range of x-values chosen, the more complete picture you'll have of the function's nature, including its curvature and vertex location.

  • Establish how x-values affect function results (y-values).
  • Identify patterns and trends in data.
  • Essential for setting appropriate scales for graphing.

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