Problem 4 - Thomas Calculus

Question

Use graphing software to determine which of the given viewing windows displays the most appropriate graph of the specified function. \begin{equation} f(x)=\sqrt{5+4 x-x^{2}} \end{equation} \begin{equation} \begin{array}{ll}{\text { a. }[-2,2] \text { by }[-2,2]} & {\text { b. }[-2,6] \text { by }[-1,4]} \\ {\text { c. }[-3,7] \text { by }[0,10]} & {\text { d. }[-10,10] \text { by }[-10,10]}\end{array} \end{equation}

Step-by-Step Solution

Answer

Viewing window b: \([-2, 6]\) by \([-1, 4]\) is the most appropriate.

Step 1: Understand the Function

The function given is \( f(x) = \sqrt{5 + 4x - x^2} \). It involves a square root, implying the expression inside must be non-negative for \( f(x) \) to be real. Analyze the quadratic expression \( 5 + 4x - x^2 \) to determine the domain of the function.

Step 2: Find the Domain of the Function

Set the expression inside the square root greater than or equal to zero: \( 5 + 4x - x^2 \geq 0 \). This can be rearranged to \( -x^2 + 4x + 5 \geq 0 \), a quadratic inequality. Using the quadratic formula, solve \( -x^2 + 4x + 5 = 0 \) to find the critical points that define the domain.

Step 3: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for \( -x^2 + 4x + 5 \):- \( a = -1, b = 4, c = 5 \)- Discriminant: \( b^2 - 4ac = 16 + 20 = 36 \)- Solutions: \( x = \frac{-4 \pm 6}{-2} \)- Roots are \( x = -1 \) and \( x = 5 \). The domain is \([-1, 5]\).

Step 4: Analyze Viewing Windows

Evaluate the available viewing windows against the domain \([-1, 5]\):- Window a: \([-2, 2] \) with y-range \([-2, 2]\).- Window b: \([-2, 6]\) with y-range \([-1, 4]\).- Window c: \([-3, 7]\) with y-range \([0, 10]\).- Window d: \([-10, 10]\) with y-range \([-10, 10]\).

Step 5: Determine the Most Appropriate Window

For the function defined on \([-1, 5]\), view window b \([-2, 6]\) by \([-1, 4]\) is best. It almost perfectly covers the domain and predicted range, allowing for visualization of both endpoints.

Key Concepts

Viewing Windows

When graphing a function, selecting the appropriate viewing window is crucial for accurately displaying the important features of the graph. A viewing window consists of an x-range and a y-range, and it determines what portion of the graph is visible on the screen.
Choosing a proper viewing window involves understanding the domain and range of the function and ensuring that these are visible within the selected x and y intervals.

• The x-range should encompass all critical points of the function.
• The y-range should cover the expected outputs of the function given its domain.

In the exercise, multiple viewing windows are considered to find the one that best fits the function's domain and likely y-values. By plotting different windows, the viewer can compare how well they display the critical features of the function.

Quadratic Inequality

Quadratic inequalities involve expressions like \(-x^2 + 4x + 5 \). These need to be solved to find where the function is defined or satisfies a certain condition. Here, the expression was \(5 + 4x - x^2 \) inside a square root, so it must be non-negative.
Quadratic inequalities can be solved through:

• Rearranging the inequality into a standard quadratic form \(ax^2 + bx + c \).
• Solving for the critical values using the quadratic formula \((b^2 - 4ac \geq 0) \).
• Determining which intervals satisfy the inequality.

In this case, solving \(-x^2 + 4x + 5 \geq 0 \) allows us to find that the function is defined within the interval \([-1, 5]\). This analysis directly influences the selection of the viewing window.

Function Domain Analysis

The domain of a function consists of all possible input values for which the function is defined. For functions involving square roots, like \(f(x) = \sqrt{5 + 4x - x^2} \), it's essential that the expression under the square root not be negative.
To find the domain:

• Set the expression under the square root \(5 + 4x - x^2 \geq 0\).
• Solve this inequality to find the range of x-values.
• Determine the domain to be the interval where the expression is non-negative.

This method ensures the function remains within the real numbers, resulting in a domain of \([-1, 5]\) for the given function. This is essential for correct graphing, as it determines the viewing window and interprets the complete behavior of the function over its domain.

Graphing Software

Graphing software tools allow users to visualize mathematical functions easily. They have built-in features to adjust viewing windows, manipulate graphs, and showcase various mathematical properties.
These tools are invaluable when:

• Testing different viewing windows to find the most appropriate setup.
• Visualizing function behavior, from simple lines to complex expressions.
• Assisting with the analysis of multiple conditions like domain restrictions or inequalities.

In this exercise, the use of graphing software aids in determining which viewing window is most suitable by quickly showing how the graph fits within each option. These visualizations help clarify how well a window displays key aspects of a function's graph.