Problem 10 - Algebra and Trigonometry

Question

\(5-10\) Use a graphing calculator or computer to decide which viewing rectangle \((a)-(\text { d) produces the most appropriate graph }\) of the equation. $$ \begin{array}{l}{y=\sqrt{8 x-x^{2}}} \\ {\text { (a) }[-4,4] \text { by }[-4,4]} \\ {\text { (b) }[-5,5] \text { by }[0,100]} \\ {\text { (c) }[-10,10] \text { by }[-10,40]} \\ {\text { (d) }[-2,10] \text { by }[-2,6]}\end{array} $$

Step-by-Step Solution

Answer

Option (d), \([-2,10] \text{ by } [-2,6]\), is the most appropriate.

Step 1: Analyze Equation

The equation given is \( y = \sqrt{8x - x^2} \). The expression inside the square root must be non-negative, so we first find the domain of \( x \) which satisfies \( 8x - x^2 \geq 0 \). The roots of this inequality will determine our domain.

Step 2: Solve Inequality

To solve \( 8x - x^2 \geq 0 \), rewrite it as \( x(8-x) \geq 0 \). The roots are \( x = 0 \) and \( x = 8 \). Utilize test intervals \( (-\infty, 0) \), \( (0, 8) \), and \( (8, \infty) \) to find where the expression is non-negative. The valid interval is \( 0 \leq x \leq 8 \).

Step 3: Check Y-values Range

For \( x \) in \([0,8]\), the maximum value of \( y = \sqrt{8x - x^2} \) occurs midway at \( x=4 \). Plugging \( x=4 \) gives \( y = \sqrt{16} = 4 \). Hence, \( y \) ranges from 0 to 4.

Step 4: Evaluate Viewing Rectangles

Now, assess which of the proposed rectangles captures the function accurately, focusing on the x-range \([0,8]\) and y-range \([0,4]\). The options are: (a) \([-4,4] \text{ by } [-4,4]\), (b) \([-5,5] \text{ by } [0,100]\), (c) \([-10,10] \text{ by } [-10,40]\), (d) \([-2,10] \text{ by } [-2,6]\).

Step 5: Determine Best Viewing Rectangle

Option (d), \([-2,10] \text{ by } [-2,6]\), includes the complete x-domain \([0,8]\) and y-range \([0,4]\) with some margin for clarity. The other rectangles include too much irrelevant space or miss parts of the domain.

Key Concepts

Viewing Rectangle

When using a graphing calculator to visualize functions, it's essential to select the correct viewing rectangle. A viewing rectangle defines the portion of the graph that will be displayed, using specified ranges for both the x-axis and the y-axis. Choosing the right one is crucial for accurately interpreting the graph's behavior.
In the given equation, we examine various viewing rectangles to identify the one that appropriately captures the function's domain and range:

• (a) [-4,4] by [-4,4]
• (b) [-5,5] by [0,100]
• (c) [-10,10] by [-10,40]
• (d) [-2,10] by [-2,6]

The correct choice provides a clear view of the graph within the x-range [0,8] and y-range [0,4], emphasizing the relevant parts without excessive empty space.

Inequality Solution

To understand the behavior of the function, solving the inequality is pivotal. The equation given is an inequality: \( 8x - x^2 \geq 0 \). Solving this helps us find the valid x-values or the domain.
We factor the inequality as \( x(8-x) \geq 0 \), which gives roots at \( x = 0 \) and \( x = 8 \). Test intervals such as \((-\infty, 0)\), \((0, 8)\), and \((8, \infty)\) identify where the expression holds. The solution, \( x \in [0, 8] \), is where the inequality is satisfied. This means only x-values between 0 and 8 give a non-negative value inside the square root, confirming the selected domain.

Domain of a Function

The domain of a function includes all possible input values that the function can accept. In the equation \( y = \sqrt{8x - x^2} \), the domain is impacted by the expression inside the square root. Since square roots require non-negative radicands, we derive the domain by ensuring \( 8x - x^2 \geq 0 \).
The solution to this inequality was found to be \( 0 \leq x \leq 8 \). Therefore, the valid x-values, or domain, are all numbers from 0 to 8. Understanding this domain is critical for defining which x-values are suitable inputs for the graphing calculator.

Coordinate Plane

The coordinate plane is a two-dimensional space where we plot graphs using axes: the x-axis (horizontal) and the y-axis (vertical). In this exercise, the coordinate plane helps us visualize the relationship described by the equation \( y = \sqrt{8x - x^2} \).
To effectively use the coordinate plane, choose an appropriate scale for both axes, identified by the viewing rectangle. The equation’s domain \([0, 8]\) for x-values and the range \([0, 4]\) for y-values suggests that a viewing rectangle extending slightly beyond these limits offers a comprehensive view of the graph. This visualization highlights the function's behavior entirely within the relevant section of the coordinate plane.